Mathematics High School

## Answers

**Answer 1**

The given **system of equations** have the same** slope** and different y intercepts. The correct option is (C).

Here, we have

to write the equation of a straight line in **slope**-intercept form:

A straight line can be written in the **slope**-intercept form as, y = mx + c.

In order to obtain the **slope**, the ratio of the difference of the coordinates are taken and c is the y-intercept which can be found by substituting x = 0 in the equation.

The given **system of equations** is as below,

y = -5x + 1 (1)

y = -5x + 10 (2)

Compare these** equation** with the general form y = mx + c to obtain,

For equation (1),

m = -5 and c = 1

And, for equation (2),

m = -5 and c = 10

Since two linear equations having the same** slope **represents two parallel lines, the given** system of equations** represent the parallel lines.

Hence, the linear** equations** are do not intersect on graph due to the sam**e slope** but different y intercepts.

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## Related Questions

Think of a proportional relationship you may see in your daily life . Make a table of data and graph the data. explain how u know that the data show a proportional relationship

### Answers

One example of a **proportional** relationship in daily life could be the relationship between the distance traveled by car and the time it takes to travel that distance, assuming a constant **speed**.

We have,

Let's say we have a car that is traveling at a constant **speed** of 60 miles per hour.

We can create a table of data to show the relationship between **distance** and **time**:

Distance (miles)Time (hours)

60 1

120 2

180 3

240 4

300 5

We can see from the table that as the distance traveled increases by a factor of 2, the time it takes to travel that distance also increases by a factor of 2.

This is a **proportional** **relationship**.

Thus,

One example of a **proportional** relationship in daily life could be the relationship between the distance traveled by car and the time it takes to travel that distance, assuming a constant **speed**.

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(1 point) (a) find a vector parametric equation for the ellipse that lies on the plane 5x 4y z=−9 and inside the cylinder x2 y2=16.

### Answers

the vector **parametric** equation for the **ellipse** is:

r(u) = (4 + sqrt(17)*cos(u), -9/4 + 4/sqrt(17)*sin(u), -5(4 + sqrt(17)*cos(u)) + 4(-9/4 + 4/sqrt(17)sin(u)) - 9), where u ranges from 0 to 2pi.

To find a vector parametric **equation** for the ellipse that lies on the plane 5x - 4y + z = -9 and inside the **cylinder** x^2 + y^2 = 16, we can use the following steps:

Solve the plane equation for z: z = -5x + 4y - 9.

Substitute this expression for z into the cylinder equation: x^2 + y^2 = 16 - (-5x + 4y - 9)^2.

Simplify this equation to get it into **standard** form for an ellipse: 41x^2 + 16y^2 - 40xy - 362x + 288y - 360 = 0.

Find the center and **axes** lengths of the ellipse using the matrix method for conic sections. The center is at the point (x0, y0) = (4, -9/4), and the axes lengths are a = sqrt(17) and b = 4/sqrt(17).

Parameterize the ellipse using the center and axes lengths: r(u) = (x0 + acos(u), y0 + bsin(u)), where u is a parameter that ranges from 0 to 2*pi.

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Use the Laplace transform to solve the given system of differential equations.d2x dt2 + d2y dt2 = t2d2x dt2 − d2y dt2 = 6tx(0) = 8, x'(0) = 0,y(0) = 0, y'(0) = 0

### Answers

To use the **Laplace transform** to solve the system of differential equations:

d2x/dt2 + d2y/dt2 = t^2

d2x/dt2 - d2y/dt2 = 6t

We first take the Laplace transform of each equation with respect to t, using the **linearity property** of the Laplace transform:

L{d2x/dt2} + L{d2y/dt2} = L{t^2}

L{d2x/dt2} - L{d2y/dt2} = 6L{t}

We know that the Laplace transform of the second derivative of a **function f(t)** with respect to t is given by:

L{d2f/dt2} = s^2F(s) - s*f(0) - f'(0)

where F(s) is the Laplace transform of f(t). Using this, we can rewrite the transformed equations as:

s^2X(s) + sY(s) - x'(0) - y(0) + s^2Y(s) + sX(s) - y'(0) - x(0) = L{t^2}

s^2X(s) - sY(s) - x'(0) + y(0) - s^2Y(s) + sX(s) + y'(0) - x(0) = 6/s

Since x(0) = y(0) = 0 and x'(0) = 0, y'(0) = 0, we can simplify the equations to:

s^2X(s) + sY(s) + s^2Y(s) = L{t^2}

s^2X(s) - sY(s) - s^2Y(s) = 6/s

Solving for X(s) and Y(s) using the elimination method, we get:

X(s) = (1/2s^3) * (sL{t^2} + 6)

Y(s) = (1/2s^3) * (sL{t^2} - 6)

Taking the inverse Laplace transform of X(s) and Y(s) using a Laplace transform table, we get:

x(t) = (t^3/6) + 4t

y(t) = (t^3/6)

Therefore, the solutions for the given system of **differential equations** are:

x(t) = (t^3/6) + 4t

y(t) = (t^3/6)

This is the final answer.

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Which set of two variables is most likely to have a cause-and-effect relationship? A. The age of a teacher and the income of the teacher B. The height of a person and the weight of a person C. The weight of a box and the postage rate we have to pay to ship the box to California D. The make of a car and the mileage of the car

### Answers

Out of the given options, the** set** of **variables** that is most likely to have a cause-and-effect relationship is A. The age of a teacher and the income of the teacher. A cause-and-effect relationship implies that one variable is the cause of changes in the other variable.

In this case, it is possible to argue that the age of a teacher can cause changes in their **income**. For example, as teachers gain more experience and seniority, they may be eligible for higher salaries. Additionally, teachers who have been in the profession for longer may have more qualifications or skills that make them more desirable to schools and therefore more likely to earn higher incomes. On the other hand, the other options do not necessarily have a clear cause- and- effect relationship. For example, while a person's **weight** may influence their **height** to some extent, it is not a direct cause-and-effect relationship. Similarly, while the make of a car may influence its mileage, it is not a direct cause-and-effect relationship. In conclusion, while it is important to consider all variables when analyzing data, it is essential to identify which variables have a direct cause-and-effect relationship. In this case, the set of variables with a cause-and-effect relationship is A. The age of a teacher and the income of the teacher. Based on the given options, the set of two variables most likely to have a cause-and-effect relationship is C. The weight of a box and the postage rate we have to pay to ship the box to California.This relationship is likely cause-and-effect because shipping companies typically charge postage rates based on the weight of the package being sent. As the weight of a box increases, the **cost** to ship it usually increases as well, demonstrating a direct cause-and-effect relationship between these two variables

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suppose f : rn → rm is a linear map. what is the derivative of f ?

### Answers

If f: rn → rm is a** linear map**, then its derivative is simply the map itself. This is because a linear map is a function that preserves vector addition and **scalar** **multiplication**.

In other words, if we take two vectors in the domain and add them together, and then apply the **linear** **map**, it is the same as applying the linear map to each vector separately and then adding the results. Similarly, if we multiply a vector in the domain by a scalar and then apply the linear map, it is the same as **multiplying **the result of applying the linear map to the original vector by the same scalar.

Formally, we can express this idea using the concept of a **Jacobian** **matrix**. The Jacobian matrix of a function describes the rate at which the function changes near a particular point. For a linear map, the Jacobian matrix is simply the matrix that represents the map. This means that the derivative of f is the matrix A such that f(x) = Ax for all x in rn.

To see why this makes sense, consider the simplest case of a linear map from R1 to R1, given by f(x) = ax, where a is a constant. The derivative of this **function** is f'(x) = a, which is just the constant coefficient of the linear map. More generally, the derivative of a linear map f: rn → rm is the matrix A such that f(x) = Ax for all x in rn.

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The student took these notes from reliable sources:

- Blood vessels - a small person can crawl through them.

.

• Eye-six times larger than the human eye

• Eye-size of a grapefruit

• Scientists have not determined if the blue whale will continue to grow larger

• Tail is split into two parts called flukes. Fluke length of two canoes

The ocean is home to a giant, the blue whale. This creature is astonishingly big. To say that the blue whale is over 100 feet long and weighs 150

tons and is bigger than the largest dinosaur that ever lived is not sufficient. A good way to understand the size of this magnificent mammal is to

examine its size compared to common objects and the size of its body parts.

.

Tongue-large enough for 50 people to stand on

Tongue the size of an elephant

• Heart-size of a small car

• Heart weight- a ton - 2000pounds

• 100 people can fit in blue whale's mouth

can grow up to 100 feet /size of two city buses/ longer than an NBA basketball court

Using information from the student's notes, write one or two paragraphs developing the idea in the last sentence of the introduction.

.

### Answers

**Answer:**

A strategy to protect whales can limit greenhouse gases and global warming. When it comes to saving the planet, one whale is worth thousands of trees.

**Step-by-step explanation:**

Verify that the Divergence Theorem is true for the vector field F = 2x^2i + 2xyj + 3zk and the region E the solid bounded by the paraboloid Z = 4 - X^2 - y^2 and the xy-plane. To verify the Divergence Theorem we will compute the expression on each side. First compute integration integration integration E div F dV div F = integration integration integration E div F dV= x2 integration x1 y2 integration y1 z2 integration dz dy dx where x1 = x2 = y1 = y2 = z1 = z2 = integration integration integration E div F dV = Now compute integration integration F dS Consider S = P D where p is the paraboloid and D is the disk integration integration p FdP = x2 integration x1 y2 integration y1 dy dx integration integration D F dD = x2 integration x1 y2 integration y1 dy dx where x1 = x1 = y1 = y2=

### Answers

The triple **integral** of the divergence over the region E is equal to the surface integral of F over the boundary surface of E, we have verified the Divergence Theorem for the given vector field F and the region E.

To verify the **Divergence** Theorem, we need to compute both sides of the equation for the given vector field F and the region E bounded by the paraboloid Z = 4 - X^2 - y^2 and the xy-plane.

First, we compute the divergence of F:

div F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

= 4x + 2

Next, we compute the triple integral of the divergence over the region E:

∫∫∫E div F dV = ∫∫∫E (4x + 2) dV

Since the region E is bounded by the xy-plane and the **paraboloid**, we can integrate over z from 0 to 4 - x^2 - y^2, over y from -√(4 - x^2) to √(4 - x^2), and over x from -2 to 2:

∫∫∫E div F dV = ∫-2^2 ∫-√(4 - x^2)√(4 - x^2) ∫0^4-x^2-y^2 (4x + 2) dz dy dx

= 128/3

Now, we compute the surface integral of F over the boundary surface of E:

∫∫S F dS = ∫∫P F dP + ∫∫D F dD

where P is the surface of the paraboloid and D is the disk at the bottom of E.

On the paraboloid, the normal vector is given by n = (∂f/∂x, ∂f/∂y, -1), where f(x,y) = 4 - x^2 - y^2. Therefore, we have:

∫∫P F dP = ∫-2^2 ∫-√(4 - x^2)√(4 - x^2) (2x^2, 2xy, 4 - x^2 - y^2) ∙ (∂f/∂x, ∂f/∂y, -1) dA

= ∫-2^2 ∫-√(4 - x^2)√(4 - x^2) (2x^2, 2xy, 4 - x^2 - y^2) ∙ (2x, 2y, 1) dA

= 16π/3

On the disk at the bottom, the normal vector is given by n = (0, 0, -1). Therefore, we have:

∫∫D F dD = ∫-2^2 ∫-√(4 - x^2)√(4 - x^2) (2x^2, 2xy, 0) ∙ (0, 0, -1) dA

= 0

Thus, we have:

∫∫S F dS = ∫∫P F dP + ∫∫D F dD = 16π/3 + 0 = 16π/3

Since the triple integral of the divergence over the region E is equal to the surface integral of F over the **boundary** surface of E, we have verified the Divergence Theorem for the given vector field F and the region E.

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The total surface** integral** is:

∫∫S F dS = ∫∫S F dS + ∫∫S F dS

= 8π/3 + 0

= 8π/3

To verify the** Divergence Theorem**, we need to show that the triple integral of the divergence of F over the region E is equal to the surface integral of F over the boundary of E.

First, let's compute the divergence of F:

div F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

= 4x + 2y + 3

Next, we'll compute the triple integral of div F over E:

∭E div F dV = ∫∫∫ (4x + 2y + 3) dz dy dx

The region E is bounded by the paraboloid Z = 4 - X^2 - y^2 and the xy-plane. To determine the limits of integration, we need to find the intersection of the paraboloid with the xy-plane:

4 - x^2 - y^2 = 0

x^2 + y^2 = 4

This is the equation of a circle with radius **2 centered** at the origin in the xy-plane.

So, the limits of integration are:

x: -2 to 2

y: -√(4 - x^2) to √(4 - x^2)

z: 0 to 4 - x^2 - y^2

∭E div F dV = ∫∫∫ (4x + 2y + 3) dz dy dx

= ∫-2^2 ∫-√(4-x^2)^(√(4-x^2)) ∫0^(4-x^2-y^2) (4x + 2y + 3) dz dy dx

= 32/3

Now, let's compute the surface integral of F over the boundary of E. The boundary of E **consists **of two parts: the top surface of the paraboloid and the circular disk in the xy-plane.

For the top surface of the paraboloid, we can use the upward-pointing normal vector:

n = (2x, 2y, -1)

For the circular disk in the xy-plane, we can use the upward-pointing normal vector:

n = (0, 0, 1)

The surface integral over the top surface of the paraboloid is:

∫∫S F dS = ∫∫D F(x, y, 4 - x^2 - y^2) ∙ n dA

= ∫∫D (4x + 2y, 2xy, 4 - x^2 - y^2) ∙ (2x, 2y, -1) dA

= ∫∫D (-4x^2 - 4y^2 + 4) dA

= 8π/3

The surface integral over the circular disk in the xy-plane is:

∫∫S F dS = ∫∫D F(x, y, 0) ∙ n dA

= ∫∫D (2x^2, 2xy, 0) ∙ (0, 0, 1) dA

= 0

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Identify the correct cross-section of the regular square pyramid.

### Answers

The **cross**-**section** of the square **pyramid **is a square.

Option D is the correct answer.

We have,

The **cross**-**section** of a square **pyramid** is the shape that is formed when a plane intersects the pyramid.

This shape can vary depending on the angle and position of the plane.

For example,

If the plane intersects the pyramid parallel to the base, the cross-section will be a **square**.

If the plane intersects the pyramid at an angle to the base, the cross-section will be a triangle or a trapezoid, depending on the angle of intersection and the height of the pyramid.

Thus,

The **cross**-**section** of the square **pyramid **is a square.

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if the assumption for using the chi-square statistic that specifies the number of frequencies in each category is violated, the researcher can:

### Answers

If the assumption for using the **chi-square statistic**, which specifies the number of frequencies in each category, is violated, the researcher has a few options to address this issue:

1. Combine categories: If some categories have very low expected frequencies, they can be combined with adjacent categories to increase the expected** frequencies**. This helps to meet the assumption of having a minimum expected frequency in each category.

2. Recategorize data: The researcher can also recategorize the **data** by collapsing categories or creating new categories that have more balanced **frequencies**. This can help to ensure an adequate number of observations in each category.

3. Use alternative statistical tests: If the assumptions for using the chi-square statistic cannot be met, the researcher can consider using alternative** statistical tests**. For example, if the **data **have a small sample size or violate the assumption of expected frequencies, Fisher's exact test or Monte Carlo simulation can be used as alternatives.

It is important for the researcher to carefully consider the specific circ*mstances and consult with a **statistician** to determine the most appropriate approach when the assumptions for using the chi-square statistic are violated.

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The table of values represents an exponential function f(x). What is the average rate of change over the interval −2≤x≤2? Enter your answer, as a decimal rounded to the nearest hundredth, in the box. −3 64 −2 16 −1 4 0 1 1 1/4 2 1/16 3 1/64

### Answers

Rounded to the nearest hundredth, the **average rate** of change over the interval -2 ≤ x ≤ 2 is approximately -3.98.

To find the average rate of change of the **exponential** function over the interval -2 ≤ x ≤ 2, we need to calculate the slope between the endpoints of the interval.

At x = -2, the **corresponding** value is 16, and at x = 2, the corresponding value is 1/16.

The change in y (Δy) between these two points is 1/16 - 16 = -255/16.

The change in x (Δx) is 2 - (-2) = 4.

The average rate of change is Δy/Δx = (-255/16)/4 = -255/64.

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Solve the linear programming problem by the method of corners.

Maximize P = x + 6y

subject to: x + y ≤ 4

2x + y ≤ 7

x ≥ 0, y ≥ 0

The maximum is P =______________ at (x, y) = (_____________)

### Answers

The **maximum** value of P is P=42 at (x,y)=(0,42).

Linear **programming** is a mathematical technique used to determine the best possible outcome from a given set of constraints. The method of corners is a technique used in linear programming to find the maximum or minimum value of a function by examining the corner points of the feasible region.

To solve the given linear programming problem using the method of **corners**, we first need to plot the two constraints on a graph. The feasible region is the shaded area bounded by the two lines x+y=42 and x+y=7. The next step is to identify the corner points of this feasible region.

The corner points of the feasible region can be found by solving the system of equations obtained by setting each of the two constraints equal to zero. Solving x+y=42 and x+y=7 simultaneously yields the corner points (0,42) and (7,0).

We can now evaluate the objective function P at each of the corner points to determine which point **maximizes** P. Substituting (0,42) and (7,0) into the objective function yields P=42 and P=7, respectively. Thus, the maximum value of P is 42, which occurs at the corner point (0,42).

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CDEF IS a trapezium

ABCF is a rectangle

BD=14

AB=7

EF=4

Perimeter of ABCF=26cm

Calculate the area of trapezium CDEF

### Answers

the **area **of the trapezium is 126cm ².

Since,

'A trapezium is a quadrilateral, which is defined as a shape with four sides and one set of parallel sides.'

According to the given problem,

AB = FC = 7cm

**Perimeter **of ABCF = 26cm

CB = FA

⇒ 26 = 7 + 7 + CB + CB

⇒ 12 = 2CB

⇒ CB = 6

Base 1 = CF = 6cm

**Base **2 = DEDE = DB + BA + AEDB = AEDE = 14 + 7 + 14 = 35

Area = 1/2 (a + b) h

h = 4a = Base 1 = 7b = DE = 28

Area of **trapezium **CDEF = 1/2 (7 + 35) 6

= 126

Hence, we can conclude that the **area **of the trapezium is 126cm ².

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nu 40 (c) A five-storey building has two floors in basem*nt and three floors above the ground. Total height of the building from basem*nt is 30 m and each floor is of the same height. One person is standing on the lowest basem*nt and another is standing at the roof of the top floor. Find at what distances both persons are standing from the ground?

### Answers

The two persons are standing at **distances** of 12 meters and 30 meters from the ground, respectively.

How to calculate the distance

Let's denote the **height** of each floor by x. Since there are 2 floors in the basem*nt and 3 floors above the ground, we have:

2x + 3x = 30

5x = 30

x = 6

Therefore, each **floor** has a height of 6 meters.

The **person** standing on the lowest basem*nt is at a distance of 2x = 12 meters from the ground.

The person **standing** at the roof of the top floor is at a distance of 5x = 30 meters from the ground.

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T/F: a dss can do a follow-up assessment on how well a solution is performing.

### Answers

True, a** decision support system **(DSS) can conduct a follow-up assessment on how well a solution is performing. This is because a DSS is designed to analyze and evaluate data, as well as provide ongoing feedback on the effectiveness of decisions made using the system.

By monitoring key performance indicators and other metrics, a DSS can provide valuable **insights** into the success of a particular **solution** or course of action, allowing users to make more informed decisions going forward.

True: A Decision Support System (DSS) can do a follow-up assessment on how well a solution is performing. This allows for continuous improvement and **optimization** of the implemented solution based on the collected data and feedback.

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The Taylor series for f(x)=ln(sec(x)) at a=0 is [infinity]∑n=0cn(x)n. Find the first few coefficients. c0, c1, c2, c3, c4Find the exact error in approximating ln(sec(−0.3)) by its fourth degree Taylor polynomial at a=0.

### Answers

The first few coefficients of the **Taylor series** for f(x) = ln(sec(x)) at a=0 are c₀ = 0, c₁ = 0, c₂ = 1/2, c₃ = 0, and c₄ = 1/8. The exact error in approximating ln(sec(-0.3)) by its fourth-degree Taylor polynomial at a=0 is ln(sec(-0.3)) - 0.0228375.

The **Taylor series** expansion of a function f(x) about a point a can be written as:

[tex]f(x) = \sum_{n=0}^{\infty} (f^n(a)/n!)(x-a)^n[/tex]

where fⁿ(a) denotes the nth **derivative** of f(x) evaluated at x = a.

In this case, we are given the **function** f(x) = ln(sec(x)) and the point a = 0. We need to find the first few coefficients of the Taylor series expansion of f(x) about a=0.

To do this, we first need to find the derivatives of f(x) up to the fourth order:

f(x) = ln(sec(x))

f'(x) = tan(x)

f''(x) = sec²(x)

f'''(x) = 2sec²(x)tan(x)

f''''(x) = 2sec⁴(x) + 4sec²(x)tan²(x)

Next, we evaluate these derivatives at a=0 to get the coefficients c₀, c₁, c₂, c₃, and c₄:

c₀ = f(0) = ln(sec(0)) = ln(1) = 0

c₁ = f'(0) = tan(0) = 0

c₂ = f''(0)/2! = sec²(0)/2 = 1/2

c₃ = f'''(0)/3! = 0

c₄ = f''''(0)/4! = (2sec⁴(0) + 4sec²(0)tan²(0))/24 = 1/8

Now, we can use the fourth-degree Taylor polynomial to approximate ln(sec(-0.3)) at a=0:

P₄(x) = c₀ + c₁(x-a) + c₂(x-a)² + c₃(x-a)³ + c₄(x-a)⁴

P₄(x) = 0 + 0(x-0) + (1/2)(x-0)² + 0(x-0)³ + (1/8)(x-0)⁴

P₄(x) = (1/2)x⁴ + (1/16)x⁴

To find the exact error in approximating ln(sec(-0.3)), we need to evaluate the remainder term:

R₄(x) = f(x) - P₄(x)

R₄(x) = ln(sec(x)) - ((1/2)x² + (1/16)x⁴)

Now, we substitute x=-0.3 into R₄(x) to get the exact error:

R₄(-0.3) = ln(sec(-0.3)) - ((1/2)(-0.3)² + (1/16)(-0.3)⁴)

R₄(-0.3) = ln(sec(-0.3)) - (0.0225 + 0.0003375)

R₄(-0.3) = ln(sec(-0.3)) - 0.0228375

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A pizza has a circumference of 10π

and the slices are cut at 30°

angles.

a. What is the radius?

b. What is the length of the crust of one slice of pizza?

c. What is the area of one slice?

Leave all answers in terms of π

.

### Answers

a) The value of **radius **= 5

b) The **length **of the crust of one slice of pizza is, 5π/6

c) The **area **of one slice is, 25π

Given that;

A **pizza **has a circumference of 10π and the slices are cut at 30° angles.

Hence, We can formulate;

**Circumference **of circle = 2πr

⇒ 10π = 2πr

⇒ r = 5

Hence, The value of radius = 5

Since, We know that;

Arc length = Radius x Central angle

Arc length = 5 x 30 x π/180

Arc length = 5π/6

And, the **area** of one slice is,

⇒ A = πr²

⇒ A = π × 5²

⇒ A = 25π

Thus,

a) The value of **radius **= 5

b) The **length **of the crust of one slice of pizza is, 5π/6

c) The **area **of one slice is, 25π

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Vanessa's parents want their child to go to the same college that they did. After talking with the college, they decided to pay a lump sum payment today so their child will have 4 years of prepaid tuition, fees, and housing for college. The college can receive 2. 8%, compounded semi-annual in an annuity and will need to have $37,000. 00 paid at the end of every six months for 4 years that Vanessa will be attending school. If Vanessa will attend school in 11 years, how much was deposited with the college?

### Answers

The **lump **sum payment that **Vanessa's parents **need to make today is $630,055.50.

To find the **lump **sum payment that Vanessa's parents need to make today, we can use the formula for the **present value **of an annuity:

PV = C * (1 - (1 + r)^(-n)) / r

where PV is the present value, C is the regular payment, r is the interest rate per period, and n is the number of periods.

We know that the regular **payment **is $37,000.00 and that it is made every six months for 4 years, or a total of 8 periods. The interest rate per period is 2.8% / 2 = 1.4% since interest is compounded semi-annually. The number of periods between today and when Vanessa starts school is 11 years * 2 = 22 semi-annual periods.

Substituting these values into the formula, we have:

PV = $37,000.00 * (1 - (1 + 0.014)^(-22)) / 0.014

PV = $630,055.50

Therefore, the lump **sum **payment that Vanessa's parents need to make today is $630,055.50.

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What is the smallest number of cells that need to be coloured in a 5 × 5 square

so that any 1 x4 or 4 x 1 rectangle lying inside the square has at least one cell

coloured?

### Answers

The smallest number of cells that need to be colored in a 5 × 5 square such that any 1 × 4 or 4 × 1 **rectangle **lying inside the **square **has at least one cell colored is five cells.

To determine the smallest number of cells that need to be colored in a 5 × 5 square such that any 1 × 4 or 4 × 1 **rectangle **lying inside the square has at least one cell colored, we need to consider the possible arrangements of these rectangles inside the **square**.

One **observation **is that any 1 × 4 or 4 × 1 rectangle must overlap with at least one of the cells in the center row or center column of the square. Therefore, we can **color **all the cells in the center row and center column of the square to ensure that any 1 × 4 or 4 × 1 rectangle has at least one colored cell. The cells in the center row and center column form a cross shape, which includes five cells. Therefore, the smallest number of cells that need to be colored in a 5 × 5 square such that any 1 × 4 or 4 × 1 rectangle lying inside the square has **at least **one cell colored is five cells.

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slope of (-9,6) and (3,6)

### Answers

Answer: Slope of 0

Step-by-step explanation: Both points are 6 on the y-axis.

If six coins are flipped, what is the probability of obtaining at least one tail?

P(obtaining at least one tail) = (Type an integer or a simplified fraction.)

### Answers

**Answer:**

**Step-by-step explanation:**

Probability of occurring at least one tail=1−641=6463

A cone has a volume of 4.5pi cubic inches and a height of 13.5 inches. which measure is closest to the radius of the cone in inches?

### Answers

**Answer: The volume of a cone can be expressed as:**

**V = (1/3)πr^2h**

**where V is the volume, r is the radius, and h is the height of the cone.**

**We are given that the volume of the cone is 4.5π cubic inches and the height is 13.5 inches. Substituting these values into the equation above, we get:**

**4.5π = (1/3)πr^2(13.5)**

**Simplifying this equation, we get:**

**r^2 = (4.5π * 3) / (13.5π)**

**r^2 = 1**

**Taking the square root of both sides, we get:**

**r = 1 inch**

**Therefore, the measure closest to the radius of the cone is 1 inch.**

Simplify: 2.4 x 10−4

0.00024

0.000024

-0.000024

-2.4000

### Answers

**Answer:**

The answer is **0.00024.**

**Step-by-step explanation:**

2.4 • 10 ^ -4 = 0.00024

Solution:

Option A

Which standard form of the equation of the hyperbola has vertices at (12, 0) and (-12, 0), and asymptotes y= + 5/12 x?

### Answers

The** standard form** of the equation of the hyperbola is:

x^2 - y^2 / (25/144) = 1

The standard form of the equation of a **hyperbola** with vertices at (h, k) and (-h, k) and asymptotes y = mx + b is:

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

In this case, the vertices are (12, 0) and (-12, 0), so the center of the hyperbola is at (0, 0). The** distance** from the center to each vertex is a = 12.

The asymptotes are y = (5/12)x and y = -(5/12)x. The slope of the asymptotes is m = 5/12.

Plugging these values into the standard form equation, we have:

(x - 0)^2 / 12^2 - (y - 0)^2 / b^2 = 1

Simplifying, we have:

x^2 / 144 - y^2 / b^2 = 1

Since the slopes of the asymptotes are equal to b / a, we can determine that b = 5a / 12.

Substituting this value into the equation, we get:

x^2 / 144 - y^2 / (25a^2 / 144) = 1

Multiplying both sides by 144 to eliminate the denominators, we have:

144x^2 - 144y^2 / (25a^2) = 144

Simplifying further, we get:

x^2 - y^2 / (25/144) = 1

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look at cliped pic please help

### Answers

**Answer:**

[tex]( { \frac{4}{7} )}^{2} = \frac{ {4}^{2} }{ {7}^{2} } = \frac{16}{49} [/tex]

The dimensions of a box are (x + 1), (4x - 2) and (3x + 4) what is the volume of the box? 1 ) 8x + 32 ) 12x ^ 3 + 22x ^ 2 + 2x - 83 ) 8x ^ 3 + 34 ) 12x ^ 3 + 22x ^ 2 - 2x - 8

### Answers

The **volume **of the **box **is [tex]12x^3 + 22x^2 + 2x - 8[/tex].

To find the **volume **of the box, we need to multiply the **length**, width, and **height **of the **box**. We have:

Length = x + 1

Width = 4x - 2

Height = 3x + 4

So, the volume of the box can be **expressed** as:

Volume = (x + 1)(4x - 2)(3x + 4)

Expanding this expression using distributive property, we get:

Volume = (12x^2 + 10x + 4)(3x + 4)

Multiplying again using **distributive property**, we get:

Volume = 36x^3 + 48x^2 + 30x + 16x^2 + 16x + 16

Simplifying, we get:

Volume = 12x^3 + 22x^2 + 2x - 8

Therefore, the volume of the box is [tex]12x^3 + 22x^2 + 2x - 8[/tex].

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The triangle above has the following measures.

s = 23 in

r = 75 in

Find the mzQ.

Round to the nearest tenth and include correct units.

Show all your work

### Answers

The measure of the **angle **of the **triangle **is ∠Q = 72.1°

Given data ,

Let the **triangle **be represented as ΔQRS

Now , the measure of sides of the **triangle **are

SQ = 75 inches

And , RS = 23 inches

From the **trigonometric **relations , we get

cos θ = adjacent / hypotenuse

On simplifying , we get

cos Q = 23/75

cos Q = 0.3066667

Taking inverse on both sides , we get

The inverse of the cosine **function **is Q = cos⁻¹ ( 0.306667 )

Q = 72.14°

Hence , the **angle **is Q = 72.1°

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1. The number of students attending summer school at a local community college has been decreasing each year by 8%.

If 864 students currently attend summer school and this rate continues, find the number of students attending

summer school in 4 years.

### Answers

**Answer:**

1140

**Step-by-step explanation:**

trust me

let p(x, y) be the terminal point on the unit circle determined by t. then sin(t) =

### Answers

We can use the parametric equation of the unit circle to find the values of sin(t) and cos(t) for a given terminal point (x, y). The parametric equation of the unit circle is given by:

x = cos(t)

y = sin(t)

Since the terminal point (x, y) is on the unit circle, we know that x^2 + y^2 = 1. Substituting the expressions for x and y from the parametric equation, we get:

cos^2(t) + sin^2(t) = 1

Simplifying this equation, we get:

sin^2(t) = 1 - cos^2(t)

Taking the square root of both sides, we get:

sin(t) = ±sqrt(1 - cos^2(t))

Since the unit circle is defined to have radius 1, we know that -1 ≤ cos(t) ≤ 1, and therefore 0 ≤ cos^2(t) ≤ 1. This means that 0 ≤ 1 - cos^2(t) ≤ 1, and therefore 0 ≤ sqrt(1 - cos^2(t)) ≤ 1.

Since we are given that the terminal point (x, y) is on the unit circle, we know that x^2 + y^2 = 1, and therefore cos^2(t) + sin^2(t) = 1. This implies that cos^2(t) = 1 - sin^2(t).

Substituting this expression for cos^2(t) into the equation for sin(t), we get:

sin(t) = ±sqrt(1 - cos^2(t)) = ±sqrt(1 - (1 - sin^2(t))) = ±sqrt(sin^2(t))

Since sin(t) is always non-negative on the unit circle, we can take the positive square root and simplify to get:

sin(t) = sqrt(sin^2(t)) = sin(t)

Therefore, the value of sin(t) is simply the y-coordinate of the terminal point (x, y) on the unit circle, which is given by sin(t).

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the linear regression model divides variation in the dependent variable y into two categories. name these two categories and briefly explain what they are.

### Answers

The **linear regression model** divides variation in the dependent variable y into two categories: explained variation and unexplained variation.

**Explained variation** refers to the portion of the total variation in the dependent variable y that can be accounted for by the linear relationship with the independent variable(s). In other words, it represents the amount of variation that the linear regression model can successfully predict or explain.

**Unexplained variation**, on the other hand, refers to the portion of the total variation in the dependent variable y that cannot be accounted for by the linear relationship with the independent variable(s). This can include factors such as measurement errors, random fluctuations, or other variables not included in the model. It represents the remaining variation that the linear regression model cannot predict or explain.

In summary, the** linear regression model** divides the variation in the dependent variable y into explained variation (the portion that can be predicted by the model) and unexplained variation (the portion that cannot be predicted by the model).

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which would be a consistent set of sentences? group of answer choices a set of sentences that contains no contradiction a set of sentences, each of which is in fact true each of the sentences is true a set of sentences that all can be true under the same assignment of truth values

### Answers

A **consistent **set of sentences refers to a group of **statements** that can all be true at the same time. Therefore, the answer is "a set of sentences that all can be true under the same assignment of truth **values**."

In other words, a **consistent** set of sentences is one where there is no contradiction among them, and they can be simultaneously true. This is an important concept in **logic **and reasoning, where **consistency** is necessary to avoid logical fallacies and ensure the validity of arguments. A consistent set of sentences helps to establish a **coherent **and reliable framework for reasoning and decision-making.

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