Two households are saving money each month to go towards the shared costs of a vacation trip. One household has $700 saved and will continue saving $400 each month. The other household has $1,000 saved and will continue saving $300 each month.

Use the system of equations, where y represents total saved and x represents number of months, to answer the question.

y = 400x + 700

y = 300x + 1,000

At what point would the two lines intersect if graphed?

a.) [tex]135 + 89 + \text{C} \geq 325[/tex]

[tex]224 + \text{C} \geq 325[/tex]

b.) [tex]224 + \text{C} \geq 325[/tex]

[tex]\text{C} \geq 325 - 224[/tex]

[tex]\text{C} \geq 101[/tex]

Answer:

These are all the 3D shapes ;cone, cylinders, cuboid,cubes, sphere, rectangular prism, pyramid

Answer: 15 square units

Step-by-step explanation:

Area of a triangle is 1/2bh

x = b = 6

H=5

1/2(6)(5)

1/2(30)

Area = 15 square units

The value of the infinite series expression [tex]\sum^{124}_{i = 1}(29a_i - 13b_i) \\[/tex] is -30.

Given information:

[tex]\sum^{124}_{i = 1}a_i = -10[/tex]  and [tex]\sum^{124}_{i = 1} b_i = -20[/tex].

The required series form can be rearranged as,

[tex]\sum^{124}_{i = 1}(29a_i - 13b_i) &= 29\sum^{124}_{i = 1}a_i - 13\sum^{124}_{i = 1}b_i[/tex]

Substitute the provided values of [tex]\sum^{124}_{i = 1}a_i = -10[/tex]  and [tex]\sum^{124}_{i = 1} b_i = -20[/tex].

We get,

[tex]\sum^{124}_{i = 1}(29a_i - 13b_i) \\[/tex] = 29(-10) - 13(-20)

[tex]\sum^{124}_{i = 1}(29a_i - 13b_i) \\[/tex] = -290+ 260

[tex]\sum^{124}_{i = 1}(29a_i - 13b_i) \\[/tex] = -30.

As a result, the required value is -30.

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Answer:

2((4.5)(6) + (4.5)(4) + (6)(4))

= 2(27 + 18 + 24) = 2(69) = 138 square cm

The number of pages that can be printed in half a minute is 90 pages

From the question, we have the following parameters that can be used in our computation:

The graph represents a  proportional relationship

Where

x = minutes

y = pages

When the time is half a minute, we have

x = 0.5

So, we have the following ordered pair

(0.5, y)

From the question. we have

(4, 720)

So, we have

(0.5, y) = (4, 720)

Express as an equation

So, we have

y/0.5 = 720/4

This gives

y = 0.5 * 720/4

Evaluate

y = 90

Hence, the number of pages is 90

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Yes , Han is correct as the a circle with a longer arc length has a larger central angle

Given data ,

Let the arc length of the circle A = 6π units

Let the radius of the circle A = 15 units

Let the arc length of circle B = 2π units

Let the radius of the circle B = 5 units

Now , The length of an arc in a circle is determined by both the radius of the circle and the central angle that subtends the arc. The formula for calculating the length of an arc in a circle is given by:

Arc length = (Central angle / 360 degrees) x (2πr)

Though two circles' radii differ, they may nevertheless have distinct arc lengths even though their centre angles are the same. Similar to this, if the radii of two circles are adjusted properly, two circles with differing central angles can nevertheless have the same arc length.

Hence , the central angle is solved

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The mean excess return of the value-weighted index portfolio is 19.458%, and the mean return of the index portfolio, adjusted for the risk-free rate, is 21.458%.

To calculate the mean excess return of the value-weighted index portfolio, we need to first calculate the total market capitalization of the three stocks:

Total Market Cap = $3,000 + $2,000 + $1,500 = $6,500

Next, we can calculate the weight of each stock in the index portfolio as follows:

Weight of Stock A = $3,000 / $6,500 = 0.4615

Weight of Stock B = $2,000 / $6,500 = 0.3077

Weight of Stock C = $1,500 / $6,500 = 0.2308

The capitalization-weighted beta of the portfolio can then be calculated as the sum of the product of each stock's weight and its beta:

Beta of Index Portfolio = (0.4615 x 0.15) + (0.3077 x 0.95) + (0.2308 x 1.7) = 1.081

The mean excess return of the index portfolio can be calculated using the single-index model as:

Mean Excess Return = Beta of Index Portfolio x Standard Deviation of Market-Index Portfolio

= 1.081 x 18%

= 19.458%

Finally, we need to adjust for the risk-free rate to obtain the actual mean return of the index portfolio. If the risk-free rate is 2%, then the mean return of the index portfolio would be:

Mean Return = Mean Excess Return + Risk-Free Rate

= 19.458% + 2%

= 21.458%

Therefore, the mean excess return of the value-weighted index portfolio is 19.458%, and the mean return of the index portfolio, adjusted for the risk-free rate, is 21.458%.

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The angle between the slant height and the base of the cone is 30.2°.

Recall the formula of the volume of a cone:

V = (1/3)πr²h

where

V = volume,

r = radius,

h = height.

To get the h, we use the volume formula and substitute:

264 = (1/3)π(6²)h

Make h the subject of the formula and simplify:

h = (3/π)(264/36) = 8.8

Therefore, h = 8.8 m.

Now, we find the slant height using the Pythagorean theorem:

s² = r² + h²

Plugging in the values, we get:

s² = 6² + 8.8² = 102.64

s = √(6² + 8.8²)

s = 10.13

To find the angle between the slant height and the base, we use trigonometry:

tan θ = r / s

r = radius of the base

s = slant height

Plugging in the values, we get:

tan θ = 6 / 10.13

tan θ = 0.5925

Find the arctan of θ

θ = tan⁻¹(0.5925)

θ = 30.2°

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Yes , y varies directly with x as a proportion of k = 2.5

Given data ,

Let the values of x = { 2 , 6 , 18 }

Let the values of y = { 5 , 15 , 45 }

Now , the proportion equation is y = kx

On simplifying , we get

k = 5/2

k = 2.5

Therefore , the proportionality constant is k = 2.5

Hence , the proportion is y = 2.5 ( x )

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The pattern that can be used to represent a triangular number is T(n) = 1 + 2 + 3 + ... + n.

A number that can take the form of an equilateral triangle is called a triangular number. By definition, it represents the sum of positive integers up to n consecutive natural numbers. The pattern for such a calculation is expressed as follows:

T(n) = 1 + 2 + 3 + ... + n.

Triangular numbers exhibit many engaging properties and are applicable across various mathematical and real-world contexts.

In conclusion, the pattern used to represent a triangular number is T (n) = 1 + 2 + 3 + ... + n.

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A = s^2, where A is the area and s is the length of one side of the square gives the area of a square base. The correct option is true

If s^2 = 144, then s = √144 = 12 feet. The correct option is false

A pyramid is a polyhedron that is created by joining an apex and a polygonal base. A triangle known as a lateral face is formed by each base edge and apex.

Therefore, s^2 = 144 is a valid equation to find the length of one side of the base. The correct option is true

Therefore, one side of the pyramid's base is 12 feet, not 72 feet.

The correct option is false

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The area of the square given in the diagram is: 25 square units.

Pythagoras Theorem is defined as the way in which we find the missing length of a right angled triangle.

The triangle has three sides, the hypotenuse (which is always the longest), Opposite (which doesn't touch the hypotenuse) and the adjacent (which is between the opposite and the hypotenuse).

Pythagoras is in the form of;

a² + b² = c²

Using Pythagoras theorem, we can find the length of a side of the square:

AB = √(3² + 4²)

AB = √(9 + 16)

AB = √25

AB = 5

Area of a square is given by the formula:

A = side * side

Thus:

A = 5 * 5

A = 25

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perimeter of the shape

Area of the shape

The shape is a composed of two simpler parts which consists of, namely:

Therefore, the perimeter of the shape includes perimeter of semicircle and the perimeter triangle

To complete the perimeter of the triangle we add two sides of the triangle. the opposite and the adjacent of the right triangle is given so we solve for the hypotenuse.

The hypotenuse of the triangle = [tex]\sqrt{ 7^2 + 6^2}[/tex] = 9.22 units

perimeter of the shape

= perimeter of triangle side + perimeter of semicircle side

= (9.22 + 6) + (3.142 * 3.5)

= 167.37 units

Area of the shape

= area of triangle + area of semicircle

= 1/2 x 7 x 6 + 1/2 x 3.142 x 3.5^2

= 21 + 19.24

= 40.24 square units

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Answer:

A

Step-by-step explanation:

sqrt 5 ( 4pi - 2pi) =   2 pi sqrt 5  for the numerator

then     2 pi sqrt (5)  / 3 sqrt (5)      the sqrt (5) 's cancel out and you have

          2pi / 3

The values of the composite functions are f(g(2)) = 5.5, h(f⁻¹(5)) = 1, x = 5 for f(h(x)) = 7, R(4) = 4 and k(x) = 2

From the question, we have the following parameters that can be used in our computation:

The functions f(x), g(x) and h(x)

Solving the composite functions, we have

f(g(2)) = f(3) i.e. g(2) = 3

This gives

f(g(2)) = 5.5

Next, we have

h(f⁻¹(5))

Here, we have

f⁻¹(5) = 4

So, we have

h(f⁻¹(5)) = h(4)

Evaluate

h(f⁻¹(5)) = 1

Next, we have

f(h(x)) = 7

From the graph, we have

f(0) = 7

This means that

h(x) = 0

From the table, we have

h(5) = 0

This means that

x = 5 for f(h(x)) = 7

Evaluating R(x) at x = 4. we have

R(x) = 2f(x) - 3g(x) + h(x)

This means that

R(4) = 2f(4) - 3g(4) + h(4)

So, we have

R(4) = 2 * 9 - 3 * 5 + 1

R(4) = 4

Evaluating k(x) at x = 5. we have

k(x) = h⁻¹(g(f⁻¹(x)))

This means that

k(x) = h⁻¹(g(f⁻¹(5)))

From the graph, f⁻¹(5) = 4

So, we have

k(x) = h⁻¹(g(4))

Next, we have

g(4) = 5

So, we have

k(x) = h⁻¹(5)

From the table, we have

h⁻¹(5) = 2

This means that

k(x) = 2

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The value of f(j) when

1. j = -1 is 11

2. j = 0 is 10

3. j = 1 is 11

4 . j = 2 is 14

A function is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable.

For example, here j is a variable and f(j) is the function of 'j'.

f(j) = j²+10

Therefore , when j = -1

f(j) = -1)² + 10

f(j) = 1+10 = 11

When j = 0

f(j) = 0²+10

= 0+10 = 10

when j = 1

f(j) = 1²+10

= 1+10 = 11

When j= 2

f(j) = 2²+10

= 4+10

= 14

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Answer:a

Step-by-step explanation:

Answer:

Step-by-step explanation:

It is a A

On solving the "equation" written as "5x² - 8x + 12 = 3x + 10" by using "factoring-method" , the "solutions" are x = 1/5 and x = 2.

To solve "5x² - 8x + 12 = 3x + 10" by using the "factoring-method", it should first be rearranged in "standard-form", where one side of the equation is = 0,

First, we combine the constant terms on the right side of the quadratic:

⇒ 5x² - 8x + 2 = 3x,

⇒ 5x² - 11x + 2 = 0,  Now, we factor the quadratic expression on the left side of the equation as :

⇒ 5x² - 10x -x + 2 = 0,

⇒ 5x(x - 2) -1(x - 2) = 0,

⇒ (5x - 1)×(x - 2) = 0

By applying the zero product property, we know that either "5x - 1 = 0" or "x - 2 = 0";

⇒ 5x - 1 = 0 , We get x = 1/5,

⇒ x - 2 = 0, We get x = 2;

Therefore, the solution are x = 2, and x = 1/5.

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The given question is incomplete, the complete question is

Solve the quadratic-equation "5x² - 8x + 12 = 3x + 10" by factoring.

The range is the difference between the maximum and minimum values in the data set. From the box plot, we can see that the maximum value is 52 and the minimum value is 16, so the range is:

Range = Maximum value - Minimum value

Range = 52 - 16

Range = 36

The IQR (interquartile range) is the difference between the first and third quartiles of the data set, which is represented by the length of the box in the box plot. From the box plot, we can see that the length of the box is:

Box length = Upper quartile - Lower quartile

Box length = 45 - 27

Box length = 18

Therefore, the range is 36, and the IQR is 18.

So the correct answer is:

The range is 36, and the IQR is 18.

Area= 69π or 216.77 square units.

Remember that the formula for the area of a circle is the following:

[tex]A=\pi r^{2}[/tex]; where "r" is the radius of the circle.

Hence, the area of the white circle is:

[tex]A_{WC} =\pi (10)^{2} =100\pi[/tex]

Here you can use the same formula, just need to change the value of the radius:

[tex]A_{RC} =\pi (13)^{2} =169\pi[/tex]

[tex]A_{RC} -A_{WC} =169\pi -100\pi =69\pi[/tex]

The area of the shaded region is 69π or 216.77 square units, approximately.

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The necessary information to carry out the hypothesis are given below.

For carrying out a hypothesis examination on the information, we need to do the following steps:

Step 1: Express the null and alternative hypotheses

Step 2: Pick the pertinent measure of value and significance level

Step 3: Compute the test statistic

Step 4: Estimate the p-value

Step 5: Draw a conclusion and interpret the results

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The values are expressed as;

1. radius = 13cm

2. Diameter = 27.2cm

3. Diameter = 24m

4.  Radius = 24. 5m

It is important to note the formula for calculating the circumference and area of a circle.

The formula for calculating area of a circle is expressed as;

A = πr²

Such that the parameters are given as;

Also, the formula for circumference is expressed as;

C = 2πr

To determine the formula for diameter is expressed as;

Diameter = 2Radius

1. Substitute the values

r = d/2

r = 26/2

r = 13cm

2. If r= 13.6m

d= 2(13.6)

d = 27.2m

3. If r = 12m

d = 2(12)

d = 24m

4. If d = 49m

Then, r = 49/2 = 24. 5m

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The correct statements regarding the scatter plot are given as follows:

The scatter plot, as shown by the image given in the problem is a series of points of the data-set in the following format:

(SAT score, GPA).

There is a positive association between the data, as the line of best fit, which gives the best approximation between the SAT score and the GPA, and can be used to predict the GPA based on the SAT score, is increasing.

For a SAT score of 1700, an expected GPA is of about 3.1, considering the intersection of y = 3.1 with the line of fit when x = 1700.

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All three values (I, II, and III) are solutions to the inequality, so the correct answer is "I, II and III."

To solve the inequality -10 + 3x ≥ -4, we can follow these steps:

Add 10 to both sides of the inequality:

3x ≥ 6

Divide both sides by 3:

x ≥ 2

Now, let's check which of the given values satisfies the inequality:

I. 8:

x ≥ 2, and 8 ≥ 2, so 8 is a solution.

II. 2:

x ≥ 2, and 2 ≥ 2, so 2 is also a solution.

III. 6:

x ≥ 2, and 6 ≥ 2, so 6 is a solution as well.

All three quantities (I, II, and III) fulfill the requirements of the inequality; hence, the right you have to select is "I, II and III."

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The statements that are always true are:

i. m∠5 + m∠6 = 180°

ii. m∠2 + m∠3 = m∠6

iii. m∠2 + m∠3 + m∠5 = 180°

A triangle is a shape which is bounded by just three sides, thus it has three internal angles whose measures add up to 180°. The internal angle of a triangle is the measure of the angle formed within the two lines forming the edge. While the exterior angles are angles that exist outside the boundary of the triangle when its sides are extended.

From the given question and information, the statements that are always true regarding the diagram are:

i. m∠5 + m∠6 =180°  

ii. m∠2 + m∠3 = m∠6

iii. m∠2 + m∠3 + m∠5 = 180°

The required diagram is attached to this answer as a guide

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The correct answer is,

C. 60(x - 3) = 45x.

This equation is based on the information that the 45 expensive wood costs $3.00 more than the 60 cheap wood. In other words, if the price of the 60 cheap wood is x, the price of the 45 expensive wood is x + 3.

Since,  Algebra involves working with numbers, equations, and variables to solve real-world problems and understand abstract relationships.

Now, To solve this equation, we must use algebra and solve for x.

First, we multiply both sides of the equation by 4/5. This gives us:

48x + 36 = 45x

-36 = 3x

Finally, we divide left side of the equation by 3, giving us the final solution:

x = -12

Therefore, the price of the 60 cheap wood is $12, and the price of the 45 expensive wood is $15.

that is, 60 (x - 3) = 45x

This equation shows the correct relationship between the two types of wood that the contractor purchased.

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Answer:

All real numbers