"if calvin buys 4 pounds of starfruit and 3 pounds of oranges, how much is his total utility"

they receive 25 per entry but 5 goes towards the cost of the race

 so 25-5 = 20 per entry is for the charity

 so 20x + 5000 = 25000

subtract 5000 from each side

20x= 20000

 divide both sides by 20

 x = 20000/ 20  

x = 1000

 they need at least 1000 entries to make 25000

 so to make more than 25000, they need 1001 entries

The student's problem involves the use of the trapezoid rule to approximate total distance traveled by a race car, finding the speed at a particular time point (Sv(6)), and calculating the car's average speed to see if it meets the qualifying speed.

To solve this problem, we need to conduct several mathematical operations. Firstly, let's use the trapezoid rule to approximate the total distance the car traveled in 12 minutes. Then we'll find an approximation for Sv(6) and explain its meaning. Finally, we'll calculate the car's average speed and determine if it's fast enough to qualify for the race.

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we have that

[tex]PR = 4x - 2\\RS = 3x - 5[/tex]

we know that

[tex]PS=PR+RS[/tex]

substitute the values of PR and RS in the formula above

[tex]PS=(4x-2)+(3x-5)[/tex]

Combine like terms

[tex]PS=(7x-7)[/tex]

therefore

the answer is

[tex]PS=(7x-7)[/tex]

The expression which represents PS is; 7x -7

According to the question;

From observation;

Read more on addition of algebraic expressions;

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Answer:

Step-by-step explanation:

The length of her lawn=64.15 feet.

Width of her lawn=22.70 meters.

we are given that:

1 foot=0.3048 meters.

Hence,

64.15 feet=21.86232 meters.

Hence, length of lawn=21.86232 meters.

Hence, width of the lawn is greater than the length of the lawn.

( since 22.70 meters> 21.862332 meters)

Also 1 meter=1.0936 yards.

so, width of lawn in yards is:

22.70×1.0936=24.82472 yards.

length of lawn is: 23.9086462752 yards.

Hence, width of lawn is 0.9160737 yards more than the length of lawn.

The options A, B, and D have a proportional relationship.

It is the relationship between two variables where their ratios are equivalent.

Consider the first equation, x = 2y.

It can be written like the ratio of x and y.

i.e              [tex]\frac{x}{y} -\frac{2}{1}[/tex]

So, option A is proportional at x = 2 and y = 1.

For option B, It can rewrite as

[tex]\frac{a}{b} =\frac{1}{3}[/tex]

So, option B is proportional when a = 1 and b = 3.

Option C is not proportional since it can't be written as a ratio.

Option D can be written as

[tex]\frac{7}{3} =\frac{n}{x}[/tex]

It is proportional when n = 7 and x = 3.

Option E is not proportional since it is not equal.

Therefore the proportional options are A,B, and D.

To learn more about proportional relationship, use the link given below:

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The surface area of the triangular prism is 468 (square cm).

The surface area A of a triangular prism is given by the formula:

[tex]\[ A = 2A_{\text{base}} + P_{\text{base}} \times h \][/tex]

where:

- [tex]\( A_{\text{base}} \)[/tex] is the area of one of the triangular bases,

- [tex]\( P_{\text{base}} \)[/tex] is the perimeter of one of the triangular bases,

- h is the distance between the bases.

The area of a right triangle is given by:

[tex]\[ A_{\text{base}} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]

And the perimeter of a right triangle is the sum of the lengths of its three sides.

Given that the legs of the right triangles forming the bases are 9 cm and 12 cm, and the distance between the bases is 10 cm, we can calculate the surface area.

1. Area of the triangular base [tex](\(A_{\text{base}})\)[/tex]:

[tex]\[ A_{\text{base}} = \frac{1}{2} \times 9 \times 12 \][/tex]

2. Perimeter of the triangular base [tex](\(P_{\text{base}})\)[/tex]:

[tex]\[ P_{\text{base}} = 9 + 12 + \sqrt{9^2 + 12^2} \][/tex]

3. Surface area of the triangular prism A:

[tex]\[ A = 2 \times A_{\text{base}} + P_{\text{base}} \times 10 \][/tex]

Calculate each part and find the total surface area.

[tex]\[ A_{\text{base}} = \frac{1}{2} \times 9 \times 12 = 54 \, \text{cm}^2 \][/tex]

[tex]\[ P_{\text{base}} = 9 + 12 + \sqrt{9^2 + 12^2} = 9 + 12 + 15 = 36 \, \text{cm} \][/tex]

[tex]\[ A = 2 \times 54 + 36 \times 10 = 108 + 360 = 468 \, \text{cm}^2 \][/tex]

So, the surface area of the triangular prism is [tex]\(468 \, \text{cm}^2\)[/tex].

Answer:

2.264 (3 d.p.)

Step-by-step explanation:

Given integral:

[tex]\displaystyle \int^0_{-1} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x[/tex]

First, evaluate the indefinite integral using the method of substitution.

[tex]\textsf{Let} \;\;u = 4x^6+2x[/tex]

Find du/dx and rewrite it so that dx is on its own:

[tex]\dfrac{\text{d}u}{\text{d}x}=24x^5+2 \implies \text{d}x=\dfrac{1}{24x^5+2}\; \text{d}u[/tex]

Rewrite the original integral in terms of u and du, and evaluate:

[tex]\begin{aligned}\displaystyle \int\left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x&=\int \left(u\right)^3\left(12x^5+1\right)\cdot \dfrac{1}{24x^5+2}\; \text{d}u\\\\&=\int \left(u\right)^3\left(12x^5+1\right)\cdot \dfrac{1}{2(12x^5+1)}\; \text{d}u\\\\&=\int \dfrac{u^3\left(12x^5+1\right)}{2(12x^5+1)}\; \text{d}u\\\\&=\displaystyle \int \dfrac{u^3}{2}\; \text{d}u\\\\&=\dfrac{u^{3+1}}{2(3+1)}+C\\\\&=\dfrac{u^4}{8}+C\end{aligned}[/tex]

Substitute back u = 4x⁶ + 2x:

                                              [tex]=\dfrac{(4x^6+2x)^4}{8}+C[/tex]

Therefore:

[tex]\displaystyle \int \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x=\dfrac{(4x^6+2x)^4}{8}+C[/tex]

To evaluate the definite integral, we must first determine any intervals within the given interval -1 ≤ x ≤ 0 where the curve lies below the x-axis. This is because when we integrate a function that lies below the x-axis, it will give a negative area value.

Find the x-intercepts by setting the function to zero and solving for x.

[tex]\left(4x^6+2x\right)^3\left(12x^5+1\right)=0[/tex]

Therefore:

[tex]\begin{aligned}\left(4x^6+2x\right)^3&=0\\4x^6+2x&=0\\x(4x^5+2)&=0\end{aligned}[/tex]

[tex]x=0[/tex]

[tex]\begin{aligned}4x^5+2&=0\\4x^5&=-2\\x^5&=-\frac{1}{2}\\x&=\sqrt[5]{-\dfrac{1}{2}}\end{aligned}[/tex]

[tex]\begin{aligned}12x^5+1&=0\\12x^5&=-1\\x^5&=-\dfrac{1}{12}\\x&=\sqrt[5]{-\dfrac{1}{12}}\end{aligned}[/tex]

Therefore, the curve of the function is:

So to calculate the total area, we need to calculate the positive and negative areas separately and then add them together, remembering that if you integrate a function to find an area that lies below the x-axis, it will give a negative value.

Integrate the function between -1 and ⁵√(-1/2).

As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:

[tex]\begin{aligned}A_1&=-\displaystyle \int_{-1}^{\sqrt[5]{-\frac{1}{2}}} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=-\left[\dfrac{(4x^6+2x)^4}{8}\right]_{-1}^{\sqrt[5]{-\frac{1}{2}}}\\\\&=-\left[\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{2}}\right)^6+2\left(\sqrt[5]{-\frac{1}{2}}\right)\right)^4}{8}\right)-\left(\dfrac{(4(-1)^6+2(-1))^4}{8}\right)\right]\\\\&=-[0-2]\\\\&=2\end{aligned}[/tex]

Integrate the function between ⁵√(-1/2) and ⁵√(-1/12).

[tex]\begin{aligned}A_2&=\displaystyle \int_{\sqrt[5]{-\frac{1}{2}}} ^{\sqrt[5]{-\frac{1}{12}}} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=\left[\dfrac{(4x^6+2x)^4}{8}\right]_{\sqrt[5]{-\frac{1}{2}}}^{\sqrt[5]{-\frac{1}{12}}}\\\\&=\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{12}}\right)^6+2\left(\sqrt[5]{-\frac{1}{12}}\right)\right)^4}{8}\right)-\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{2}}\right)^6+2\left(\sqrt[5]{-\frac{1}{2}}\right)\right)^4}{8}\right)\\\\\end{aligned}[/tex]

     [tex]\begin{aligned}&=\dfrac{625}{648\sqrt[5]{12^4}}-0\\\\&=0.132117398...\end{aligned}[/tex]

Integrate the function between ⁵√(-1/12) and 0.

As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:

[tex]\begin{aligned}A_3&=-\displaystyle \int_{\sqrt[5]{-\frac{1}{12}}}^0 \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x\\\\&=-\left[\dfrac{(4x^6+2x)^4}{8}\right]_{\sqrt[5]{-\frac{1}{12}}}^0\\\\&=-\left[\left(\dfrac{(4(0)^6+2(0))^4}{8}\right)-\left(\dfrac{\left(4\left(\sqrt[5]{-\frac{1}{12}}\right)^6+2\left(\sqrt[5]{-\frac{1}{12}}\right)\right)^4}{8}\right)\right]\\\\&=-\left[0-\dfrac{625}{648\sqrt[5]{12^4}}\right]\\\\&=\dfrac{625}{648\sqrt[5]{12^4}}\\\\&=0.132117398...\\\\\end{aligned}[/tex]

To evaluate the definite integral, sum A₁, A₂ and A₃:

[tex]\begin{aligned}\displaystyle \int^0_{-1} \left(4x^6+2x\right)^3\left(12x^5+1\right)\;\text{d}x&=2+2\left( \dfrac{625}{648\sqrt[5]{12^4}}\right)\\\\&=2+ \dfrac{625}{324\sqrt[5]{12^4}}\right}\\\\&=2.264\; \sf (3\;d.p.)\end{aligned}[/tex]

the correct answer is - 1/2

Answer: The value of the scale factor is [tex]-\frac{1}{2}[/tex].

Explanation:

It is given that the △E′F′G′ is the image of △EFG under a dilation through point C.

We know that the length of sides of image are in the proportion of length of sides of preimage. That proportion factor is also known as scale factor.

It is given that the length of FG is 8 cm and the length of F'G' is 4 cm.

[tex]k=\frac{F'G'}{FG} =\frac{4}{8} =\frac{1}{2}[/tex]

Therefore the side of image and preimage are in proportion of [tex]\frac{1}{2}[/tex].

Since the image and preimage are not on the same side of the center of dilation it means the factor must be negative. So,

[tex]k=-\frac{1}{2}[/tex].

Therefore, [tex]-\frac{1}{2}[/tex] is the scale factor of dilation.

Answer:

N - 8 = 22

Step-by-step explanation:

Final answer:

Elga's age is determined by setting up an equation E + 3E = 32 and solving for E. After simplifying, we find that Elga is 8 years old.

Explanation:

To solve this problem, we can use algebra to set up two equations based on the information given that Alvin's age is three times Elga's age and the sum of their ages is 32.

Let's let E represent Elga's age. According to the problem, Alvin's age will be 3E because it is three times Elga's age. We can write the following equation to represent the sum of their ages:

E + 3E = 32

Combining like terms, we have:

4E = 32

Dividing both sides by 4 to solve for E, we get:

E = 32 / 4

Therefore, Elga's age is:

E = 8

Elga is 8 years old.

we know that

Vertical angles are a pair of opposite and congruent angles formed by intersecting lines

In this problem

[tex](7x-8)=(6x+11)[/tex] --------> by vertical angles

Solve for x

Combine like terms

[tex](7x-6x)=(11+8)[/tex]

[tex]x=19\ degrees[/tex]

therefore

the answer is

the value of x is [tex]19\ degrees[/tex]

Answer:

Feet is the appropriate unit.

Step-by-step explanation:

Feet will be an appropriate measure to describe the depth of a lake.

Miles is a very large unit usually used to describe distance between two places.

Cubic centimeters is a unit of volume.

Millimeters is a very small unit used to describe small objects like the diameter of a penny etc.

Therefore, feet is the most appropriate unit to measure the depth of the lake.