Consider the parametric equations
x=cos(t)−sin(t);y=cos(t)+sin(t) 0≤t≤2π
a) Eliminate the parameter t to find a Cartesian equation for the parametric curve.
b) Sketch this parametric curve, indicating with arrows the direction in which the curve is traced.

The volume of water remaining in Abigail’s cup can be found to be 25. 14 cm³ .

First, find the volume of water in the cup when it is full. This would be the volume of the cup which is the formula of the volume of a cone :

Volume = ( 1 / 3 ) × π × r² × h

Volume = ( 1 / 3 ) × π × ( 3 cm )² × ( 8 cm )

Volume = 24π cm³

If Abigail too 2 / 3 to slosh on her face, the amount of water left would be :

= 24π cm³ - ( 1 - 2 / 3 )

= 24π cm³ - 1 / 3

= 8π cm³

= 25. 14 cm³

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La suma de dos números es 15 y la suma de sus cuadrados es 113. Por lo tanto, los dos números son 7 y 8.

Para resolver este problema, podemos utilizar el método de sustitución. Si llamamos a los dos números "x" e "y", podemos plantear dos ecuaciones con la información que nos dan:

x + y = 15 (ecuación 1)
x² + y² = 113 (ecuación 2)

De la primera ecuación, podemos despejar a "y" para obtener:

y = 15 - x

Ahora, podemos sustituir este valor de "y" en la segunda ecuación:

x² + (15 - x)² = 113

Expandiendo y simplificando:

x² + 225 - 30x + x² = 113
2x^2 - 30x + 112 = 0

Esta es una ecuación cuadrática que podemos resolver utilizando la fórmula general:

x = (-b ± sqrt(b² - 4ac)) / 2a

Donde:

a = 2
b = -30
c = 112

Sustituyendo:

x = (-(-30) ± sqrt((-30)² - 4(2)(112))) / 2(2)
x = (30 ± sqrt(900 - 896)) / 4
x = (30 ± 2) / 4

Esto nos da dos posibles valores para "x":

x₁ = 8
x₂ = 7

Para encontrar los valores correspondientes de "y", podemos utilizar la ecuación que obtuvimos antes:

y = 15 - x

Así que:

y₁ = 15 - 8 = 7
y₂ = 15 - 7 = 8

Por lo tanto, los dos números son 7 y 8.

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Gray's $3,500 tax-deductible contribution to his IRA will save him $980 on his taxes.

When Gray contributes $3,500 to his individual retirement account (IRA), it is considered a tax-deductible contribution. This means that the amount contributed is deducted from his taxable income, reducing the amount of taxes he owes.

Since Gray is in the 28 percent tax bracket, this means that for every dollar of taxable income, he pays 28 cents in taxes. To calculate the tax savings from his $3,500 IRA contribution, we need to multiply the contribution amount by his tax rate:

$3,500 (contribution) x 0.28 (tax rate) = $980 (tax savings)

In this case, Gray's $3,500 tax-deductible contribution to his IRA will save him $980 on his taxes. By contributing to his IRA, Gray not only invests in his future retirement but also takes advantage of the tax benefits associated with these accounts. In the end, he reduces his taxable income and, consequently, the amount of taxes he needs to pay.

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The calculated cost of the dryer is $475.

Let's assume that the cost of the dryer is x dollars.

According to the problem, the cost of the washer is $54 more than the dryer.

Therefore, the cost of the washer is (x + $54).

We are given that the combined cost of the washer and the dryer is $1004. So we can set up the equation:

x + (x + $54) = $1004

Simplifying the equation:

2x + $54 = $1004

2x = $950

x = $475

Therefore, the cost of the dryer is $475.

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The final value is f'(-1) = 16,777,2160.

We can use the chain rule and the power rule of differentiation to find f'(-1).

Recall that the chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) h'(x). Applying this rule to f(x) = (h(x))^10, we get:

f'(x) = 10(h(x))^9 h'(x)

Now, we can substitute x = -1 into the above equation, since we are asked to find f'(-1). Thus, we have:

f'(-1) = 10(h(-1))^9 h'(-1)

We are given that h(-1) = 3 and h'(-1) = 6, so we can substitute these values to get:

f'(-1) = 10(3)^9 (6)

Simplifying, we get:

f'(-1) = 16,777,2160

Therefore, f'(-1) = 16,777,2160

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The initial water level is given as follows:

28 units.

The slope-intercept representation of a linear function is given by the equation shown as follows:

y = mx + b

The coefficients m and b have the meaning presented as follows:

The function for this problem is defined as follows:

n = 4.5t + 28.

The intercept is of b = 28, representing the initial amount of water.

The graph is given by the image presented at the end of the answer.

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Answer:

unfortunately there's no questions to be answered

To find the perimeter of a square, you simply add up the lengths of all four sides. If we let "s" be the length of one side of the square, then the perimeter P can be found using the formula:

P = 4s

Since all four sides of a square are equal, we can simplify this expression to:

P = s + s + s + s = 4s

Therefore, the perimeter of the square is equal to 4 times the length of one side. If the length of one side is given in simplified radical form (such as √2 or √3), then the perimeter should also be expressed in simplified radical form.

For example, if the length of one side is 2√2 units, then the perimeter would be:

P = 4s = 4(2√2) = 8√2 units

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A theater can make at least $675 from ticket sales by selling at least 100 adult tickets and up to 50 child tickets.

To solve this problem, we can use a system of equations. Let's define:

- x as the number of adult tickets sold
- y as the number of child tickets sold

We know that:

- x + y ≤ 150 (due to the limit on seating)
- 12x + 7.5y ≥ 675 (they want to make at least $675)

We can solve this system of equations using substitution or elimination. Let's use elimination:

- Multiply the second equation by 2 to get rid of the decimals: 24x + 15y ≥ 1350
- Multiply the first equation by 15: 15x + 15y ≤ 2250
- Subtract the second equation from the first: 9x ≥ 900
- Divide both sides by 9: x ≥ 100

So they need to sell at least 100 adult tickets to make at least $675. Let's see if that's possible:

- If they sell 100 adult tickets, that leaves 50 tickets for children
- 100 adult tickets * $12 = $1200
- 50 child tickets * $7.50 = $375
- Total ticket sales = $1575 (more than $675)

So they can make at least $675 from ticket sales by selling at least 100 adult tickets and up to 50 child tickets.

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There would be   1.302 x 10¹² cells in koala, and  there are  3.04 x 10^17 cells in the average hippo.

To find the number of cells in the average koala, we first need to find the volume of the koala in cubic centimeters, since we know the density of the cell and can use that to find the mass of the koala in grams.

The average weight of a koala is 6 kilograms, which is equivalent to 6,000 grams. We can use the density of the cell to find the volume of the koala:

Density = Mass / Volume

1.1 g/cm³ = 6,000 g / Volume

Volume = 6,000 g / 1.1 g/cm³

Volume = 5,454.54 cm³

Next, we need to find the volume of one cell:

Volume of cell = 4/3 * π * (0.001 cm)³

Volume of cell = 4.188 x 10⁻⁹ cm³

Finally, we can divide the volume of the koala by the volume of one cell to find the number of cells in the average koala:

Number of cells = 5,454.54 cm³ / (4.188 x 10⁻⁹ cm³)

Number of cells = 1.302 x 10¹²

Therefore, there are approximately 1.302 x 10¹² cells in the average koala.

To find the number of cells in the average hippo, we can follow the same process. The average weight of a hippo is 1,400 kilograms, which is equivalent to 1,400,000 grams. Using the density of the cell, we can find the volume of the hippo:

Density = Mass / Volume

1.1 g/cm^3 = 1,400,000 g / Volume

Volume = 1,400,000 g / 1.1 g/cm³

Volume = 1,272,727.27 cm³

Dividing the volume of the hippo by the volume of one cell, we get:

Number of cells = 1,272,727.27 cm³ / (4.188 x 10⁻⁹ cm³)

Number of cells = 3.04 x 10¹⁷

Therefore, there are approximately 3.04 x 10¹⁷ cells in the average hippo.

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The equation of the tangent line to the lemniscuses 2(x²+y²) = 25 (z²-y²) at the point (-3, -1) is y = (16/25)x + 23/25.

To find the equation of the tangent line to a curve, we need to take the derivative of the equation of the curve and evaluate it at the given point.

First, let's rewrite the equation of the lemniscate in terms of x and y:

2(x² + y²) = 25(z² - y²)

Dividing both sides by 25, we get:

(x² + y²) / (25/2) = (z² - y²) / 12.5

Now, we can take the partial derivatives with respect to x and y:

∂/∂x [(x² + y²) / (25/2)] = (2x) / (25/2) = (4x) / 25
∂/∂y [(x² + y²) / (25/2)] = (2y) / (25/2) = (4y) / 25

Next, we need to find the value of z at the point (-3, -1). To do this, we can substitute x = -3 and y = -1 into the equation of the lemniscate:

2((-3)² + (-1)²) = 25(z² - (-1)²)
20 = 25(z² + 1)
z^2 = 19/25
z = ±sqrt(19)/5

Since we want the tangent line at the point (-3, -1), we'll use z = -sqrt(19)/5.

Now, we can evaluate the partial derivatives at (-3, -1, -sqrt(19)/5):

(4(-3)) / 25 = -12/25
(4(-1)) / 25 = -4/25

So, the slope of the tangent line is:

m = ∂z/∂x × -12/25 + ∂z/∂y × -4/25
m = (2x / (25/2)) × (-12/25) + (2y / (25/2)) × (-4/25)
m = -24x/125 - 8y/125
m = -24(-3)/125 - 8(-1)/125
m = 72/125 + 8/125
m = 80/125
m = 16/25

Finally, we can use the point-slope form of a line to find the equation of the tangent line:

y - (-1) = (16/25)(x - (-3))
y + 1 = (16/25)(x + 3)
y = (16/25)x + 48/25 - 25/25
y = (16/25)x + 23/25

So the equation of the tangent line to the lemniscuses 2(x²+y²) = 25 (z²-y²) at the point (-3, -1) is y = (16/25)x + 23/25.

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Answer:

Step-by-step explanation:

Required Answer :

x(2x + 4)= x²+ 4 (9 - 3x)

2x² + 4x = x² + 36 - 12x

2x² - x² + 4x + 12x - 36=0

x² + 16x - 36=0

Quadratic equation in Standard form,

Using Quadratic formula,

[tex] \implies \sf x = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} \\ \\ \implies \sf x = \dfrac{ - 16 \pm \sqrt{ {16}^{2} - 4(1)( - 36)} }{2 \times 1} \\ \\ \implies \sf x = \dfrac{ - 16\pm \sqrt{256 - ( - 144)} }{2} \\ \\ \implies \sf x = \frac{ - 16 \pm \sqrt{256 + 144} }{2} \\ \\ \implies \sf x = \frac{ - 16 \pm \sqrt{400} }{2} \\ \\ \implies \sf x = \dfrac{ - 16 \pm20}{2} \\ \\ \implies \sf x = \dfrac{ - 16 \pm 20}{2} \\ \\ \implies \sf x = \dfrac{4}{2} \: or \: \dfrac{ - 36}{2} \\ \\ \implies \sf x = 2 \: or \: - 18[/tex]

5 bonds were traded during this session.

Based on the provided bond listing for Pacific Bell, the number of bonds traded during this session is 5. Here's the breakdown of the information in the 5-column table:

- Column 1 (Bonds): PacBell 6 5/8 34
- Column 2 (Current Yield): 6.55
- Column 3 (Volume): 5
- Column 4 (Close): 99 1/4
- Column 5 (Net Change): +1/8

The "Volume" column indicates the number of bonds traded during the session. In this case, the volume entry is 5. Therefore, 5 bonds were traded during this session.

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Answer:

The population proportion is given as 58% or 0.58 in decimal form.

The sample size is given as 50 Americans.

Yes, the normal approximation can be used with this distribution because the sample size is sufficiently large (n=50) and the underlying population is assumed to be large enough to satisfy the independence requirement.

The mean of the sampling proportion (o) can be calculated using the formula:

mean = population proportion = 0.58

Therefore, the mean of the sampling proportion is 0.58 or 58%.

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101 is not in the sequence of 3n-2 because it cannot be obtained by multiplying a positive integer n by 3 and subtracting 2 from the product.

The sequence 3n-2 is a set of numbers obtained by taking a positive integer n, multiplying it by 3 and then subtracting 2 from the product. For example, if n = 1, then 3n-2 = 1. If n = 2, then 3n-2 = 4. If n = 3, then 3n-2 = 7, and so on.

Now, you may wonder why the number 101 is not in the sequence of 3n-2. To understand this, we need to determine whether there exists a positive integer n such that 3n-2 is equal to 101.

Let's start by assuming that such an n exists. Then we can write:

3n-2 = 101

Adding 2 to both sides, we get:

3n = 103

Dividing both sides by 3, we get:

n = 103/3

This means that n is not a whole number, which contradicts our assumption that n is a positive integer. Therefore, there cannot exist any positive integer n such that 3n-2 equals 101.

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The calculated value of the expression 3 - (-1) + (-1) - 3 using a calculator is 0

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

The expression 3 - (-1) + (-1) - 3

We can add the numbers using a calculator

So, we have the following representation

Value = 3 - (-1) + (-1) - 3

Using the above as a guide, we have the following:

Value = 0

This means that the value of the expression 3 - (-1) + (-1) - 3 using a calculator is 0

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To solve for the surface area of a rectangular prism, you'll need to find the area of each of its six faces and add them together.

A rectangular prism has three pairs of faces: two each for length (L), width (W), and height (H).

First, find the area of the two faces with dimensions L x W. The area is calculated by multiplying length by width: A₁ = L * W. Since there are two such faces, the total area for these is 2A₁ = 2(L * W).

Next, find the area of the two faces with dimensions L x H. The area is calculated by multiplying length by height: A₂ = L * H. The total area for these faces is 2A₂ = 2(L * H).

Finally, find the area of the two faces with dimensions W x H. The area is calculated by multiplying width by height: A₃ = W * H. The total area for these faces is 2A₃ = 2(W * H).

To find the total surface area of the rectangular prism, add the areas of all six faces together: Surface Area = 2A₁ + 2A₂ + 2A₃ = 2(L * W) + 2(L * H) + 2(W * H).

So, the formula for the surface area of a rectangular prism is Surface Area = 2(L * W + L * H + W * H).

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Answer:

This relation is a function.

Algebraically, this function is written as

y = 5x.

Answer:

Step-by-step explanation:

The base of the parallelogram is approximately equal to the circumference of the circle.

The following statements could help Josie justify the area formula for a circle:

Therefore, the area of the circle can be calculated using the formula A = C × r, or in terms of the radius, A = πr².

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To use the Integral Test, we need to find an integral that is comparable to the series. We can do this by using a basic comparison test and comparing it to the p-series with p=2.

n^2 / (n^3 + 3) < n^2 / n^3 = 1/n
The series 1/n is a divergent p-series with p=1, so we can conclude that the original series is also divergent.
To find the corresponding integral, we can integrate the function 1/n^2:
∫(n=1 to ∞) 1/n^2 dn = [-1/n] (n=1 to ∞) = 1/1 - 0 = 1
Since the improper integral converges to 1, we can conclude that the infinite series is divergent by the Integral Test.
Hi there! To use the Integral Test to determine whether the given infinite series is convergent, first rewrite the series as a function:
f(x) = x^2 / (x^3 + 3)

Next, we need to check that the function is continuous, positive, and decreasing on the interval [1, ∞). This function satisfies these conditions.
Now, we will calculate the improper integral:
∫(from 1 to ∞) (x^2 / (x^3 + 3)) dx
Let's use substitution: u = x^3 + 3, so du = 3x^2 dx, and x^2 dx = (1/3)du.
Now, the integral becomes:
(1/3) ∫(from 1 to ∞) (1/u) du
This integral is the same as the integral of 1/u from 1 to ∞, which is a well-known improper integral that diverges (ln(u) evaluated from 1 to ∞ results in ∞).

Therefore, by the Integral Test, the infinite series ∑(from n=1 to ∞) (n^2 / (n^3 + 3)) diverges. So the correct answer is B. Diverges.

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Deion increased the distance he ran by 20%.

To discover the percentage increase within the distance Deion ran, we need to first calculate the amount of increase.

The increase in distance that Deion ran this week compared to final week is:

6.4 miles

To find the proportion increase, we need to divide the increase by means of the original value (the distance he ran last week),

Then multiply by using a hundred to express the result as a percent.

The original price (last week's distance) is:

32 miles

Therefore, the percentage increase within the distance he ran is:

(6.4 miles / 32 miles) x 100% = 20%

So, Deion increased the distance he ran by 20%.

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The answer of the given question based on the graph could be used to determine the variance levels that would result in a package of vegetables is histogram.

To determine the variance levels that would result in a package of vegetables being rejected because of its weight, X, consider the following:

1. The company has a tolerance level of less than or equal to y grams for a 250-gram package. This means that the graph must represent a relationship between the weight of the package (X) and the tolerance level (y).
2. Since the package is rejected if it weighs more than the allowed tolerance, the graph should show that as the weight (X) increases, the acceptance range decreases (y decreases).
3. The graph should ideally have a boundary line that represents the maximum tolerance level (y). Any points above this line would represent rejected packages.

Based on these criteria, you should select the graph that best represents this relationship between the weight of the package (X) and the tolerance level (y), where packages with a weight exceeding the tolerance level are rejected.

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The value of the sides are;

x = 20.4

x = 13.84

There are six different trigonometric identities. They include;

Given that the ratios are;

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

Using the tangent identity, we have;

tan 64 = 42/x

cross multiply the values

x = 42/2. 050

x = 20. 4

Using the sine identity;

sin 70 = 13/x

cross multiply the values

x = 13/0. 939

x = 13. 84

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Hannah mixed 6.83 lb of pretzels with 3.57 lb of popcorn to make 10.4 lb of mixture. She then filled up 6 bags with an average of 1.68 lb of mixture per bag, leaving her with 0.35 lb of mixture left over.

In this problem, Hannah mixed 6.83 lb of pretzels with 3.57 lb of popcorn. This means that she had a total of 10.4 lb of mixture. She then filled up 6 bags that were the same size with the mixture, which means that each bag had approximately 1.73 lb of mixture (10.4 lb / 6 bags).

After filling up all 6 bags, Hannah had 0.35 lb of the mixture left over. This means that she used a total of 10.05 lb of mixture for the bags (10.4 lb - 0.35 lb).

To find out how much mixture was used per bag, we can divide the total amount of mixture used (10.05 lb) by the number of bags (6). This gives us an average of approximately 1.68 lb per bag.

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Consistent practice is key to improving your math skills over time.

If you feel like you're really bad at math, it's important to start with the basics. This might mean reviewing concepts like arithmetic, fractions, decimals, and percentages. You can find resources online or in books that can help you with this. Once you have a solid foundation, try to identify your strengths and weaknesses so you can focus your efforts on the areas where you need the most improvement. Find a learning style that works best for you, whether it's working independently, with a tutor, or in a study group. Finally, remember that consistent practice is key to improving your math skills over time. Don't give up, and don't be afraid to ask for help when you need it.

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The integral ∫8(1-tan²(x)/sec² dx Note Use an upper-case "C" for the constant of integration is ∫8(1-tan²(x)/sec²(x)) dx = 8 tan(x) + C where C is the constant of integration.

To evaluate the integral ∫8(1-tan²(x)/sec²(x)) dx, we need to use trigonometric identities to simplify the integrand.

First, we use the identity tan²(x) + 1 = sec²(x) to rewrite the integrand as follows:

8(1 - tan²(x)/sec²(x)) = 8(sec²(x)/sec²(x) - tan²(x)/sec²(x))

Simplifying this expression by canceling out the common factor of sec²(x), we get:

8(sec²(x) - tan²(x))/sec²(x)

Next, we use the identity sec²(x) = 1 + tan²(x) to simplify the expression further:

8(sec²(x) - tan²(x))/sec²(x) = 8((1 + tan²(x)) - tan²(x))/sec²(x)

Simplifying the expression inside the parentheses, we obtain:

8/ sec²(x)

Therefore, the integral simplifies to:

∫8(1-tan²(x)/sec²(x)) dx = ∫8/ sec²(x) dx

We can now use the substitution u = cos(x) and du/dx = -sin(x) dx to transform the integral into a simpler form:

∫8/ sec²(x) dx = ∫8/cos²(x) dx = 8∫cos(x)² dx

Using the power-reducing formula cos²(x) = (1 + cos(2x))/2, we get:

8∫cos(x)² dx = 8/2 ∫(1 + cos(2x))/2 dx = 4(x + 1/2 sin(2x)) + C

Substituting back u = cos(x), we obtain:

∫8(1-tan²(x)/sec²(x)) dx = 8 tan(x) + C

where C is the constant of integration.

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By evaluating square root and using Pythagorean Theorem the value of a is a= 3.33 (rounded to two decimal places)

The given expression is 5/3, which can be simplified as follows:

a = 5/3

To evaluate the square root of a, we can rewrite it in terms of exponents:

a = (5/3)¹/₂

Using a calculator, we get:

a ≈ 1.83

Next, we can use the Pythagorean Theorem to find the value of c, given that a = 3.33 and b = 4.66:

a² + b² = c²

(3.33)² + (4.66)² = c²

11.0889 + 21.7156 = c²

32.8045 = c²

c ≈ 5.72

Therefore, the final answer is a = 3.33 and c ≈ 5.72.

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Since the expression is the same, we can conclude that the binary operation ⋆ is associative.

To show that the binary operation ⋆ is commutative, we need to demonstrate that a ⋆ b is equal to b ⋆ a for any integers a and b.

Let's start by evaluating a ⋆ b:

a ⋆ b = 2ab + 5.

Now let's evaluate b ⋆ a:

b ⋆ a = 2ba + 5.

By comparing the expressions for a ⋆ b and b ⋆ a, we can see that they are indeed equal:

2ab + 5 = 2ba + 5.

Since the expression is the same, we can conclude that the binary operation ⋆ is commutative.

To show that the binary operation ⋆ is associative, we need to demonstrate that (a ⋆ b) ⋆ c is equal to a ⋆ (b ⋆ c) for any integers a, b, and c.

Let's evaluate (a ⋆ b) ⋆ c:

(a ⋆ b) ⋆ c = (2ab + 5) ⋆ c = 2(2ab + 5)c + 5 = 4abc + 10c + 5.

Now let's evaluate a ⋆ (b ⋆ c):

a ⋆ (b ⋆ c) = a ⋆ (2bc + 5) = 2a(2bc + 5) + 5 = 4abc + 10a + 5.

By comparing the expressions for (a ⋆ b) ⋆ c and a ⋆ (b ⋆ c), we can see that they are indeed equal:

4abc + 10c + 5 = 4abc + 10a + 5.

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