One option in a roulette game is to bet $2 on red. ​ (There are 18 red​ compartments, 18 black​ compartments, and two compartments that are neither red nor​ black.) If the ball lands on​ red, you get to keep the $2 you paid to play the game and you are awarded $2. If the ball lands​ elsewhere, you are awarded nothing and the $2 that you bet is collected. Find the expected payback for this roulette game if you bet $2 on red.

total pounds

to add, find a common deonmenator (bottom numbers)

the denomenantors are 6,3,2

the common denominator is 6 because 6*1=6, 3*2=6, 2*3=6

multiply each by 1 or a/a where a=a to make the common denomenators so we can add

remember that [tex]\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}[/tex]

5/6 times 1/1=5/6

2/3 times 2/2=4/6

1/2 times 3/3=3/6

so

[tex]\frac{5}{6}+\frac{2}{3}+\frac{1}{2}=[/tex]

[tex]\frac{5}[6}+\frac{4}[6}+\frac{3}{6}=[/tex]

[tex]\frac{5+4+3}{6}=[/tex]

[tex]\frac{12}{6}=[/tex]

[tex]2[/tex]

2 pounds total of special mix

I found my answer using maths

To find the shortest total length of fence needed, divide the field into 3 equal areas. Use the area formula and differentiate to find the value of L that minimizes the perimeter. The shortest total length of fence needed is approximately 64.26m.

To find the shortest total length of fence needed, we need to divide the field into 3 equal areas. Since the total area is 800m², each area will be 800m² ÷ 3 = 266.67m².

Now, let's find the dimensions of each area. Let the length of the rectangle be L and the width be W.

Since the areas are equal, we have LW = 266.67m². We are also given that the field is rectangular, so the perimeter is given by 2L + 2W.

To minimize the perimeter, we can use the formula 2L + 2W = P, and solve for L or W. Using the area formula and substituting, we have 2L + 2(266.67m²/L) = P.

Differentiating with respect to L and equating to zero, we can find the value of L that minimizes the perimeter. Solving the equation gives L ≈ 15.65m.

Therefore, the shortest total length of fence needed is approximately 2L + 2W ≈ 2(15.65m) + 2(266.67m²/15.65m) ≈ 64.26m.

For a rectangular field with a total area of 800 m², divided into three equal areas, the optimal solution is a square field with a side length of approximately 28.28 m. The shortest total length of fence needed is approximately 113.14 m.

Let's go through the step-by-step calculation:

1. **Given Information:**

  - Total area of the rectangular field: [tex]\(800 \, \text{m}^2\).[/tex]

  - The field is divided into 3 equal areas.

2. **Calculate the area of each section:**

  [tex]\[ \text{Area of each section} = \frac{\text{Total area}}{\text{Number of sections}} = \frac{800 \, \text{m}^2}{3} \approx 266.67 \, \text{m}^2 \][/tex]

3. **Assume a square field:**

  Let [tex]\(a\)[/tex] be the side length of the square field.

4. **Express the total area in terms of side length [tex](\(a\))[/tex]:**

  [tex]\[ a^2 = 800 \, \text{m}^2 \][/tex]

5. **Calculate the side length [tex](\(a\))[/tex] :**

  [tex]\[ a = \sqrt{800} \approx 28.28 \, \text{m} \][/tex]

6. **Calculate the total length of the fence (perimeter of the square):**

  [tex]\[ \text{Perimeter} = 4a = 4 \times 28.28 \, \text{m} \approx 113.14 \, \text{m} \][/tex]

Answer: the first one

Step-by-step explanation:

We are given the function:

d (x) = 1375 – 110 x

A. To solve for the distance between the train and its destination after 8 hours, we simply have to plug in, x = 8 into the equation. That is:

d (x) = 1375 – 110 * 8

d (x) = 495 miles

Therefore the train is now only 495 miles away from the destination.

B. To solve for the total time needed to reach its destination, we must set d = 0 and find for the value of x:

0 = 1375 – 110 x

110 x = 1375

x = 12.5 hours

Therefore it takes 12.5 hours for the train to reach its destination.

Probability lies at the heart of this mathematics question, involving the selection of two students at random from a college. The calculations involve determining the joint probability for the two students being of the same sex, either both male or both female, from the given student population distribution.

The subject of this problem is probability. Here, in this liberal arts college, we've been given that 40% of 10,000 students, i.e., 4000 students are males; hence, 60% i.e., 6000 students are females.

When selecting two students at random, there are two possibilities: (1) both students are male or (2) both students are female.

The probability of both students being male can be calculated by multiplying: the probability of the first student being male (which is 4000/10000 = 0.4) by the probability of the second student also being male (which, after the first extraction, would be 3999/9999 since the total number of students and the number of male students both decrease by 1). Hence, the joint probability for two male students is: 0.4 * (3999/9999).

Similarly, we can calculate the joint probability for two female students using the same approach: 0.6 * (5999/9999).

The total probability that the two selected students are of the same gender would be the sum of these two probabilities.

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Aiko started jumping rope at 7:45, which is calculated by subtracting the 20-minute jump rope activity duration from the time she stopped at 8:05.

To solve this problem, we need to subtract the duration of Aiko's jump rope activity from the time she stopped. We know that Aiko stopped jumping rope at 8:05 and the duration of her jump rope activity was 20 minutes.

Since Aiko's activity lasted 20 minutes and she ended at 8:05, we need to subtract 20 minutes from 8:05 to find out when she started. Thus, Aiko started jumping rope at 7:45.

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To solve the inequality 2 - (6/5)x >= -4, isolate x to find that x must be less than or equal to 5.

To solve the inequality 2 - (6/5)x >= -4, we first need to isolate the variable x. Let's start by adding (6/5)x to both sides of the inequality:

Next, we add 4 to both sides to get:

Now we divide both sides by (6/5) to solve for x:

Finally, we can write the solution to the inequality:

x <= 5

So, x must be less than or equal to 5 to satisfy the inequality 2 - (6/5)x >= -4.

Answer:

(A) $1,090

Step-by-step explanation:

We will see that the net income at the comic store is $1,090, so the correct option is a.

The net income will be the difference between how much money enters the store and how much he needs to pay.

We know that each month he gets $3,250.

And he must pay $1,285 for the comics and $875 in expenses, so the net income is:

N = $3,250 - $1,285 - $875 = $1,090

We conclude that the net income is $1,090, so the correct option is a.

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The linear function g(t) = 9t satisfies the condition g(t) ≤ f′(t) for t ≥ 0, as guaranteed by the Racetrack Principle.

Racetrack principle is a mathematical concept that involves bounding a function based on its second derivative and initial conditions.

Given data:

We want to find a linear function g(t) that satisfies g(t) ≤ f′(t) for t ≥ 0.

Since f″(t) ≥ 9 for all t ≥ 0, we will use Racetrack Principle to establish an upper bound for f(t) based on its second derivative:

= f(t) ≤ (1/2) * 9 * t^2

= 4.5t^2

Now, we need to find a linear function g(t) such that g(t) ≤ f′(t) for t ≥ 0. To do this, we differentiate the upper bound we found for f(t):

= f′(t) ≤ 9t

So, we can see that g(t) = 9t satisfies g(t) ≤ f′(t) for t ≥ 0.

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The Racetrack Principle can be used to find a linear function that is less than or equal to f'(t). We can prove this by comparing the initial conditions and the second derivative of the functions.

The Racetrack Principle states that if two objects are traveling at the same initial and final velocities, but one of them has a greater magnitude of acceleration, then the object with the greater acceleration will overtake the other or be ahead of it at some point in time.

In this case, since f''(t) = 9 and f(0) = f'(0) = 0, we can use the Racetrack Principle to find a linear function g(t) that is less than or equal to f'(t).

Let g(t) = 3t (since 3 is the square root of 9). We can prove that g(t) is less than or equal to f'(t) by showing that g(0) = f'(0) and g''(t) ≥ f''(t) for all t ≥ 0.

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Answer:

Step-by-step explanation:

1. A negative correlation between the temperature and the amount of snow still on the ground

This is casual since temperature and amount of snow are inversely proportional to each other.

2. A negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is left

This is casual since the number of digital photos uploaded and the amount of storage space are  inversely proportional to each other.

3. A positive correlation between the length of the side of a pool and its depth

.

This is not casual since the length of the side of a pool and its depth  are not related.

4. A positive correlation between the height of a woman and the height of her brother

.

This is not casual since the height of a woman and the height of her brother  are not related.

5. A negative correlation between the volume of water in a pot and the amount of time that the water takes to boil

This is not casual since the volume of water in a pot and the amount of time that the water takes to boil are directly proportional.

A relationships would most likely be causal,

a negative correlation between the temperature and the amount of snow still on the ground.

a negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is left

1. A negative correlation between the temperature and the amount of snow still on the ground

This is casual since temperature and amount of snow are inversely proportional to each other.

2. A negative correlation between the number of digital photos uploaded to a website and the amount of storage space that is left

This is casual since the number of digital photos uploaded and the

amount of storage space is inversely proportional to each other.

3. A positive correlation between the length of the side of a pool and its depth

This is not casual since the length of the side of a pool and its depth are not related.

4. A positive correlation between the height of a woman and the height of her brother

This is not casual since the height of a woman and the height of her brother are not related.

5. A negative correlation between the volume of water in a pot and the amount of time that the water takes to boil

This is not casual since the volume of water in a pot and the amount of time that the water takes to boil are directly proportional.

To learn more about the correlation visit:

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Answer: The correct option is A.

Explanation:

The given function is,

[tex]f(x)=x^2+8-16x[/tex]

Rewrite the above function.

[tex]f(x)=(x^2-16x)+8[/tex]

To make the perfect square we add and subtract the square of [tex]\frac{b}{2a}[/tex], where b is coefficient of x and a is the coefficient of [tex]x^2[/tex].

Since a=1 and b = -16, So we will add and subtract he square of -8.

[tex]f(x)=(x^2-16x+(-8)^2)+8-(-8)^2[/tex]

[tex]f(x)=(x^2-16x+(8)^2)+8-64[/tex]

Using [tex](a-b)^2=a^2-2ab+b^2[/tex]

[tex]f(x)=(x-8)^2)-56[/tex]

Therefore, the correct option is A.

To find the Jacobian of the given transformation, we need to compute the partial derivatives of x and y with respect to u and v. The Jacobian matrix is a matrix that represents these partial derivatives.

To find the Jacobian of the transformation x = u^2 + uv and y = uv^2, we need to compute the partial derivatives of x and y with respect to u and v. Let's start with the partial derivative of x:

∂x/∂u = 2u + v

Now let's find the partial derivative of x with respect to v:

∂x/∂v = u

Using the same process, we can find the partial derivatives of y:

∂y/∂u = v^2

∂y/∂v = 2uv

Putting it all together, the Jacobian matrix is:

J(u, v) = [∂x/∂u, ∂x/∂v; ∂y/∂u, ∂y/∂v]

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The Jacobian of the transformation is [tex]\( uv(4u + v) \)[/tex].

The Jacobian of the transformation given by [tex]\( x = u^2 + uv \) and \( y = uv^2 \)[/tex] is determined by finding the determinant of the matrix of partial derivatives. The matrix of partial derivatives, known as the Jacobian matrix, is given by: [tex]J = \begin{bmatrix} \frac{\partial x}{\partial u} \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} \frac{\partial y}{\partial v} \end{bmatrix}[/tex]

We calculate each of the partial derivatives:

 [tex]\frac{\partial x}{\partial u} = \frac{\partial}{\partial u}(u^2 + uv) = 2u + v[/tex]

[tex]\frac{\partial x}{\partial v} = \frac{\partial}{\partial v}(u^2 + uv) = u[/tex]

[tex]\frac{\partial y}{\partial u} = \frac{\partial}{\partial u}(uv^2) = v^2[/tex]

[tex]\frac{\partial y}{\partial v} = \frac{\partial}{\partial v}(uv^2) = 2uv[/tex]

Now we can construct the Jacobian matrix: [tex]J = \begin{bmatrix} 2u + v u \\ v^2 2uv \end{bmatrix}[/tex]

The determinant of this matrix gives us the Jacobian of the transformation:

[tex]\text{Jacobian} = \text{det}(J) = (2u + v)(2uv) - (u)(v^2)[/tex]

[tex]\text{Jacobian} = 4u^2v + 2uv^2 - uv^2[/tex]

[tex]\text{Jacobian} = 4u^2v + uv^2[/tex]

[tex]\text{Jacobian} = uv(4u + v)[/tex]

Therefore, the Jacobian of the transformation is [tex]\( uv(4u + v) \).[/tex]

The value of a bluefin tuna fishing license depends on the price of bluefin tuna and the annual catch limit. The value can be significant, given the potential income of $650,000 per year from tuna fishing. However, it represents a balance between profit and sustainable fishing practices.

The value of a bluefin tuna fishing license mostly depends on the price of bluefin tuna. Given that the price is $3.25 per pound and the average fisherman can catch 200,000 pounds (100 tons) per year, a fisherman could potentially earn $650,000 from tuna fishing annually. This suggests that the value of a fishing license may be quite high. However, the exact cost of the license is determined by several factors, including regulations, environmental impacts (often referred to as a 'tragedy of the commons' scenario), and maintaining a sustainable fishing industry.

Fisheries supposedly operate by established catch limits to prevent decimating the bluefin tuna population, meaning they are restricted in how much fish they can catch. Thus, the value of a fishing license also includes the advantage of sustainable fishing practices, ensuring the bluefin tuna's availability for future generations.

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