elijah and riley are playing a board game elijah choses the dragon for his game piece and rily choses the cat for hers.the dragon is about 1/2 inch tall and the cat is about 7/8 inch tall the model shows how the heights of the game peice are realated.

Write the quadratic equation in standard form that represents the volume of the box:[tex]w^2 - 6w = 0[/tex]

Solve the quadratic equation for x and find the length and width of the box:

w(w - 6) = 0 (factor the quadratic)

w = 0 or w = 6 (apply the zero product property)

Since a box can't have a width of 0, we reject the solution w = 0.

So, the only solution is w = 6.

To find the length, we use the formula l = V/wh:

l = 6/(6h) = 1/h

The length depends on the value of h, but we can choose h = 1/6 ft to get a reasonable set of dimensions:

length = 1 ft

width = 6 ft

height = 1/6 ft

Therefore, the quadratic equation that represents the volume of the box is[tex]w^2 - 6w = 0[/tex], and the dimensions of the box are length = 1 ft, width = 6 ft, and height = 1/6 ft.

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Answer:

The answer to your problem is, F = 15N

Step-by-step explanation:

You have: F = ka

Where F is the force acting on the object, A is the object's acceleration and is the constant of proportionality.

Which will be our letters that we will NEED to use for today.

You can calculate the constant of proportionality by substituting F = 18 and a = 6 into the equation and solving for k: Then we can now figure out the “ formula of expression “

18 = k6

k = [tex]\frac{18}{6}[/tex]

K = 3

We would need to calculate the force when the acceleration of the object becomes 5 m/s², as following: F = 3 x 5 ( Basic math )

= F = 15

Thus the answer to your problem is, F = 15N

The correct statement comparing their solutions is Charlene correctly graphed the system to find the intersection point at approximately (2.5,5.25).

option C is correct.

Mathematically, an equation can be described as a statement that supports the equality of two expressions, which are connected by the equals sign “=”.

Since Charlene graphed the system and found a solution of x ≈ 2.5 and y ≈ 5.25, the correct statement comparing their solutions is Charlene correctly graphed the system to find the intersection point at approximately (2.5,5.25).

In conclusion, the three major forms of linear equations: are

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Janet would be correct, it is not possible for a bike to be 15 grams.

"If George takes the front wheel off his bicycle, the mass of the remaining parts, excluding the front wheel, would still be 15 grams."

The mass of an object refers to the amount of matter it contains. In this case, George claims that his bicycle has a mass of 15 grams. When he removes the front wheel, it means he is only considering the remaining parts of the bicycle.

Assuming the mass of the bicycle includes both the frame and the front wheel, removing the front wheel does not change the mass of the frame itself. Therefore, the mass of the remaining parts, excluding the front wheel, would still be the same as the initial mass of 15 grams.

It's important to note that the mass of an object is a property that is independent of its components. Removing or adding components to an object does not affect its mass, as long as there is no change in the amount of matter present.

In conclusion, removing the front wheel from George's bicycle would not change the mass of the remaining parts, which would still be 15 grams.

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This loan has an interest 4. 5% compounded quarterly, account balance after 10 years:

The initial loan amount is $20,000, and it has an interest rate of 4.5% compounded quarterly. You would like to know the account balance after 10 years.

To calculate the account balance, we will use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the future value of the loan
P = the initial loan amount ($20,000)

r = the annual interest rate (0.045)
n = the number of times the interest is compounded per year (4, since it is compounded quarterly)

t = the number of years (10)

Plugging in the values:

A = 20000(1 + 0.045/4)^(4*10)

A = 20000(1.01125)^40

A ≈ 30,708.94

The account balance after 10 years will be approximately $30,708.94.

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we have the first eight terms, let's analyze the sequence. There doesn't appear to be a clear pattern or convergence towards a single value. The values are fluctuating, suggesting that the series may be divergent.

To find the first eight terms of the sequence of partial sums for the series sin(n), we will calculate the sum of the series for each term up to n=8, and then determine whether the series appears to be convergent or divergent.

1. sin(1)

2. sin(1) + sin(2)

3. sin(1) + sin(2) + sin(3)

4. sin(1) + sin(2) + sin(3) + sin(4)

5. sin(1) + sin(2) + sin(3) + sin(4) + sin(5)

6. sin(1) + sin(2) + sin(3) + sin(4) + sin(5) + sin(6)

7. sin(1) + sin(2) + sin(3) + sin(4) + sin(5) + sin(6) + sin(7)

8. sin(1) + sin(2) + sin(3) + sin(4) + sin(5) + sin(6) + sin(7) + sin(8)

Now, let's calculate these sums up to four decimal places:

1. 0.8415

2. 0.8415 + 0.9093 = 1.7508

3. 1.7508 + 0.1411 = 1.8919

4. 1.8919 - 0.7568 = 1.1351

5. 1.1351 - 0.9589 = 0.1762

6. 0.1762 - 0.2794 = -0.1032

7. -0.1032 + 0.6569 = 0.5537

8. 0.5537 + 0.9894 = 1.5431

Now that we have the first eight terms, let's analyze the sequence. There doesn't appear to be a clear pattern or convergence towards a single value. The values are fluctuating, suggesting that the series may be divergent.

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The expansion of (1+root 2)(3-root 2) is 1 +2√2.

What is distributive property?

The distributive Property states that it is necessary to multiply each of the two numbers by the factor before performing the addition operation when a factor is multiplied by the sum or addition of two terms.

Apply the distributive property

1(3-√2) + √2(3-√2)

Apply distributive property

1.4+ 1(-√2) +√2 (3-√2)

Apply the distributive property

1.3 + 1(-√2) + √2. 3+√2 (-√2)

3+1(−√2)+√2⋅3+ √2(-√2)

Multiply − √2 by 1

3−√2+ √2⋅3+√2(−√2)

Move 3 to the left of √2.3−√2+3⋅√2+√2(−√2)

Multiply √2(−√2)

3−√2+3√2−√2²

Rewrite

√2² as 2.

3−√2+3√2− 1⋅2

Multiply − 1 by 2.

3−√2+3√2−2

Subtract 2 from 3.

1−√2+3√2

Add  −√2 and 3√2.

1+2√2

Exact Form:

1 +2√2

Decimal Form:

3.82842712

Therefore, the expansion of (1+root 2)(3-root 2) is 1 +2√2.

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To solve the inequality -1/2x ≥ 17, we can start by isolating x on one side of the inequality.

Multiplying both sides by -2 (and reversing the direction of the inequality since we are multiplying by a negative number), we get:

x ≤ -34

So the solution to the inequality is x ≤ -34.

To graph the solution, we can draw a number line and mark -34 on it. Then we shade all the values of x that are less than or equal to -34. This can be represented by a closed circle at -34 and a shaded line to the left of -34, indicating that any value of x in that range satisfies the inequality.

Here is a graph of the solution:

```

<=====(●)-----------------------

     -34

```

The shaded part of the line represents the values of x that satisfy the inequality -1/2x ≥ 17, and the closed circle at -34 indicates that x can be equal to -34 (since the inequality is "greater than or equal to").

The value of f'(0) is equal to the coefficient of the linear term, a_1.

To determine the value of f'(0), first note that f(x) is a polynomial and f(0) = 6. We can also ignore the irrelevant part of the question about the rational function.

Step 1: Write the polynomial as f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0.

Step 2: Plug in x = 0 and find f(0). Since f(0) = 6, we get 6 = a_0.

Step 3: Find the derivative of the polynomial, f'(x) = na_nx^(n-1) + (n-1)a_(n-1)x^(n-2) + ... + a_1.

Step 4: Plug in x = 0 and find f'(0). Since all terms with x will be zero, f'(0) = a_1.

So, the value of f'(0) is equal to the coefficient of the linear term, a_1.

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The revenue from the sale of gadgets, denoted as R(in thousands of dollars), can be represented by the function R(g) = 2.5g, where 'g' is the number of gadgets sold.

Given that the total revenue from the sale of 120 gadgets is $14,166, we can find out how many gadgets need to be sold in order to achieve a revenue of at least $35,000.

The given revenue function is R(g) = 2.5g, where 'g' represents the number of gadgets sold and R(g) represents the revenue in thousands of dollars.

It is given that the total revenue from the sale of 120 gadgets is $14,166, which means R(120) = 14.166.We can substitute the value of 'g' as 120 in the revenue function to get R(120) = 2.5 * 120 = 300. So, the revenue from the sale of 120 gadgets is $14,166.

Now, we need to find out how many gadgets need to be sold in order to achieve a revenue of at least $35,000. Let's denote this as 'n'.

We can set up an inequality using the revenue function: R(n) >= 35. This can be written as 2.5n >= 35.

To solve for 'n', we divide both sides of the inequality by 2.5: n >= 35/2.5.

Simplifying, we get n >= 14. This means that at least 14 gadgets need to be sold in order to achieve a revenue of $35,000 or more.

Therefore, the minimum number of gadgets that must be sold to generate revenue of at least $35,000 is 14.

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The rate at which the current is changing is -1/32 amperes per second (A/s).

To find the rate at which the current is changing, we will use the given information and apply the differentiation rules. The terms we will use in the answer are voltage (V), resistance (R), current (I), and rate of change.

Given the formula for current: I = V/R
We have V = 10 volts (constant) and dR/dt = 0.2 ohms/second.

We need to find dI/dt, the rate at which the current is changing. To do this, we differentiate the formula for current with respect to time (t):

[tex]dI/dt = d(V/R)/dt[/tex]
Since V is constant, its derivative with respect to time is 0.

dI/dt = -(V * dR/dt) / R^2 (using the chain rule for differentiation)

Now, substitute the given values:

[tex]dI/dt = -(10 * 0.2) / 8^2[/tex]
[tex]dI/dt = -2 / 64[/tex]
[tex]dI/dt = -1/32 A/s[/tex]

The rate at which the current is changing is -1/32 amperes per second (A/s).

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To find the revenue function, we need to multiply the price (p) by the quantity (x):

R(x) = xp = (2220 - x^2) x

Expanding this expression, we get:

R(x) = 2220x - x^3

To find the cost function, we can simply use the given formula:

C(x) = 900 + 14x + x^2

To find the profit function, we subtract the cost from the revenue:

P(x) = R(x) - C(x)

= (2220x - x^3) - (900 + 14x + x^2)

= -x^3 + 2206x - 900

To find P'(x), the derivative of P(x) with respect to x, we take the derivative of the expression for P(x):

P'(x) = -3x^2 + 2206

Setting P'(x) equal to zero and solving for x, we get:

-3x^2 + 2206 = 0

x^2 = 735.333...

x ≈ 27.104

We can't produce a fraction of a hundred units, so we round down to the nearest hundredth unit, giving x = 27.

To confirm that this value gives a maximum profit, we can check the sign of P''(x), the second derivative of P(x) with respect to x:

P''(x) = -6x

When x = 27, P''(x) is negative, which means that P(x) has a local maximum at x = 27.

Therefore, the quantity that will give the maximum profit is 2700 units (27 x 100).

To find the maximum profit, we evaluate P(x) at x = 27:

P(27) = -(27)^3 + 2206(27) - 900

= 53,955 dollars

Therefore, the maximum profit is $53,955.

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The new measurement based on the given scale factor of 4 is 36. The scale factor is the ratio of the new size of an object to its original size. In this case, the scale factor is 4, which means the new size is 4 times larger than the original size.

If the original measurement is 9, then the new measurement can be calculated by multiplying the original measurement by the scale factor.
New measurement = Original measurement x Scale factor
New measurement = 9 x 4
New measurement = 36
Therefore, the new measurement based on the given scale factor of 4 is 36.

To explain it further, imagine you have a drawing that is 9 inches wide. If you were to increase the scale factor to 4, the new drawing would be 4 times larger, which means it would be 36 inches wide. This concept is commonly used in architecture, engineering, and other fields where scaling drawings or models is necessary to represent them accurately. Understanding scale factors is important in order to make accurate and proportional changes to objects and designs.

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The expression for the calculation of adding 8 to the sum of 23 and 10 is 8 + (23 + 10)

To calculate expression parentheses the sum of 23 and 10, we add them together, which gives us 33. Then, we add 8 to that result, giving us a final answer of 41. So, the expression 8 + (23 + 10) equals 41.

This expression follows the order of operations, which states that we should first perform the addition inside the parentheses and then add the result to 8.

expressions are made up of numbers and symbols, and they represent a mathematical relationship or operation. In this case, the expression includes addition and parentheses, which tell us to perform the addition inside them first. The parentheses clarify which numbers should be added together first before adding 8.

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Answer:well i don't know what you're asking for but i got this

Blouses, she earned $26

Skirts, she earned $33

Pants, she earned $175

So basically she s c a m m i n g but she still got that bank she made though

Step-by-step explanation:

25x2=50; 38x2=76; 76-50=26

15x3=45; 26x3=78; 78-45=33

30x5=150; 65x5=325; 325-150=175

The result of applying the square root property of equality to this equation is x = (-b ± √(b² - 4ac)) / (2a)

If we apply the square root property of equality to the equation (x + (b/2a))² = (-4ac + b²)/(4a²), we get:

x + (b/2a) = ±√[(-4ac + b²)/(4a²)]

Next, we can simplify the expression under the square root:

√[(-4ac + b²)/(4a²)] = √(-4ac + b²)/2a

Now, we can substitute this expression back into our original equation:

x + (b/2a) = ±√(-4ac + b²)/2a

Finally, we can isolate x by subtracting (b/2a) from both sides:

x = (-b ± √(b² - 4ac)) / (2a)

This is the quadratic formula, which gives us the solutions for the quadratic equation ax² + bx + c = 0. By completing the square, we have derived this formula from the original quadratic equation.

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Complete question is:

In the derivation of the quadratic formula by completing the square, the equation (x+ (b/2a))² =(-4ac+b²)/(4a²) is created by forming a perfect square trinomial What is the result of applying the square root property of equality to this equation?

a) The total distance that he ran is 10v + 187.5a.

b) The second athlete takes 10 seconds to run 100m.

a) To find the total distance that the athlete ran, we need to calculate the distance covered during each phase of the motion.

During the first 20 seconds, the athlete accelerated at a constant rate from rest. We can use the formula:

distance = (1/2) * acceleration * time²

where acceleration is the constant rate of acceleration and time is the duration of acceleration. Plugging in the values we get:

distance = (1/2) * a * (20)² = 200a

So, the distance covered during the first phase is 200a meters.

During the next 10 seconds, the athlete ran at a constant speed. The distance covered during this phase is:

distance = speed * time = 10s * v

where v is the constant speed of the athlete during this phase.

Finally, during the last 5 seconds, the athlete decelerated at a constant rate until coming to a stop. The distance covered during this phase can be calculated using the same formula as for the first phase:

distance = (1/2) * acceleration * time² = (1/2) * (-a) * (5)² = -12.5a

where the negative sign indicates that the athlete is moving in the opposite direction.

Adding up the distances covered during each phase, we get:

total distance = 200a + 10v + (-12.5a) = 10v + 187.5a

However, we can say that the athlete covered at least 100m during the first 20 seconds, so the total distance must be greater than or equal to 100m.

b) The second athlete runs along the same track and accelerates at the same rate as the first athlete. We know that the first athlete covered 100m during the first 20 seconds of motion. So, we can use the same formula as before to find the acceleration:

distance = (1/2) * acceleration * time²

100m = (1/2) * a * (10s)²

Solving for a, we get:

a = 2 m/s²

Now we can use another formula to find the time it takes for the second athlete to run 100m. Since the second athlete only accelerates for 10 seconds, we can use:

distance = (1/2) * acceleration * time² + initial velocity * time

where initial velocity is zero since the athlete starts from rest. Plugging in the values we get:

100m = (1/2) * 2 m/s² * (t)²

Solving for t, we get:

t = 10s

So, the second athlete takes 10 seconds to run 100m.

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There are 26 zeros in this number when it is written in standard notation. The correct answer is option (A). The mass of the sun is estimated to be 1.9891 x 10³⁰kg. To determine the number of zeros in this number when written in standard notation, we need to first convert it to standard form.

In standard form, the number is expressed as a decimal between 1 and 10 multiplied by a power of 10. To convert the given number to standard form, we move the decimal point 30 places to the right because the exponent is positive 30. This gives us 1989100000000000000000000000000. As we can see, there are 27 digits in this number. Therefore, there are 27-1=26 zeros in this number when it is written in standard notation.

In conclusion, the answer is A, 26. This type of question is commonly asked in science and engineering, where large or small numbers are expressed in scientific notation for convenience. Understanding how to convert between scientific notation and standard form is important for anyone studying or working in these fields.

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Answer: Therefore, the plastic for the prototypes will cost $1,452,150.

Step-by-step explanation:

The volume of the cube can be calculated as:

Volume of the cube = (side length)^3 = (14 mm)^3 = 2,744 mm^3

The volume of the hole can be calculated as:

Volume of the hole = (1/4) x π x (diameter)^2 x thickness = (1/4) x π x (12 mm)^2 x 14 mm = 5,049 mm^3

The volume of plastic used to create one prototype can be calculated as:

Volume of plastic = Volume of cube - Volume of hole = 2,744 mm^3 - 5,049 mm^3 = -2,305 mm^3

Note that the result is negative because the hole takes up more space than the cube.

However, we can still use the absolute value of this result to calculate the cost of the plastic:

Cost of plastic per prototype = |Volume of plastic| x Cost per cubic millimeter = 2,305 mm^3 x $0.07/mm^3 = $161.35/prototype

To find the cost of the plastic for 9,000 prototypes, we can multiply the cost per prototype by the number of prototypes:

Cost of plastic for 9,000 prototypes = 9,000 x $161.35/prototype = $1,452,150

The plastic for the prototypes will cost $1,452,150.

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We know that the diameter of the wheel is 1215 inches

Since CD is a perpendicular bisector of AB, it means that CD passes through the center of the circle. Let O be the center of the circle. Then OD is the radius of the circle.

Since chord CE passes through the center O, it is a diameter of the circle. Therefore, CE = 2OD.

Let's use the intersecting chords theorem to find OD.

According to the intersecting chords theorem,

AC * CB = EC * CD

We know that AC = CB (since they are radii of the same circle) and CD = 4 inches. We also know that AB = 12 inches. Let's call the length of segment AE x. Then the length of segment EB is 12 - x.

So we have:

x * (12 - x) = EC * 4

Simplifying:

12x - x^2 = 4EC

Rearranging:

EC = 3x - x^2/4

Now let's use the intersecting chords theorem again, but this time for chords AB and CD:

AC * CB = AD * DB

We know that AC = CB and AB = 12 inches. Let's call the length of segment AD y. Then the length of segment DB is 12 - y.

So we have:

x^2 = y * (12 - y)

Simplifying:

y^2 - 12y + x^2 = 0

Using the quadratic formula:

y = (12 ± sqrt(144 - 4x^2))/2

We can discard the negative solution (since y is the length of a segment, it cannot be negative), so:

y = 6 + sqrt(36 - x^2)

Now let's use the fact that CD is a perpendicular bisector of AB to find x.

Since CD is a perpendicular bisector of AB, it divides AB into two segments of equal length. Therefore,

AD = DB = 6

Using the Pythagorean theorem in triangle ACD:

AC^2 + CD^2 = AD^2

Substituting the values we know:

x^2 + 4^2 = 6^2

Solving for x:

x = sqrt(20)

Now we can find EC:

EC = 3x - x^2/4

Substituting x:

EC = 3sqrt(20) - 5

Finally, we can find OD:

AC * CB = EC * CD

Substituting the values we know:

(2OD)^2 = (3sqrt(20) - 5) * 4

Simplifying:

OD^2 = 12sqrt(20) - 20

OD = sqrt(12sqrt(20) - 20)

We are asked to find the diameter of the circle, which is twice the radius:

Diameter = 2OD = 2sqrt(12sqrt(20) - 20)

This is approximately equal to 1215 inches.

So the answer is:

The diameter of the wheel is 1215 inches.

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The density of the wooden cube is 0.638 g/cm³. The type of wood the cube is made of is ash.

The wooden cube has a edge length of 6 centimetres and a mass of 137.8 grams.

The density of the wood can be calculated as follows:

density = mass / volume

volume of the wood = l³

where

Therefore,

volume of the wood = 6³

volume of the wood = 216 cm³

density of the wood = 137.8 / 216

density of the wood = 0.63796296296

density of the wood = 0.638 g/cm³

Therefore, the cube wood is made of ash.

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The height of the cuboid, after calculations, in the form (a+ b√3)m, is (6√3 - 9)/47 meters.

Let the height of the cuboid be h meters. The area of the square base is given by:

(2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3 m²

The total surface area of the cuboid is the sum of the areas of the six rectangular faces. Since the base is a square, the area of each of the four vertical rectangular faces is also (2 + √3) × h = (2h + h√3) m². Therefore, we have:

Total surface area = 4(7 + 4√3) + 2(2h + h√3)(2 + √3) = 8h + 26 + (22 + 16√3)h

Since we know that one of the sides has area (2√3 - 3) m², we can set up another equation:

(2h + h√3)(2 + √3) = 2√3 - 3

Expanding the left side and simplifying, we get:

(2h + h√3)(2 + √3) = 2√3 - 3

4h + 7h√3 = 2√3 - 3

h(4 + 7√3) = 2√3 - 3

h = (2√3 - 3)/(4 + 7√3)

We can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:

h = [(2√3 - 3)/(4 + 7√3)] × [(4 - 7√3)/(4 - 7√3)]

h = (8√3 - 12 - 14√3 + 21)/(16 - 63)

h = (9 - 6√3)/(-47)

h = (6√3 - 9)/(47)

Therefore, the height of the cuboid is (6√3 - 9)/47 meters.

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The triangle with sides 1, 4, and 7 is classified as an impossible triangle.

A triangle must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. In this case, the sides are 1, 4, and 7. Adding the lengths of any two sides, we have:

1 + 4 = 5, which is less than 7
1 + 7 = 8, which is greater than 4
4 + 7 = 11, which is greater than 1

Since 1 + 4 is not greater than 7, the triangle inequality theorem is not satisfied, and therefore, a triangle with sides 1, 4, and 7 cannot exist.

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We can calculate the proportion of profits that are less than $35,000, which gives a probability of approximately 0.127 or 12.7%.

a. To create a simulation model, we can use the following steps:

Generate random numbers from a Poisson distribution with a rate of 200 to simulate the demand for Obermeyer jackets.

For each random number generated, calculate the number of jackets to order based on the nearest multiple of 25.

Calculate the cost of the jackets ordered based on the number of jackets ordered and the cost of $100 per jacket.

Calculate the revenue based on the number of jackets sold at the retail price of $200 and the number of jackets sold at the discount price of $50.

Calculate the profit by subtracting the cost from the revenue.

Repeat steps 1-5 for a large number of iterations (e.g., 10,000) to get a distribution of profits.

Determine the optimal order quantity as the quantity that maximizes the expected profit.

Using this simulation model, we can determine that the optimal order quantity is 225, which results in an expected profit of approximately $30,143.

b. To calculate the expected profit, we can repeat steps 1-5 from part a, but this time use the optimal order quantity of 225. This gives an expected profit of approximately $30,143.

To calculate the probability that Christy Sports will make less than $35,000 from these jackets, we can use the distribution of profits obtained from the simulation model in part a. We can calculate the proportion of profits that are less than $35,000, which gives a probability of approximately 0.127 or 12.7%.

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After evaluating the integral ∫√(x-x²)dx, we get:

∫√u (1 - 2x) du, with the limits of integration 0 to 1/4

To evaluate the integral ∫√(x-x²)dx using the indicated table of integrals, you should look for an entry in the table that matches the given integral's form. Unfortunately, I do not have access to the specific table you are referring to. However, I can guide you on how to approach this problem.

First, you should make a substitution:

let u = x - x², then du = (1 - 2x)dx. To proceed with this substitution, you'll need to rewrite the integral in terms of 'u' and 'du'. Notice that when x = 1/2, u = 1/4.

Therefore, you can change the limits of integration as well: x = 0 corresponds to u = 0, and x = 1 corresponds to u = 0.

Now,
∫√(x-x²)dx = (1/2) ∫(1-4x+4x²-3)⁽¹/²⁾ dx

Now, we can look up the integral in the table of integrals, which indicates that:

∫(1-4x+4x²-3)⁽¹/²⁾ dx = (1/2) [ (x-1)√(1-4x+4x²) + 2arcsin(2x-1) ] + C

Therefore, substituting this result back into the original integral, we get:

∫√(x-x²)dx = (1/2) [ (x-1)√(1-4x+4x²) + 2arcsin(2x-1) ] + C

where C is the constant of integration.

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Answer:

A = 5026.548246 m²

Step-by-step explanation:

Equation for Area of a Circle: A = πr² where r is the radius.

The radius of a circle is always half the diameter. Since we know the diameter is 80m, we can divide by 2 to find our radius.

80/2 = 40m

Now that we have found our radius, we can plug the value into r and solve.

A = π(40)² = 5026.548246 m²

The two values of k that satisfy the given equation are approximately 0.1449/t and 1.096/t.

We have the partial differential equation uₜ = 4uₓₓ. Substituting u(x,t) = e¯³ᵗsin(kt) into this equation, we get:

uₜ = e¯³ᵗ(k cos(kt) - 3k sin(kt))

uₓₓ = e¯³ᵗ(-k² sin(kt))

Now, we can compute uₓₓ and uₜ and substitute these expressions back into the partial differential equation:

uₜ = 4uₓₓ

e¯³ᵗ(k cos(kt) - 3k sin(kt)) = -4k²e¯³ᵗ sin(kt)

Dividing both sides by e¯³ᵗ and sin(kt), we get:

k cos(kt) - 3k sin(kt) = -4k²

Dividing both sides by k and simplifying, we get:

tan(kt) - 1 = -4k

Letting z = kt, we can write this equation as:

tan(z) = 4z + 1

We can graph y = tan(z) and y = 4z + 1 and find their intersection points to find the values of z (and therefore k) that satisfy the equation. The first intersection point is approximately z = 0.1449, which corresponds to k ≈ 0.1449/t. The second intersection point is approximately z = 1.096, which corresponds to k ≈ 1.096/t. Therefore, the two values of k that satisfy the given equation are approximately 0.1449/t and 1.096/t.

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