Gymnast Clothing manufactures expensive soccer cleats for sale to college bookstores in runs of up to 500. Its cost (in dollars) for a run of x pairs of cleats is

C(x) = 3000 + 9x + 0.1x2 (0 ≤ x ≤ 500).
Gymnast Clothing sells the cleats at $130 per pair. Find the revenue and profit functions. How many should Gymnast Clothing manufacture to make a profit?

Revenue:
Profit:

At Least ____pairs

I don't know how to find the Revenue and Profit function with the formula given, how do I determine it?

Thanks

Revenue is

R(x)=130x since it sells for $130 per pair
Profit function is
P(x)=R(x)-C(x)

To make a profit, P(x) must be positive.

To find the revenue function, we need to multiply the number of pairs of cleats (x) by the selling price per pair ($130). So, the revenue function (R) is given by:

R(x) = 130x

To find the profit function, we subtract the cost function (C) from the revenue function (R). So, the profit function (P) is given by:

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

Substituting the revenue function and the cost function, we get:

P(x) = 130x - (3000 + 9x + 0.1x^2)

Simplifying this expression, we have:

P(x) = 130x - 3000 - 9x - 0.1x^2

Combining like terms, we get:

P(x) = -0.1x^2 + 121x - 3000

Now, to determine the number of pairs Gymnast Clothing should manufacture to make a profit, we need to find the value of x for which the profit (P(x)) is greater than zero. This is because a positive profit represents making money or making a profit.

Therefore, we need to solve the inequality:

P(x) > 0

-0.1x^2 + 121x - 3000 > 0

To solve this inequality, we can either use factoring or the quadratic formula. However, since the quadratic equation does not factor nicely, we will use the quadratic formula. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = -0.1, b = 121, and c = -3000. Substituting these values into the quadratic formula, we get:

x = (-121 ± √(121^2 - 4(-0.1)(-3000))) / (2(-0.1))

Simplifying this expression, we have:

x = (-121 ± √(14641 - 1200)) / (-0.2)

x = (-121 ± √(13441)) / (-0.2)

Calculating the square root, we have:

x = (-121 ± 116) / (-0.2)

Now, evaluating both the positive and negative square root separately:

x₁ = (-121 + 116) / (-0.2) = -5 / (-0.2) = 25

x₂ = (-121 - 116) / (-0.2) = -237 / (-0.2) = 1185

Since the number of pairs cannot be negative, we discard the negative root. Therefore, Gymnast Clothing should manufacture at least 25 pairs of cleats to make a profit.

To summarize:

Revenue function: R(x) = 130x
Profit function: P(x) = -0.1x^2 + 121x - 3000
At least 25 pairs should be manufactured to make a profit.

To find the revenue function, we need to multiply the selling price by the number of pairs sold. In this case, the selling price is $130 per pair.

So, the revenue function (R) can be calculated by multiplying the selling price ($130) by the number of pairs (x):

R(x) = 130x

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

Profit = Revenue - Cost

Let's calculate each step separately:

1. Revenue Function:
R(x) = 130x

2. Cost Function:
C(x) = 3000 + 9x + 0.1x^2

3. Profit Function:
Profit(x) = R(x) - C(x)
= (130x) - (3000 + 9x + 0.1x^2)
= 130x - 3000 - 9x - 0.1x^2

Simplifying further, we get:
Profit(x) = -0.1x^2 + 121x - 3000

To determine how many pairs Gymnast Clothing should manufacture to make a profit, we need to find the x value where Profit(x) is positive. In other words, we need to find the x intercept where Profit(x) is greater than zero.

We can solve this by setting the Profit(x) equation equal to zero and finding its roots:
-0.1x^2 + 121x - 3000 = 0

Now, we can factor or use the quadratic formula to solve for x:

Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = -0.1, b = 121, and c = -3000. Plugging these values into the formula, we can calculate the two possible values for x.

Once we find the positive value for x, we will have the number of pairs Gymnast Clothing should manufacture to make a profit.