SOLUTION: To maximize yearly revenue, a company wishes to determine the best-selling price for their new line of shirts. They realize that if the price is too high or too low, they will not

Question 1110688: To maximize yearly revenue, a company wishes to determine the best-selling price for their new line of shirts. They realize that if the price is too high or too low, they will not maximize profit. The equation y=-16(x-30)^2+10 000 models the scenario, where x represents the selling price per shirt and y represents the number of shirts sold
Determine the selling price that maximizes the revenue

Determine the maximum number of shirts sold

What is the maximum revenue

What are the x-intercepts for the function?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the equation is y = -16 * (x-30)^2 + 10,000

simplify to get y = -16 * (x^2 - 60x + 900) + 10,000

simplify further to get y = -16x^2 + 960x - 14,400 + 10,000

combine like terms to get y = -16x^2 + 960x - 4,400

set y = 0 to get -16x^2 + 960x - 4,400 = 0

since this is in standard quadratic form, then:

a = coefficient of x^2 term = -16
b = coefficient of x term = 960
c = constant term = -4,400

the maximum value is when x = -b/2a.

solve for x to get x = -960 / -32 = 30

when x = 30, the original equation becomes y = -16 * (30-30) + 10,000

solve for y to get y = 10,000

looks like the maximum number of shirts to be sold is 10,000 when the price per shirt is equal to 30.

here's the graph of the original equation.

$$$

the maximum revenue will be 10,000 * 30 = 300,000.

the x intercepts for the function will be when the value of y is equal to 0.

the original equation becomes -16(x-30)^2 + 10,000 = 0

subtract 10,000 from both sides to get -16(x-30)^2 = -10,000

divide both sides by -16 to get (x-30)^2 = -10,000 / -16

take the square root of both sides to get x-30 = plus or minus sqrt(10,000/16)

simplify to get x = 30 = plus or minus 25.

solve for x to get x = 5 or 55.

y = 0 when x = 5 or when x = 55.

the graph shows that as well.

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