Question

The cost, in cents, to produce x cups of Mountain Thunder Lemonade at Junior's Lemonade Stand is C(x)=18x+240,x≥0 and the price-demand function, in cents per cup, is p(x)=90-3x,0≤x≤30. Find the maximum profit.

Answer 1

$192

Explanation:

The cost function is given as:

C(x)=18x+240

The price function is given as:

p(x)= 90 - 3x

The revenue R(x) is the product of the price and the number of products. It is given by:

R(x) = xp(x) = x(90 - 3x) = 90x - 3x²

The profit P(x) is the difference between the revenue and the cost of production. Therefore:

P(x) = R(x) - C(x) = 90x - 3x² - (18x + 240) = 90x - 3x² - 18x - 240

P(x) = -3x² + 72x - 240

The standard equation of a quadratic equation is ax² + bx + c. The function has a maximum value at x = -b/2a

Since P(x) = -3x² + 72x - 240, the maximum profit is at:

x = -72/2(-3) = 12

at x = 12, the profit is:

P(12) = -3(12)² + 72(12) - 240 = -432 + 864 - 240 = $192