Question

The price that a company charged for a basketball hoop is given by the equation mc012-1.jpg where x is the number of hoops that are produced, in millions. It costs the company $30 to make each basketball hoop. The company recently reduced its production to 1 million hoops but maintained its profit of 15 million dollars. Approximately how many basketball hoops did the company previously produce to make the same profit?

Answer 1

$\sf{\underline{Correction:}}$

The price that a company charged for a basketball hoop is given by the equation $\boxed{\sf{50-5x^{2}}}$ where x is the number of hoops that are produced, in millions. It costs the company $30 to make each basketball hoop. The company recently reduced its production to 1 million hoops but maintained its profit of 15 million dollars. Approximately how many basketball hoops did the company previously produce to make the same profit?

$\sf{\underline{Answer:}}$ The number of basketball hoops did the company previously produce to make the same profit is 1.3 million hoops.

Let the number of basketball hoops be x.

$\boxed{\sf{Total \: revenue = Price \times Quantity}}$

The total revenue in million dollars is:

$\sf{\underline{Total\:revenue:}}$

$\boxed{\sf{(50 - {5x}^{2} )x}}$

$\sf{\underline{That\:is:}}$

$\boxed{\sf{50x - {5x}^{3}}}$

$\sf{\underline{The\:total\:cost\:in\:million\:dollars\:is:}}$

$\sf{Total \: cost = 1 \: unit \: cost \times Quantity}$

$\boxed{\sf{Total \: cost = 30x}}$

$\sf{\underline{And:}}$

$\sf{Profit = Total \: revenue - Total \: cost}$

$\sf{Profit = 50x - {5x}^{3} - 30x}$

$\boxed{\sf{Profit = 20x - {5x}^{3}}}$

$\sf{\underline{Note:}}$ The profit is 15 million.

Let's see how.

$\implies$ $\sf{15 = 20x - {5x}^{3}}$

$\implies$ $\sf{ {5x}^{3} - 20x + 15 = 0}$

$\implies$ $\sf{ {x}^{2} - 4x + 3 = 0}$

$\implies$ $\sf{(x - 1)( {x}^{2} + x - 3) = 0}$

$\sf{\underline{The\:value\:of\:x\:is:}}$

$\implies$ 1

$\implies$ $\frac{ - 1 \pm \sqrt{13} }{2} = 1.3$

The production is always positive.

$\sf{\underline{Therefore:}}$

The value of x is either 1 or 1.3.

Answer 2

The correct answer is A, or 1.3 million hoops

Just got it right on the edge 2020 test, hope this helps!! :)