Need help solving this question PLEASE

Question

The per-unit price p (in dollars) of a popular toy when x units (in thousands) are produced is modeled by the function p = 12 - 0.025x. The revenue (in thousands of dollars) is the product of the price per unit and the number of units (in thousands) produced.

So the revenue function is given by r(x) = xp, or rather:

r(x) = x (12 - 0.025x)

According to this function, how many single units should be produced for the revenue to equal $440,000? There will be two answers, type the least answer into the box. Be careful with units, remember that x represents thousands of units and r(x) represents thousands of dollars.

Answer 1

The function is r(x) = x (12 - 0.025x) and we want to find x when r(x) = $440,000.

Graphically, this is two functions, y = x (12 - 0.025x) and y = 440 and we need to find where they intersect. The latter is a straight line, the former is a quadratic (or parabola) as it has an x² term.

To solve, we have 440 = x (12 - 0.025x)

=> 440 = 12x - 0.025x²

=> 0.025x² - 12x + 440 = 0

=> ¹/₄₀ (x² - 480x + 17600) = 0

I know 17600 = 440 x 40 which looks like 480, so it factorizes in this case

=> ¹/₄₀ (x - 40) (x - 440) = 0

hence x = 40 or x = 440 (two values, as expected).

The least value is x = 40 or 40,000 units.

You can double-check with Desmos Graphing to see the two graphs above, or use a Quadratic Equation Solver to find the roots of the quadratic equation. Or, more simply, we can plug 40 into the price equation p = 12 - 0.025 (40) = 11$. 11 x 40,000 units = $440,000 revenue as required.