Question

The revenue a company generates from selling x units of a product is given by the expression 100x – 0.0005x2. The cost for the company to produce x units of the product is given by the expression 5x + 112,500. Profit is revenue minus cost.

What is the smallest number of units that the company can sell to make a profit of $4.2 million?
A. 60,000
B. 75,000
C. 115,000
D. 140,000

Answer 1

To find the smallest number of units to make a profit of $4.2 million, set the profit equation to $4,200,000 and solve the resulting quadratic equation. Take the smallest positive integer solution that satisfies the required profit level.

Step-by-step explanation:

To determine the smallest number of units a company must sell to achieve a profit of $4.2 million, the profit equation is set equal to $4,200,000, where profit equals revenue minus cost. First calculate the profit function by combining the revenue and cost functions, resulting in the equation Profit(x) = 100x - 0.0005x2 - (5x + 112,500). Simplify this to get Profit(x) = 95x - 0.0005x2 - 112,500. Now set the profit function equal to $4,200,000 and solve for x:

95x - 0.0005x2 - 112,500 = 4,200,000.

After solving the quadratic equation, you will find two solutions for x, from which you take the smallest positive integer value that yields the required profit. The answer options indicate the number of units (x) must be an integer, so the aim is to find the smallest value among the given choices that satisfies the equation for the desired profit level.