Final answer:
To calculate the smallest number of units for a $4.2 million profit, subtract the total cost from the total revenue to find the profit function, set it equal to $4.2 million, and solve for X.
Step-by-step explanation:
To find the smallest number of units the company must sell to make a profit of $4.2 million, we need to find the profit function by subtracting the total cost from the total revenue. After that, we set this profit function equal to $4,200,000 and solve for X.
The revenue function is R(X) = 100X - 0.0005X2 and the cost function is C(X) = 5X + 112,500. The profit function P(X) is the revenue function minus the cost function: P(X) = R(X) - C(X) = (100X - 0.0005X2) - (5X + 112,500).
To find when the profit equals $4.2 million, we set the profit function equal to 4,200,000:
P(X) = (100X - 0.0005X2) - (5X + 112,500) = 4,200,000.
Solving this quadratic equation, we can find the value of X that gives a profit of $4.2 million. The smallest integer solution from the options provided that would give the company a profit of at least $4.2 million can be determined by testing the options or solving the equation.