Question

solve the problem.
find the number of units that must be produced and sold in order to yield the maximum profit, given the following equations for revenue and cost:
$r(x) = 40x - 0.5x^2$
$c(x) = 7x + 3$.
\bigcirc 33 units
\bigcirc 36 units
\bigcirc 34 units
\bigcirc 47 units

Explanation:

Step1: Recall Profit Formula

Profit \( P(x) = R(x) - C(x) \). Substitute \( R(x) = 40x - 0.5x^2 \) and \( C(x) = 7x + 3 \).
\( P(x) = (40x - 0.5x^2) - (7x + 3) = -0.5x^2 + 33x - 3 \).

Step2: Find Vertex of Quadratic

For \( ax^2 + bx + c \), vertex \( x = -\frac{b}{2a} \). Here, \( a = -0.5 \), \( b = 33 \).
\( x = -\frac{33}{2(-0.5)} = \frac{33}{1} = 33 \).

Answer:

33 units