Question

reba makes 3d - printed toys and sells them online. she predicts that if she sells each toy for d dollars, then she will earn a total profit of $60d - 4d^{2}$ dollars. how much should reba sell each toy for in order to maximize her profit? $\square$

Explanation:

Step1: Identify the profit function

The profit function is given by \( P(d) = 60d - 4d^2 \), which is a quadratic function in the form \( ax^2 + bx + c \) (here \( x = d \), \( a = -4 \), \( b = 60 \), \( c = 0 \)).

Step2: Find the vertex of the parabola

For a quadratic function \( y = ax^2 + bx + c \), the \( x \)-coordinate of the vertex (which gives the maximum or minimum value) is given by \( x = -\frac{b}{2a} \). Since \( a = -4 < 0 \), the parabola opens downward, so the vertex is a maximum point.

Substitute \( a = -4 \) and \( b = 60 \) into the formula:
\( d = -\frac{60}{2(-4)} \)

Step3: Calculate the value of \( d \)

Simplify the expression:
\( d = -\frac{60}{-8} = \frac{60}{8} = 7.5 \)

Answer:

\( 7.5 \)