Question

Question 15 A)The function P(L) = −4L2+12L +25P(L) = −4L2+12L +25 gives the profit in thousands of producing LL units of lip gloss. What is the maximum profit that can be made? (1 point) Responses 34,000 34,000 25,000 25,000 68,000 68,000 4,000 4,000

Answer 1

To find the maximum profit from the function \( P(L) = -4L^2 + 12L + 25 \), we need to determine the vertex of this quadratic equation. The vertex of a parabola given by \( y = ax^2 + bx + c \) occurs at \( L = -\frac{b}{2a} \).

In this case, the coefficients are:
- \( a = -4 \)
- \( b = 12 \)

Calculating the \( L \) value at the vertex:

\[
L = -\frac{12}{2 \times -4} = -\frac{12}{-8} = \frac{3}{2} = 1.5
\]

Now we substitute this value back into the profit function \( P(L) \) to find the maximum profit:

\[
P(1.5) = -4(1.5)^2 + 12(1.5) + 25
\]

Calculating \( (1.5)^2 \):

\[
(1.5)^2 = 2.25
\]

Now substituting back into the profit function:

\[
P(1.5) = -4(2.25) + 12(1.5) + 25
\]

Calculating each term:

- \( -4(2.25) = -9 \)
- \( 12(1.5) = 18 \)

So,

\[
P(1.5) = -9 + 18 + 25 = 34
\]

Since the profit function \( P(L) \) is given in thousands, the maximum profit is:

\[
34,000
\]

Thus, the maximum profit that can be made is **34,000**.