To determine both the number of pairs of sunglasses that should be sold to maximize profits and the actual maximum profits, we need to analyze the given profit function. The profit function provided is:
[tex]\[ P(q) = -0.02q^2 + 5q - 46 \][/tex]
This is a quadratic function of the form [tex]\( P(q) = aq^2 + bq + c \)[/tex], where [tex]\( a = -0.02 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -46 \)[/tex]. The graph of this quadratic function is a parabola that opens downward (since [tex]\( a \)[/tex] is negative), indicating it has a maximum point at its vertex.
A) To find the value of [tex]\( q \)[/tex] (the number of thousands of pairs of sunglasses) that maximizes profit, we calculate the vertex of the parabola. The formula to find the vertex [tex]\( q \)[/tex] of a parabola [tex]\( ax^2 + bx + c \)[/tex] is:
[tex]\[ q = -\frac{b}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ q = -\frac{5}{2 \times -0.02} \][/tex]
Performing the calculation:
[tex]\[ q = -\frac{5}{-0.04} \][/tex]
[tex]\[ q = 125.0 \][/tex]
Therefore, [tex]\( 125.0 \)[/tex] thousand pairs of sunglasses should be sold to maximize profits.
Answer: [tex]\( 125.0 \)[/tex] thousand pairs of sunglasses need to be sold.
B) To determine the actual maximum profits that can be expected, we substitute [tex]\( q = 125.0 \)[/tex] back into the profit function [tex]\( P(q) \)[/tex]:
[tex]\[ P(125.0) = -0.02(125.0)^2 + 5(125.0) - 46 \][/tex]
Performing the calculation:
[tex]\[ P(125.0) = -0.02 \times 15625 + 625 - 46 \][/tex]
[tex]\[ P(125.0) = -312.5 + 625 - 46 \][/tex]
[tex]\[ P(125.0) = 266.5 \][/tex]
Therefore, the actual maximum profits that can be expected are [tex]\( 266.5 \)[/tex] thousand dollars.
Answer: [tex]\( 266.5 \)[/tex] thousand dollars of maximum profits can be expected.
