suppose the sunglasses hut company has a profit function given by $p(q)=-0.01q^{2}+5q - 46$, where $q$ is…

suppose the sunglasses hut company has a profit function given by $p(q)=-0.01q^{2}+5q - 46$, where $q$ is the number of thousands of pairs of sunglasses sold and produced, and $p(q)$ is the total profit, in thousands of dollars, from selling and producing $q$ pairs of sunglasses.\na) how many pairs of sunglasses (in thousands) should be sold to maximize profits? (if necessary, round your answer to three decimal places.)\nanswer: 250.00 thousand pairs of sunglasses need to be sold.\nb) what are the actual maximum profits (in thousands) that can be expected? (if necessary, round your answer to three decimal places.)\nanswer: 585.00 thousand dollars of maximum profits can be expected.\nquestion help: message instructor\nsubmit question

suppose the sunglasses hut company has a profit function given by $p(q)=-0.01q^{2}+5q - 46$, where $q$ is the number of thousands of pairs of sunglasses sold and produced, and $p(q)$ is the total profit, in thousands of dollars, from selling and producing $q$ pairs of sunglasses.\na) how many pairs of sunglasses (in thousands) should be sold to maximize profits? (if necessary, round your answer to three decimal places.)\nanswer: 250.00 thousand pairs of sunglasses need to be sold.\nb) what are the actual maximum profits (in thousands) that can be expected? (if necessary, round your answer to three decimal places.)\nanswer: 585.00 thousand dollars of maximum profits can be expected.\nquestion help: message instructor\nsubmit question

Answer

Explanation:

Step1: Identify the profit - function form

The profit function $P(q)=-0.01q^{2}+5q - 46$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a=-0.01$, $b = 5$, and $c=-46$.

Step2: Find the value of $q$ for maximum profit

For a quadratic function $y = ax^{2}+bx + c$, the $x$ - value (in our case $q$) of the vertex (which gives the maximum or minimum) is $q=-\frac{b}{2a}$. Substituting $a=-0.01$ and $b = 5$ into the formula, we get $q=-\frac{5}{2\times(-0.01)}=\frac{5}{0.02}=250$.

Step3: Calculate the maximum profit

Substitute $q = 250$ into the profit - function $P(q)=-0.01q^{2}+5q - 46$. $P(250)=-0.01\times(250)^{2}+5\times250 - 46$. First, calculate $-0.01\times(250)^{2}=-0.01\times62500=-625$. Then, calculate $5\times250 = 1250$. So, $P(250)=-625+1250 - 46=579$.

Answer:

A. 250.000 thousand pairs of sunglasses need to be sold. B. 579.000 thousand dollars of maximum profits can be expected.