suppose the sunglasses hut company has a profit function given by $p(q)=-0.01q^{2}+5q - 40$, where $q$ is…

suppose the sunglasses hut company has a profit function given by $p(q)=-0.01q^{2}+5q - 40$, 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: \nthousand 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: \nthousand dollars of maximum profits can be expected.\nquestion help: message instructor\nsubmit question\nquestion 10\nthis is a question where you will do the work on paper and upload a picture in the add work box. follow the directions below. show as much work as possible.\na box with a square base and no top is to be made from a square piece of cardboard by cutting 4 in. squares from each corner and folding up the sides. the box is to hold 8464 in³. how big a piece of cardboard is needed?\nquestion help: video message instructor\nadd work

suppose the sunglasses hut company has a profit function given by $p(q)=-0.01q^{2}+5q - 40$, 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: \nthousand 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: \nthousand dollars of maximum profits can be expected.\nquestion help: message instructor\nsubmit question\nquestion 10\nthis is a question where you will do the work on paper and upload a picture in the add work box. follow the directions below. show as much work as possible.\na box with a square base and no top is to be made from a square piece of cardboard by cutting 4 in. squares from each corner and folding up the sides. the box is to hold 8464 in³. how big a piece of cardboard is needed?\nquestion help: video message instructor\nadd work

Answer

Explanation:

Step1: Identify the profit - function form

The profit function is $P(q)=-0.01q^{2}+5q - 40$, which is a quadratic function in the form $y = ax^{2}+bx + c$ where $a=-0.01$, $b = 5$, and $c=-40$.

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

For a quadratic function $y = ax^{2}+bx + c$, the vertex (which gives the maximum or minimum) occurs at $q=-\frac{b}{2a}$. 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 $P(q)=-0.01q^{2}+5q - 40$. $P(250)=-0.01\times(250)^{2}+5\times250 - 40$. First, calculate $-0.01\times(250)^{2}=-0.01\times62500=-625$. Then, $5\times250 = 1250$. So, $P(250)=-625+1250 - 40=585$.

Answer:

A. 250.000 B. 585.000