Question

select the correct answer from each drop-down menu.
a store owner buys leather for his handmade shoes at a wholesale warehouse. he finds that when he sells each pair of shoes for $75, the average number of customers per week is 30 people. later, he finds that for each $5 decrease in the price per pair of shoes, the number of customers increases by an average of 5 people per week.
the weekly revenue from his sales is given by the polynomial below, where ( x ) is the number of $5 decreases in the price per pair of shoes over $75.
( r(x) = 2,250 + 225x - 25x^2 )
use the information given above to determine how to best complete the following sentences.
the constant of the polynomial represents the (\boldsymbol{
abla}) in the price per pair of shoes.
the binomial ( (30 + 5x) ) is a factor of polynomial ( r(x) ). does this binomial represent revenue, number of customers, or price per pair of shoes? does this binomial take into account the price decreases?
the binomial represents the average (\boldsymbol{
abla}) per week, (\boldsymbol{
abla}) the change due to price decreases.

Explanation:

First Drop - down (Constant of the Polynomial)

The revenue function is \(R(x)=2250 + 225x-25x^{2}\). When \(x = 0\) (which means there are no \(\$5\) price decreases), the price per pair of shoes is \(\$75\) and the number of customers is \(30\). The revenue when \(x = 0\) is \(R(0)=2250+225(0)-25(0)^{2}=2250\). This revenue is calculated as (price per pair)\(\times\) (number of customers) \(=75\times30 = 2250\). So the constant term represents the weekly revenue when there are no decreases (i.e., when the price is \(\$75\) and number of customers is \(30\)). But in terms of what it represents in the context of the first sentence "the constant of the polynomial represents the \(\underline{}\) in the price per pair of shoes" - wait, actually, when \(x = 0\), the price is \(75\) and revenue is \(75\times30=2250\). The constant term is the revenue when \(x = 0\) (no price decreases). But maybe the first sentence is about the revenue when there are no price decreases (since revenue \(=\) price \(\times\) number of customers). So the constant term \(2250\) is the weekly revenue when the price per pair of shoes is \(\$75\) (no \(\$5\) decreases), so it represents the weekly revenue with no decreases in the price per pair of shoes.

Second and Third Drop - downs (Binomial \((30 + 5x)\))

The number of customers starts at \(30\) and for each \(x\) (number of \(\$5\) decreases), the number of customers increases by \(5\) per decrease. So the number of customers as a function of \(x\) is \(30 + 5x\). The revenue function is \(R(x)=\text{(price per pair)}\times\text{(number of customers)}\). The price per pair when there are \(x\) \(\$5\) decreases is \(75-5x\), and the number of customers is \(30 + 5x\). So the binomial \((30 + 5x)\) represents the number of customers per week. And it does take into account the change due to price decreases (because the \(5x\) term is the increase in customers due to \(x\) price decreases of \(\$5\) each).

Final Answers for the Drop - downs:

1. The constant of the polynomial represents the \(\boldsymbol{\text{weekly revenue with no decreases}}\) in the price per pair of shoes.
2. The binomial \((30 + 5x)\) represents the average \(\boldsymbol{\text{number of customers}}\) per week, \(\boldsymbol{\text{including}}\) the change due to price decreases.

(Note: The exact wording for the first drop - down may be adjusted based on the options in the drop - down menu. If the options are like "revenue with no price decreases", "price with no price decreases", etc., the first answer should match the option. For the second part, the binomial \(30 + 5x\) is the number of customers, and it includes the change from price decreases (the \(5x\) is the increase from price decreases).)

Answer:

The number of customers starts at \(30\) and for each \(x\) (number of \(\$5\) decreases), the number of customers increases by \(5\) per decrease. So the number of customers as a function of \(x\) is \(30 + 5x\). The revenue function is \(R(x)=\text{(price per pair)}\times\text{(number of customers)}\). The price per pair when there are \(x\) \(\$5\) decreases is \(75-5x\), and the number of customers is \(30 + 5x\). So the binomial \((30 + 5x)\) represents the number of customers per week. And it does take into account the change due to price decreases (because the \(5x\) term is the increase in customers due to \(x\) price decreases of \(\$5\) each).

Final Answers for the Drop - downs:

1. The constant of the polynomial represents the \(\boldsymbol{\text{weekly revenue with no decreases}}\) in the price per pair of shoes.
2. The binomial \((30 + 5x)\) represents the average \(\boldsymbol{\text{number of customers}}\) per week, \(\boldsymbol{\text{including}}\) the change due to price decreases.

(Note: The exact wording for the first drop - down may be adjusted based on the options in the drop - down menu. If the options are like "revenue with no price decreases", "price with no price decreases", etc., the first answer should match the option. For the second part, the binomial \(30 + 5x\) is the number of customers, and it includes the change from price decreases (the \(5x\) is the increase from price decreases).)