Question

debbie is president of a company that produces garbage cans. the company has developed a new type of garbage can that is animal proof, and debbie wants to use the demand function to help set a price. she surveys ten retailers to get an approximation of how many garbage cans would be demanded at each price, and creates a table.
part a
using the table above, determine the demand function that predicts the quantity of garbage cans demanded by the retailers, q, in terms of the wholesale price, p. explain how you determined the equation.
part b
explain the meaning of the slope of the equation in the context of the situation.
part c
how many garbage cans would be demanded at a wholesale price of $18.00? if the company sold this many garbage cans at $18.00, how much money would the company receive from the garbage can sales?

Explanation:

Step1: Calculate the slope \(m\)

Use the formula \(m=\frac{q_2 - q_1}{p_2 - p_1}\). Let \((p_1,q_1)=(13,744)\) and \((p_2,q_2)=(14,630)\).
\(m=\frac{630 - 744}{14 - 13}=\frac{- 114}{1}=-114\)

Step2: Find the \(y -\)intercept \(b\)

Substitute \(m=-114\), \(p = 13\), and \(q = 744\) into \(q=mp + b\).
\(744=-114\times13 + b\)
\(744=-1482 + b\)
\(b=744 + 1482=2226\)

Step3: Write the demand function

The demand function is \(q=-114p + 2226\)

Step4: Interpret the slope for Part B

The slope \(m=-114\) means that for every \(\$1\) increase in the wholesale price \(p\), the quantity demanded \(q\) (in hundreds) decreases by \(114\) units.

Step5: Calculate for Part C

When \(p = 18\), substitute into \(q=-114p + 2226\)
\(q=-114\times18+2226=-2052 + 2226 = 174\) (in hundreds). So the number of garbage - cans is \(174\times100 = 17400\)
The money received is \(18\times17400=\$313200\)

Answer:

• Part A: The demand function is \(q=-114p + 2226\). We used the two - point formula for the slope \(m=\frac{q_2 - q_1}{p_2 - p_1}\) with two points \((p_1,q_1)=(13,744)\) and \((p_2,q_2)=(14,630)\) to find \(m=-114\), then substituted one point and \(m\) into \(q = mp + b\) to find \(b = 2226\).
• Part B: The slope \(-114\) implies that for each \(\$1\) increase in price, the quantity demanded (in hundreds) decreases by \(114\) units.
• Part C: The number of garbage - cans demanded is \(17400\) and the money received is \(\$313200\)