Q. 67 - Calculus

Question

You work for a company that sells velvet Elvis paintings. The function $N\left(p\right)=9.2p^2-725p+16333$ predicts the number N of velvet Elvis paintings that your company will sell if they are priced at p dollars each, and is shown in the following graph. The Presley estate does not allow you to charge more than $50 per painting.

(a) Use a graphing utility to estimate the price your company should charge per painting if it wishes to sell 6000 velvet Elvis paintings. 

(b) Find the range of prices that would enable your company to sell between 5000 and 7000 velvet Elvis paintings. 

(c) Interpret this problem in terms of delta and epsilon ranges. Specifically, what is c? What is L? What is epsilon? What is the corresponding value of delta? Illustrate these values of c, L, epsilon, and delta on a graph of N(p).

Step-by-Step Solution

Answer

Part(a) The estimated price is $18.68

Part(b) The range for the price is $\left[16.20,21.50\right]$

Part(c) $c=18.68, L=6000, p\approx 18.68, \delta \approx 18.68$

Part(a) Step 1. Given Information

The given function is $N\left(p\right)=9.2p^2-725p+16333$ and the given graph is

Part(b) Step 2. Explanation

Trace the value N equals to 6000 in the graph to find p, we get,

Hence, the estimated price is $\$18.68$

Part(b) Step 1. Explanation

Tracing the value $N=5000, N=7000$ in the graph to get the range of the prices,

Hence, the range for the price is $\left[16.20,21.50\right]$

Part(c) Step 1. Explanation

Considering the statement from part(a), the limit expression can be made as,

$\underset{p\to 18.68}{lim}N\left(p\right)=6000, \epsilon =1000$

Here, we have,

$$\begin{gathered} N\left(p\right)=9.2p^2-725p+16333 \\ c=18.68, \\ L=6000 \end{gathered}$$

And the largest value of delta is given by,

$$\begin{gathered} 9.2p^2-725p+16333=6000 \\ 9.2p^2-725p+10333=0 \\ p=60.12,18.68 \end{gathered}$$

Hence, $\delta \approx 18.68$