Jonathan T. answered • 10/05/23
10+ Years of Experience from Hundreds of Colleges and Universities!
Let's solve this problem step by step:
Given information:
- The weekly revenue of the business selling gummy bears is a continuous function of price.
- The revenue changes continuously per price increase at a decay rate of 2.5%.
- When the price is $2, the weekly revenue is $8493.
**(a) What is the revenue when the price is $4.80?**
To find the revenue when the price is $4.80, we can use the information given and the concept of exponential decay. The decay rate of 2.5% per price increase means that the revenue decreases by 2.5% for each dollar increase in price.
Let R(p) be the weekly revenue when the price is p dollars. We can express the relationship as follows:
\[R(p) = R(2) \times \left(1 - \frac{2.5}{100}\right)^{(p-2)}\]
We know that when the price is $2, the weekly revenue is $8493:
\[R(2) = 8493\]
Now, we can calculate R(4.80):
\[R(4.80) = 8493 \times \left(1 - \frac{2.5}{100}\right)^{(4.80-2)}\]
\[R(4.80) = 8493 \times \left(1 - 0.025\right)^{2.80}\]
\[R(4.80) = 8493 \times 0.975^{2.80}\]
Calculating this, we find:
\[R(4.80) \approx 8493 \times 0.9055\]
\[R(4.80) \approx 7683.62\]
So, the weekly revenue when the price is $4.80 is approximately $7683.62.
**(b) How fast is the revenue changing when the price is $4.80?**
To find how fast the revenue is changing with respect to the price when the price is $4.80, we need to find the derivative of the revenue function with respect to price and evaluate it at \(p = 4.80\).
We already have the revenue function:
\[R(p) = 8493 \times \left(1 - \frac{2.5}{100}\right)^{(p-2)}\]
Now, find the derivative \(R'(p)\) using the chain rule:
\[R'(p) = 8493 \times \left(-\frac{2.5}{100}\right) \times \left(1 - \frac{2.5}{100}\right)^{(p-2-1)} \times 1\]
Simplify this:
\[R'(p) = -212.325 \times \left(0.975\right)^{(p-3)}\]
Now, evaluate \(R'(4.80)\) to find the rate of change of revenue at \(p = 4.80\):
\[R'(4.80) = -212.325 \times \left(0.975\right)^{(4.80-3)}\]
\[R'(4.80) \approx -212.325 \times 0.975^{1.80}\]
\[R'(4.80) \approx -212.325 \times 0.9531\]
\[R'(4.80) \approx -202.47\]
So, the rate at which the revenue is changing when the price is $4.80 is approximately -$202.47 per dollar increase in price. Since this value is negative, it indicates that the revenue is decreasing as the price increases.
