Suppose that the weekly sales volume y (in thousands of units sold) depends on the price per unit (in dollars) of the product according to the following formula.
y = 29(3p + 1)−2/5,  p > 0
(a) What is the rate of change in sales volume when the price is $23? (Round your answer to three decimal places.)
dy
dp
 = 
-0.074
 
Changed: Your submitted answer was incorrect. Your current answer has not been submitted.


(b) Interpret your answer to part (a). (Round your answer to the nearest whole number.)
If the price increases $1, the sales volume will decrease by 
74
 
Incorrect: Your answer is incorrect.
 units.

To solve this problem, we need to calculate the rate of change of the sales volume with respect to the price $ p $. This is done by finding the derivative of the given formula for $ y $ with respect to $ p $.

Given:
\[
y = 29(3p + 1)^{-\frac{2}{5}}
\]

Let's solve the problem step-by-step:

### Part (a): Rate of Change in Sales Volume

To find the rate of change of $ y $ with respect to $ p $, we need to compute $ \frac{dy}{dp} $.

1. **Differentiate $ y $ with respect to $ p $:**

\[
y = 29(3p + 1)^{-\frac{2}{5}}
\]

Using the chain rule for differentiation, we get:

\[
\frac{dy}{dp} = 29 \cdot \left(-\frac{2}{5}\right)(3p + 1)^{-\frac{2}{5} - 1} \cdot (3)
\]

Simplify the expression:

\[
\frac{dy}{dp} = 29 \cdot \left(-\frac{2}{5}\right) \cdot (3) \cdot (3p + 1)^{-\frac{7}{5}}
\]

\[
\frac{dy}{dp} = -\frac{174}{5} (3p + 1)^{-\frac{7}{5}}
\]

2. **Calculate $ \frac{dy}{dp} $ at $ p = 23 $:**

Substitute $ p = 23 $ into the derivative:

\[
\frac{dy}{dp} \bigg|_{p = 23} = -\frac{174}{5} (3 \times 23 + 1)^{-\frac{7}{5}}
\]

\[
\frac{dy}{dp} \bigg|_{p = 23} = -\frac{174}{5} (69 + 1)^{-\frac{7}{5}}
\]

\[
\frac{dy}{dp} \bigg|_{p = 23} = -\frac{174}{5} (70)^{-\frac{7}{5}}
\]

Let's compute this value.

### Calculations:

Let's calculate the numerical value now.The rate of change in sales volume when the price is $23 is approximately $ -0.091 $ (rounded to three decimal places).

### Part (b): Interpretation of the Answer

The derivative $ \frac{dy}{dp} $ represents the rate of change of sales volume $ y $ with respect to the price $ p $. The negative value means that as the price increases, the sales volume decreases.

1. **Interpreting the value:**  
   If the price increases by $1, the sales volume will decrease by approximately $ 0.091 \times 1000 = 91 $ units (rounded to the nearest whole number).

Thus, for every $1 increase in the price, the sales volume is expected to decrease by about 91 units.

Would you like more details or have any questions? Here are some related questions to consider:

1. How would the rate of change in sales volume vary for a different price, say $30?
2. What happens to the sales volume if the price decreases instead of increases?
3. How can we find the price that maximizes the sales volume?
4. What is the elasticity of demand at $ p = 23 $?
5. How does the sales volume behave as the price approaches zero or very high values?

**Tip:** Understanding the derivative's sign and magnitude helps in interpreting how sensitive the sales volume is to price changes.

Suppose that the weekly sales volume y (in thousands of units sold) depends on the price per unit (in dollars) of the product according to the following formula.
y = 29(3p + 1)−2/5,  p > 0
(a) What is the rate of change in sales volume when the price is $23? (Round your answer to three decimal places.) dy/dp = -0.074
Changed: Your submitted answer was incorrect. Your current answer has not been submitted.
(b) Interpret your answer to part (a). (Round your answer to the nearest whole number.) If the price increases $1, the sales volume will decrease by 74 units.

Explore how the weekly sales volume of a product depends on its price using differentiation. Calculate the rate of change in sales volume when the price is $23, and interpret the results to understand the impact of price increases. This problem uses chain rule differentiation and is suitable for high school calculus students.

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