A company estimates that the weekly sales q of its product is related to the product's price
by the function q=13900/p^12/5 where p is in dollars. Currently, each unit of the product is selling for $3. Determine the point elasticity of demand of this product. (Do not round e; answer exactly.)
The point elasticity of demand is e=_. Round these next two parts to the nearest whole number.
Determine the effect on revenue if the price were decreased by 1%.
If the price were decreased by 1%, the revenue will (increase/decrease)
Select an answer
by $_.

To solve this problem, we'll break it down into parts:

### 1. Point Elasticity of Demand (e)
The point elasticity of demand, $ e $, is given by the formula:
\[
e = \frac{dq/dp \cdot p}{q}
\]
Where:
- $ q = \frac{13900}{p^{12/5}} $ (the sales function)
- $ p $ is the price of the product

#### Step 1: Differentiate $ q $ with respect to $ p $
To find $ \frac{dq}{dp} $, we differentiate $ q = 13900p^{-12/5} $ with respect to $ p $:
\[
\frac{dq}{dp} = 13900 \cdot \left(-\frac{12}{5}\right)p^{-\frac{12}{5}-1}
= -\frac{166800}{5}p^{-\frac{17}{5}}
\]

#### Step 2: Substitute $ p = 3 $ and $ q = \frac{13900}{3^{12/5}} $
Now substitute $ p = 3 $ into both $ \frac{dq}{dp} $ and $ q $.

- $ q = \frac{13900}{3^{12/5}} $
- $ \frac{dq}{dp} = -\frac{166800}{5} \cdot 3^{-17/5} $

#### Step 3: Calculate the point elasticity $ e $
Using the formula for $ e $:
\[
e = \frac{\frac{dq}{dp} \cdot p}{q}
= \frac{-\frac{166800}{5} \cdot 3^{-17/5} \cdot 3}{\frac{13900}{3^{12/5}}}
= -\frac{500400 \cdot 3^{-17/5}}{5 \cdot 13900 \cdot 3^{-12/5}}
= -\frac{500400}{69500} = -7.2
\]

Thus, the point elasticity of demand is:
\[
e = -7.2
\]

### 2. Effect on Revenue When Price Decreases by 1%
Revenue, $ R $, is given by:
\[
R = p \cdot q = p \cdot \frac{13900}{p^{12/5}} = 13900 \cdot p^{-7/5}
\]
To determine the effect of a 1% price decrease on revenue, we use the elasticity of demand:
- $ \Delta R \approx e \cdot \Delta p \cdot R $

Since $ e = -7.2 $, and a 1% price decrease means $ \Delta p = -0.01 $, the change in revenue is:
\[
\Delta R \approx -7.2 \cdot (-0.01) \cdot R = 0.072 \cdot R
\]
So, revenue will increase by approximately 7.2%.

Finally, if the price were decreased by 1%, revenue will **increase** by approximately 7%.

Would you like more details on any step? Here are some related questions:

1. What is the formula for revenue in terms of price and sales?
2. How does elasticity affect revenue when prices change?
3. Can you explain how differentiation helps find elasticity?
4. What happens if elasticity is greater than -1 (inelastic demand)?
5. How does elasticity vary between different types of goods?

**Tip:** The point elasticity helps businesses understand how sensitive their sales are to price changes—critical for pricing strategies!

A company estimates that the weekly sales q of its product is related to the product's price by the function q=13900/p^12/5 where p is in dollars. Currently, each unit of the product is selling for $3. Determine the point elasticity of demand of this product. (Do not round e; answer exactly.) The point elasticity of demand is e=_. Round these next two parts to the nearest whole number. Determine the effect on revenue if the price were decreased by 1%. If the price were decreased by 1%, the revenue will (increase/decrease) by $_.

Learn how to determine the point elasticity of demand using the function q = 13900/p^(12/5) when the price of a product is $3. Understand how to calculate the exact elasticity and determine the effect on revenue if the price is decreased by 1%. Suitable for students studying economics or business math.

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