Math / Algebra

QuestionGiven the demand function $D(p)=375-3 p^{2}$ , Find the Elasticity of Demand at a price of \ $2$ \square$ At this price, we would say the demand is: Elastic Unitary Inelastic
Based on this, to increase revenue we should: Keep Prices Unchanged Lower Prices Raise Prices

Studdy Solution

STEP 1

What is this asking? We need to figure out how sensitive customer demand is to price changes at $\$2$ and then suggest a revenue-boosting strategy. Watch out! Don't mix up the elasticity formula with other formulas!
Also, remember the sign of the elasticity matters for interpreting the result.

STEP 2

1. Define the elasticity of demand formula
2. Calculate $D'(p)$
3. Evaluate $D(p)$ and $D'(p)$ at $p=2$
4. Compute the elasticity
5. Interpret the elasticity and recommend a pricing strategy

STEP 3

The elasticity of demand, $E(p)$ , is given by the formula: $E(p) = \frac{p}{D(p)} \cdot D'(p)$ Where $D(p)$ is the demand function and $D'(p)$ is its derivative with respect to price $p$ .
This formula tells us how much the quantity demanded changes (as a percentage) for a small change in price (as a percentage).

STEP 4

We are given the demand function $D(p) = 375 - 3p^2$ .
To find its derivative, we apply the power rule: $D'(p) = \frac{d}{dp}(375 - 3p^2) = -6p$ So, $D'(p) = -6p$ .
This tells us the instantaneous rate of change of demand with respect to price.

STEP 5

We need to find the demand and its rate of change at the specific price of $\$2$ .
Let's plug in $p=2$ into our demand function and its derivative: $D(2) = 375 - 3 \cdot (2^2) = 375 - 3 \cdot 4 = 375 - 12 = 363$ $D'(2) = -6 \cdot 2 = -12$ So, at a price of $\$2$ , the demand is 363 units, and the rate of change of demand with respect to price is -12 units per dollar.

STEP 6

Now, let's plug our values into the elasticity formula: $E(2) = \frac{2}{D(2)} \cdot D'(2) = \frac{2}{363} \cdot (-12) = -\frac{24}{363} = -\frac{8}{121}$ Therefore, $E(2) \approx -0.066$ .

STEP 7

Since $|E(2)| < 1$ , the demand is inelastic at a price of $\$2$ .
This means that a small percentage change in price will lead to a smaller percentage change in quantity demanded.

STEP 8

To increase revenue when demand is inelastic, we should raise prices.
Since the change in price is proportionally larger than the change in quantity demanded, the increase in price will more than offset the decrease in quantity sold, leading to higher revenue.

STEP 9

At a price of $\$2$ , the elasticity of demand is approximately $-0.066$ .
The demand is inelastic.
To increase revenue, we should raise prices.

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QuestionGiven the demand function $D(p)=375-3 p^{2}$ , Find the Elasticity of Demand at a price of \ $2$ \square$ At this price, we would say the demand is: Elastic Unitary Inelastic
Based on this, to increase revenue we should: Keep Prices Unchanged Lower Prices Raise Prices

Studdy Solution

STEP 1

What is this asking? We need to figure out how sensitive customer demand is to price changes at $\$2$ and then suggest a revenue-boosting strategy. Watch out! Don't mix up the elasticity formula with other formulas!
Also, remember the sign of the elasticity matters for interpreting the result.

STEP 2

1. Define the elasticity of demand formula
2. Calculate $D'(p)$
3. Evaluate $D(p)$ and $D'(p)$ at $p=2$
4. Compute the elasticity
5. Interpret the elasticity and recommend a pricing strategy

STEP 3

STEP 4

We are given the demand function $D(p) = 375 - 3p^2$ .
To find its derivative, we apply the power rule: $D'(p) = \frac{d}{dp}(375 - 3p^2) = -6p$ So, $D'(p) = -6p$ .
This tells us the instantaneous rate of change of demand with respect to price.

STEP 5

STEP 6

Now, let's plug our values into the elasticity formula: $E(2) = \frac{2}{D(2)} \cdot D'(2) = \frac{2}{363} \cdot (-12) = -\frac{24}{363} = -\frac{8}{121}$ Therefore, $E(2) \approx -0.066$ .

STEP 7

Since $|E(2)| < 1$ , the demand is inelastic at a price of $\$2$ .
This means that a small percentage change in price will lead to a smaller percentage change in quantity demanded.

STEP 8

To increase revenue when demand is inelastic, we should raise prices.
Since the change in price is proportionally larger than the change in quantity demanded, the increase in price will more than offset the decrease in quantity sold, leading to higher revenue.

STEP 9

At a price of $\$2$ , the elasticity of demand is approximately $-0.066$ .
The demand is inelastic.
To increase revenue, we should raise prices.

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QuestionGiven the demand function D(p)=375−3p2D(p)=375-3 p^{2}D(p)=375−3p2, Find the Elasticity of Demand at a price of \22 2\square$ At this price, we would say the demand is: Elastic Unitary InelasticBased on this, to increase revenue we should: Keep Prices Unchanged Lower Prices Raise Prices

Studdy Solution

STEP 1

What is this asking? We need to figure out how sensitive customer demand is to price changes at $2\$2$2 and then suggest a revenue-boosting strategy. Watch out! Don't mix up the elasticity formula with other formulas!Also, remember the *sign* of the elasticity matters for interpreting the result.

STEP 2

1. Define the elasticity of demand formula2. Calculate D′(p)D'(p)D′(p)3. Evaluate D(p)D(p)D(p) and D′(p)D'(p)D′(p) at p=2p=2p=24. Compute the elasticity5. Interpret the elasticity and recommend a pricing strategy

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

Since ∣E(2)∣<1|E(2)| < 1∣E(2)∣<1, the demand is **inelastic** at a price of $2\$2$2.This means that a small percentage change in price will lead to a smaller percentage change in quantity demanded.

STEP 8

To increase revenue when demand is inelastic, we should **raise prices**.Since the change in price is proportionally larger than the change in quantity demanded, the increase in price will more than offset the decrease in quantity sold, leading to higher revenue.

STEP 9

At a price of $2\$2$2, the elasticity of demand is approximately −0.066-0.066−0.066.The demand is **inelastic**.To increase revenue, we should **raise prices**.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionGiven the demand function $D(p)=375-3 p^{2}$ , Find the Elasticity of Demand at a price of \ $2$ \square$ At this price, we would say the demand is: Elastic Unitary Inelastic
Based on this, to increase revenue we should: Keep Prices Unchanged Lower Prices Raise Prices

What is this asking? We need to figure out how sensitive customer demand is to price changes at $\$2$ and then suggest a revenue-boosting strategy. Watch out! Don't mix up the elasticity formula with other formulas!
Also, remember the sign of the elasticity matters for interpreting the result.

1. Define the elasticity of demand formula
2. Calculate $D'(p)$
3. Evaluate $D(p)$ and $D'(p)$ at $p=2$
4. Compute the elasticity
5. Interpret the elasticity and recommend a pricing strategy

Since $|E(2)| < 1$ , the demand is inelastic at a price of $\$2$ .
This means that a small percentage change in price will lead to a smaller percentage change in quantity demanded.

To increase revenue when demand is inelastic, we should raise prices.
Since the change in price is proportionally larger than the change in quantity demanded, the increase in price will more than offset the decrease in quantity sold, leading to higher revenue.

At a price of $\$2$ , the elasticity of demand is approximately $-0.066$ .
The demand is inelastic.
To increase revenue, we should raise prices.