Business Calculus Quiz: Price Elasticity Of Demand

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Price Elasticity Of DemandQuestion 1 of 20

The demand function for a brand of gourmet chocolate is given by $q = \frac{1800}{p+5} - 40$ , where $p$ is the price in dollars per bar and $q$ is the number of bars sold per week. At what price is the demand for the chocolate unit elastic?

p = \

p = \

10$

p = \

20$

p = \

40$

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Business Calculus Quiz

Business Calculus Quiz: Price Elasticity Of Demand

Practice Price Elasticity Of Demand in Business Calculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Price Elasticity Of Demand, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Calculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$p = \$ 5$
$p = \$ 10$ (correct answer)
$p = \$ 20$
$p = \$ 40$

Explanation: The price elasticity of demand is given by the formula

E(p) = -\frac{p}{q} \cdot \frac{dq}{dp}

. First, we find the derivative of the demand function:

q'(p) = \frac{d}{dp}(1800(p+5)^{-1} - 40) = -1800(p+5)^{-2} = -\frac{1800}{(p+5)^2}

. Next, we set up the elasticity formula:

E(p) = -\frac{p}{\frac{1800}{p+5} - 40} \cdot \left(-\frac{1800}{(p+5)^2}\right) = \frac{1800p}{(\frac{1800 - 40(p+5)}{p+5})(p+5)^2} = \frac{1800p}{(1600 - 40p)(p+5)}

. Demand is unit elastic when

E(p) = 1

. So we set the expression equal to 1 and solve for

p

1 = \frac{1800p}{(1600 - 40p)(p+5)}

(1600 - 40p)(p+5) = 1800p

1600p + 8000 - 40p^2 - 200p = 1800p

-40p^2 - 400p + 8000 = 0

p^2 + 10p - 200 = 0

(p+20)(p-10) = 0

Since price must be positive, we have

p=10

Question 2

The weekly demand for a smart home device is modeled by the function $q = 200e^{-0.05p}$ , where $q$ is the number of units sold and $p$ is the price in dollars. The current price is $p = \$ 30$. To increase total revenue, how should the company adjust the price?

Lower the price, because the demand is elastic at this price point. (correct answer)
Raise the price, because the demand is elastic at this price point.
Lower the price, because the demand is inelastic at this price point.
Raise the price, because the demand is inelastic at this price point.

Explanation: First, calculate the price elasticity of demand

E(p)

. The derivative of the demand function is

q'(p) = 200e^{-0.05p} \cdot (-0.05) = -10e^{-0.05p}

. The elasticity formula is

E(p) = -\frac{p}{q} \cdot q'(p)

E(p) = -\frac{p}{200e^{-0.05p}} \cdot (-10e^{-0.05p}) = \frac{10p}{200} = 0.05p

. Now, evaluate the elasticity at the current price

p = 30

E(30) = 0.05(30) = 1.5

. Since

E(30) = 1.5 > 1

, demand is elastic. When demand is elastic, a price decrease leads to an increase in total revenue. Therefore, the company should lower the price.

Question 3

The total revenue $R(p)$ from selling a product at price $p$ is given by the function $R(p) = 300p^2 - 2p^3$ for the valid price range $0 \le p \le 150$ . For what range of prices is the demand for this product elastic?

$0 < p < 50$
$0 < p < 100$
$p > 50$
$100 < p \le 150$ (correct answer)

Explanation: Demand is elastic when the price elasticity

E(p) > 1

. A key relationship in business calculus is that demand is elastic if and only if revenue decreases as price increases, which means the derivative of the revenue function,

R'(p)

, is negative. First, find the derivative of the revenue function:

R'(p) = 600p - 6p^2

. Next, find the price range where

R'(p) < 0

600p - 6p^2 < 0

6p(100 - p) < 0

Since price

p

must be positive, the term

6p

is positive. For the entire expression to be negative, we must have

(100 - p) < 0

, which implies

p > 100

. Combining this with the given valid price range

0 \le p \le 150

, the range for which demand is elastic is

100 < p \le 150

Question 4

A manufacturer determines that the demand for its product is linear, given by the demand equation $q = a - bp$ for positive constants $a$ and $b$ . Which statement accurately describes the price elasticity of demand, $E(p)$ , for this product?

Elasticity is constant for all prices along the demand curve.
Elasticity is equal to the negative of the slope of the demand curve.
Elasticity decreases as the price increases along the demand curve.
Elasticity increases as the price increases along the demand curve. (correct answer)

Explanation: For the linear demand function

q = a - bp

, the derivative is

\frac{dq}{dp} = -b

. The price elasticity of demand is

E(p) = -\frac{p}{q} \cdot \frac{dq}{dp} = -\frac{p}{a-bp}(-b) = \frac{bp}{a-bp}

. To see how

E(p)

changes as

p

increases, we can analyze the expression. As price

p

increases, the numerator (

bp

) increases and the denominator (

a-bp

) decreases (since

b

is positive). An increasing numerator and a decreasing positive denominator result in the overall fraction increasing. Thus, elasticity increases as price increases. At

p=0

E(0)=0

(perfectly inelastic). As

p

approaches

a/b

(the price at which quantity is zero),

E(p)

approaches infinity (perfectly elastic).

Question 5

An analyst determines that the price elasticity of demand for a monthly software subscription is $0.75$ at the current price. The company is considering increasing the price. Which of the following is the most accurate conclusion that can be drawn from this information alone?

Total revenue will increase, and total profit will also increase proportionally.
Total revenue will increase, but the effect on profit cannot be determined without cost information. (correct answer)
Total revenue will decrease, and total profit will also decrease proportionally.
Both total revenue and the number of subscriptions sold will increase simultaneously.

Explanation: The price elasticity of demand is

E = 0.75

. Since

E < 1

, the demand is inelastic. For inelastic demand, an increase in price leads to an increase in total revenue. This is because the percentage decrease in quantity demanded will be smaller than the percentage increase in price. So, we can conclude that total revenue will increase. However, profit is calculated as Total Revenue - Total Cost. While revenue will increase, a price increase will also cause the quantity sold to decrease. The change in total cost depends on how cost varies with quantity (i.e., the marginal cost). Without information on the cost structure, we cannot determine whether the increase in revenue will outweigh the change in total production costs. Therefore, the effect on profit is uncertain.

Question 6

A tech company's subscription service has demand $p = 50e^{-0.02q}$ , where $p$ is monthly price and $q$ is thousands of subscribers. The marketing team claims that at their current subscriber level of 25,000, demand is unit elastic. If true, what should be their revenue-maximizing strategy?

Maintain current pricing since unit elasticity confirms they are already maximizing revenue at this point (correct answer)
Increase price slightly since unit elasticity suggests small price increases will maintain revenue while improving margins
Decrease price since their calculation is incorrect and actual elasticity is likely different from -1
Focus on cost reduction rather than pricing changes since unit elasticity means revenue cannot be increased through price adjustments

Explanation: Unit elastic demand (elasticity = -1) occurs exactly at the revenue-maximizing point. When elasticity equals -1, the percentage change in price is exactly offset by the percentage change in quantity, keeping total revenue constant for small price changes. This is the theoretical maximum of the revenue function. Any price increase or decrease from this point will decrease total revenue. Choice B misunderstands that unit elasticity means revenue stays constant with price changes, not that margins improve. Choice C incorrectly suggests verifying the calculation rather than accepting the given information. Choice D misses that unit elasticity identifies the revenue maximum.

Question 7

A restaurant chain analyzes three menu items with different demand elasticities during economic uncertainty. Item X has elasticity -0.6, Item Y has elasticity -1.2, and Item Z has elasticity -2.5. If they must increase all prices by 10% due to inflation, what should be their primary strategic concern?

Item X will lose the most customers proportionally, requiring the strongest marketing support to maintain sales volume
Item X will generate the largest revenue increase but may alienate price-sensitive customers in the long term
Item Y will face the most unpredictable demand response since it's near unit elasticity, requiring careful monitoring
Item Z will experience the largest revenue decline, potentially requiring menu restructuring or cost reduction focus (correct answer)

Explanation: When you encounter price elasticity questions, focus on how elasticity values predict both quantity changes and revenue impacts. Elasticity of demand measures how responsive quantity demanded is to price changes, calculated as the percentage change in quantity divided by percentage change in price. With a 10% price increase, you can predict each item's quantity change: Item X will see a 6% decrease in quantity (

-0.6 \times 10\% = -6\%

), Item Y will drop 12% (

-1.2 \times 10\% = -12\%

), and Item Z will plummet 25% (

-2.5 \times 10\% = -25\%

). For revenue impact, remember that when demand is inelastic (elasticity between 0 and -1), price increases boost revenue because the price effect outweighs the quantity decrease. When demand is elastic (elasticity less than -1), price increases hurt revenue because customers flee proportionally more than prices rise. Option A incorrectly identifies Item X as losing the most customers—Item Z actually loses 25% versus X's 6%. Option B misses that Item X (inelastic) will indeed increase revenue, but this isn't the primary concern when other items face severe losses. Option C wrongly suggests Item Y is unpredictable—elasticity of -1.2 clearly indicates it's elastic and will lose revenue. Option D correctly identifies the core strategic issue: Item Z's high elasticity (-2.5) means the 10% price increase will devastate both quantity (down 25%) and revenue, requiring immediate attention through menu changes or cost reductions. Remember: inelastic goods (elasticity between 0 and -1) gain revenue from price increases, while elastic goods (elasticity beyond -1) lose revenue and need special strategic attention.

Question 8

An airline's demand for business class seats is $q = \frac{500}{p^{0.8}}$ and for economy seats is $q = \frac{2000}{p^{1.3}}$ . During a fuel cost crisis requiring across-the-board price increases, which cabin class should receive priority for revenue protection strategies?

Business class should receive priority since its lower elasticity (-0.8) means price increases will generate net revenue gains
Both classes need equal attention since the elasticity difference is not economically significant for revenue management
Economy class should receive priority since its higher elasticity (-1.3) means it's more sensitive to competitive pricing pressures (correct answer)
Business class should receive priority since its higher absolute demand means larger total revenue impact from pricing changes

Explanation: When you encounter demand functions and pricing decisions, you need to understand price elasticity of demand - how responsive quantity demanded is to price changes. The elasticity is the exponent in these power functions. For business class, the demand function

q = \frac{500}{p^{0.8}}

has an elasticity of -0.8 (inelastic demand). For economy class,

q = \frac{2000}{p^{1.3}}

has an elasticity of -1.3 (elastic demand). Since |-1.3| > 1, economy demand is highly sensitive to price changes, while business class demand with |-0.8| < 1 is relatively insensitive. During a fuel crisis requiring price increases, economy class should receive priority for revenue protection because its elastic demand means passengers will dramatically reduce purchases when prices rise. This makes the cabin vulnerable to revenue losses and competitive pressure from other airlines. Choice A is wrong because while business class has lower elasticity, this actually means it's less vulnerable to price increases, not that it should receive priority protection. Choice B incorrectly suggests the elasticity difference isn't significant - the gap between -0.8 and -1.3 represents the crucial boundary between inelastic and elastic demand. Choice D focuses on absolute demand levels rather than price sensitivity, missing the key insight that elasticity determines how demand responds to price changes. Remember: when facing across-the-board price increases, prioritize protecting revenue streams with elastic demand (|elasticity| > 1) since these are most vulnerable to customer defection and competitive pressure.

Question 9

The weekly demand for a new electric scooter is given by $q = 10\sqrt{400 - p^2}$ , where $p$ is the price in hundreds of dollars and $0 < p < 20$ . What is the price elasticity of demand when the price is $1600 (i.e., $p=16$)?

$16/9$ (correct answer)
$9/16$
$4/3$
$1$

Explanation: First, note that the price

p

in the formula is in hundreds of dollars, so a price of $1600 corresponds to $p=16

. Step 1: Find the quantity

at

p=16

.

q(16) = 10\sqrt{400 - 16^2} = 10\sqrt{400 - 256} = 10\sqrt{144} = 10(12) = 120

. Step 2: Find the derivative

\frac{dq}{dp}

.

q(p) = 10(400 - p^2)^{1/2}

. Using the chain rule:

q'(p) = 10 \cdot \frac{1}{2}(400 - p^2)^{-1/2} \cdot (-2p) = -10p(400 - p^2)^{-1/2} = -\frac{10p}{\sqrt{400 - p^2}}

. Step 3: Evaluate the derivative at

p=16

.

q'(16) = -\frac{10(16)}{\sqrt{400 - 16^2}} = -\frac{160}{12} = -\frac{40}{3}

. Step 4: Calculate the elasticity

E(p) = -\frac{p}{q} \cdot \frac{dq}{dp}

.

E(16) = -\frac{16}{120} \cdot \left(-

\frac{40}{3}\right) = \frac{16 \cdot 40}{120 \cdot 3} = \frac{640}{360} = \frac{64}{36} = \frac{16}{9}

Question 10

The demand function for a product is given by $q = \frac{k}{p}$ , where $q$ is the quantity demanded, $p$ is the price, and $k$ is a positive constant. Which of the following correctly describes the price elasticity of demand for this product?

The elasticity is 1 at all prices. (correct answer)
The elasticity is equal to the constant $k$ .
The elasticity is equal to $p/k$ .
The elasticity cannot be determined without knowing the value of $k$ and $p$ .

Explanation: To find the price elasticity of demand, we use the formula

E(p) = -\frac{p}{q} \cdot \frac{dq}{dp}

. The demand function is

q = kp^{-1}

. First, we find the derivative with respect to price:

\frac{dq}{dp} = -1 \cdot kp^{-2} = -\frac{k}{p^2}

. Now, we substitute

q

and

\frac{dq}{dp}

into the elasticity formula:

E(p) = -\frac{p}{k/p} \cdot \left(-\frac{k}{p^2}\right)

. Simplifying the expression:

E(p) = -\frac{p^2}{k} \cdot \left(-\frac{k}{p^2}\right) = \frac{p^2k}{kp^2} = 1

. The elasticity is equal to 1 for all values of

p

and

k

. This special type of demand curve is a rectangular hyperbola, for which total revenue (

R = p \cdot q = p \cdot (k/p) = k

) is constant. When revenue is constant, demand is always unit elastic.

Question 11

The demand for a product is given by the equation $p = \sqrt{400 - 2q}$ , where $p$ is the price per unit and $q$ is the number of units demanded. What is the price elasticity of demand when the price is $p = \$ 10$?

$1/3$
$2/3$ (correct answer)
$1$
$3/2$

Explanation: To find the price elasticity of demand,

E(p) = -\frac{p}{q} \cdot \frac{dq}{dp}

, we need

p

q

, and

\frac{dq}{dp}

. We are given

p=10

. We find the corresponding quantity

q

by plugging this price into the demand equation:

10 = \sqrt{400 - 2q} \implies 100 = 400 - 2q \implies 2q = 300 \implies q = 150

. To find

\frac{dq}{dp}

, we can first express

q

as a function of

p

p^2 = 400 - 2q \implies 2q = 400 - p^2 \implies q(p) = 200 - 0.5p^2

. Now, we differentiate with respect to

p

\frac{dq}{dp} = -p

. At

p=10

\frac{dq}{dp} = -10

. Finally, we calculate the elasticity:

E(10) = -\frac{10}{150} \cdot (-10) = \frac{100}{150} = \frac{2}{3}

Question 12

The demand function for a particular model of tablet computer is given by $q = \frac{5000}{p^2}$ , where $p$ is the price per tablet. If the current price is $p = \$ 20$, what is the approximate percentage change in demand that would result from a 1% increase in the price?

A decrease of approximately 2% (correct answer)
An increase of approximately 2%
A decrease of approximately 0.5%
A decrease of approximately 1%

Explanation: The relationship between elasticity and percentage change is

E \approx -\frac{\%\Delta q}{\%\Delta p}

. We first need to find the price elasticity of demand

E(p)

. The demand function is

q = 5000p^{-2}

. The derivative is

q'(p) = -10000p^{-3} = -\frac{10000}{p^3}

. The elasticity is

E(p) = -\frac{p}{q} \cdot q'(p) = -\frac{p}{5000/p^2} \cdot \left(-\frac{10000}{p^3}\right) = \frac{p^3}{5000} \cdot \frac{10000}{p^3} = 2

. The elasticity of demand is constant at

E(p)=2

. Now, we use the approximation formula with

E=2

and

\%\Delta p = +1\%

2 \approx -\frac{\%\Delta q}{1\%}

. Solving for

\%\Delta q

, we get

\%\Delta q \approx -2\%

. Therefore, a 1% price increase will lead to an approximate 2% decrease in demand.

Question 13

The price elasticity of demand for a product at a price of $p_0$ is $E(p_0) = 0.8$ . At this price, the quantity demanded is $q_0 = 500$ units. Which of the following best describes the instantaneous rate of change of revenue with respect to price, $R'(p_0)$ ?

Revenue is increasing at a rate of $100 per dollar increase in price. (correct answer)
Revenue is decreasing at a rate of $100 per dollar increase in price.
Revenue is increasing at a rate of $400 per dollar increase in price.
Revenue is decreasing at a rate of $400 per dollar increase in price.

Explanation: The derivative of the revenue function,

R'(p)

, is related to the price elasticity of demand,

E(p)

, by the formula

R'(p) = q(1 - E(p))

. We are given

q_0 = 500

and

E(p_0) = 0.8

. We can substitute these values into the formula to find the rate of change of revenue at this price point:

R'(p_0) = 500(1 - 0.8) = 500(0.2) = 100

. Since

R'(p_0) = 100

is positive, revenue is increasing at this price point. The value means that for a small increase in price, revenue increases by $100 times that change. Thus, revenue is increasing at a rate of $100 per dollar increase in price.

Question 14

The relationship between price $p$ and quantity demanded $q$ for a certain commodity is given by the implicit equation $p^2 + 2q^2 = 1100$ . What is the price elasticity of demand when the price is $p = \$ 30$?

$1.50$
$2.25$
$4.50$ (correct answer)
$0.22$

Explanation: We need to find

E(p) = -\frac{p}{q} \cdot \frac{dq}{dp}

. First, find the value of

q

when

p=30

30^2 + 2q^2 = 1100 \implies 900 + 2q^2 = 1100 \implies 2q^2 = 200 \implies q^2 = 100

. Since quantity must be positive,

q=10

. Next, find

\frac{dq}{dp}

using implicit differentiation with respect to

p

\frac{d}{dp}(p^2 + 2q^2) = \frac{d}{dp}(1100)

2p + 4q \frac{dq}{dp} = 0

\frac{dq}{dp} = -\frac{2p}{4q} = -\frac{p}{2q}

. Now, evaluate

\frac{dq}{dp}

at the point

(p,q) = (30,10)

\frac{dq}{dp} = -\frac{30}{2(10)} = -\frac{30}{20} = -1.5

. Finally, calculate the elasticity:

E(30) = -\frac{p}{q} \cdot \frac{dq}{dp} = -\frac{30}{10} \cdot (-1.5) = -3(-1.5) = 4.5

Question 15

The relationship between the price $p$ (in dollars) and the quantity demanded $q$ (in thousands of units) for a product is given by the equation $p^2 + 2q^2 = 1100$ . What is the price elasticity of demand when the price is $p=\$ 30$?

0.22
2.25
4.5 (correct answer)
9.0

Explanation: This problem requires three steps. First, find the quantity

q

when

p=30

. Substitute into the equation:

(30)^2 + 2q^2 = 1100 \implies 900 + 2q^2 = 1100 \implies 2q^2 = 200 \implies q^2 = 100 \implies q=10

(since quantity must be positive). Second, find the derivative

\frac{dq}{dp}

using implicit differentiation:

2p + 4q\frac{dq}{dp} = 0 \implies \frac{dq}{dp} = -\frac{2p}{4q} = -\frac{p}{2q}

. Third, calculate the price elasticity of demand,

E(p) = -\frac{p}{q} \frac{dq}{dp}

E(p) = -\frac{p}{q} \left(-\frac{p}{2q}\right) = \frac{p^2}{2q^2}

. Finally, substitute the values

p=30

and

q=10

E(30) = \frac{(30)^2}{2(10)^2} = \frac{900}{2(100)} = \frac{900}{200} = 4.5

Question 16

The demand for a product is given by the function $q(p) = \frac{5000}{p^2+16}$ . A manager calculates the price elasticity of demand at the current price of $p = \$ 6$. Based on this calculation, which statement is correct?

Demand is elastic, and a small price increase will cause total revenue to decrease. (correct answer)
Demand is inelastic, and a small price increase will cause total revenue to increase.
Demand is elastic, and a small price increase will cause total revenue to increase.
Demand is inelastic, and a small price decrease will cause total revenue to decrease.

Explanation: First, find the derivative of the demand function,

q'(p)

. Using the chain rule,

q(p) = 5000(p^2+16)^{-1}

, so

q'(p) = -5000(p^2+16)^{-2}(2p) = \frac{-10000p}{(p^2+16)^2}

. Next, use the formula for price elasticity of demand,

E(p) = -\frac{p \cdot q'(p)}{q(p)}

E(p) = -\frac{p}{ (\frac{5000}{p^2+16}) } \cdot \frac{-10000p}{(p^2+16)^2} = \frac{p(p^2+16)}{5000} \cdot \frac{10000p}{(p^2+16)^2} = \frac{2p^2}{p^2+16}

. Now, evaluate at

p=6

E(6) = \frac{2(6^2)}{6^2+16} = \frac{2(36)}{36+16} = \frac{72}{52} = \frac{18}{13} \approx 1.38

. Since

E(6) > 1

, demand is elastic at this price. When demand is elastic, price and total revenue move in opposite directions. Therefore, a small price increase will cause total revenue to decrease.

Question 17

The demand function for a particular style of athletic shoe is given by $q(p) = 400(25-p^2)$ , where $q$ is the number of pairs sold per week and $p$ is the price in dollars, for $0 < p < 5$ . At what price is the demand for the shoes unit elastic?

$p = \frac{5\sqrt{2}}{2}$
$p = \frac{5\sqrt{3}}{3}$ (correct answer)
$p = 2.50$
$p = 5.00$

Explanation: Demand is unit elastic when the price elasticity of demand

E(p) = 1

. First, find

q'(p) = 400(-2p) = -800p

. The elasticity function is

E(p) = -\frac{p \cdot q'(p)}{q(p)} = -\frac{p(-800p)}{400(25-p^2)} = \frac{800p^2}{400(25-p^2)} = \frac{2p^2}{25-p^2}

. Set

E(p) = 1

\frac{2p^2}{25-p^2} = 1

. This gives

2p^2 = 25-p^2

, which simplifies to

3p^2 = 25

. Solving for

p

p^2 = \frac{25}{3}

, so

p = \sqrt{\frac{25}{3}} = \frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}

Question 18

The price elasticity of demand for a product is given by the function $E(p) = \frac{2p}{300-p}$ . At a price of $p=\$ 100 $, the demand is$ q=8000 $units. Which of the following represents the demand function$ q(p)$?

$q(p) = 40(300-p)$
$q(p) = 0.2(300-p)^2$ (correct answer)
$q(p) = 8000 + 100(300-p)^2$
$q(p) = 80(300-p)^2$

Explanation: The formula for elasticity is

E(p) = -\frac{p}{q}\frac{dq}{dp}

. We are given

E(p)

, so we can set up a differential equation:

\frac{2p}{300-p} = -\frac{p}{q}\frac{dq}{dp}

. Assuming

p \neq 0

, we can divide by

p

\frac{2}{300-p} = -\frac{1}{q}\frac{dq}{dp}

. This is a separable differential equation. Rearranging gives

\int \frac{2}{300-p} dp = \int -\frac{1}{q} dq

. Integrating both sides yields:

-2 \ln(300-p) = -\ln(q) + C

. (Assuming

p<300

, so absolute values are not needed). Rearranging to solve for

q

\ln(q) = 2 \ln(300-p) - C = \ln((300-p)^2) - C

. Exponentiating both sides:

q = e^{\ln((300-p)^2) - C} = e^{-C}(300-p)^2

. Let

k = e^{-C}

. The demand function has the form

q(p) = k(300-p)^2

. We use the point

(p=100, q=8000)

to find the constant

k

8000 = k(300-100)^2 = k(200)^2 = 40000k

. Solving for

k

k = \frac{8000}{40000} = \frac{1}{5} = 0.2

. Thus, the demand function is

q(p) = 0.2(300-p)^2

Question 19

The current price of a product is $p=\$ 40 $, and the price elasticity of demand is calculated to be$ E(40) = 1.25$. Based on this information, which action should the company take to increase total revenue?

Decrease the price, because demand is elastic, so the percentage increase in quantity will be greater than the percentage decrease in price. (correct answer)
Increase the price, because demand is elastic, and the higher price will more than compensate for the resulting decrease in quantity sold.
Keep the price the same, because revenue is already at its maximum when elasticity is greater than 1.
Decrease the price, because demand is inelastic, and the company will capture a larger market share with a lower price.

Explanation: The price elasticity of demand is

E(40) = 1.25

. Since

|E(p)| > 1

, the demand is elastic at this price. For elastic demand, total revenue moves in the opposite direction of price. To increase total revenue, the company should decrease the price. A price decrease will lead to a proportionally larger increase in quantity demanded, thus increasing total revenue (

R = p \cdot q

). Choice A correctly identifies the demand as elastic and prescribes the correct action with the correct reasoning.

Question 20

A movie theater determines that its daily demand function for tickets is $q(p) = 1600 - p^2$ for a price $p$ . The theater is currently charging $p = \$ 20$ per ticket. To increase total revenue from ticket sales, what action should the theater take based on the price elasticity of demand?

Decrease the price, because demand is elastic at $p=\$ 20$.
Decrease the price, because demand is inelastic at $p=\$ 20$.
Increase the price, because demand is elastic at $p=\$ 20$.
Increase the price, because demand is inelastic at $p=\$ 20$. (correct answer)

Explanation: When you encounter price elasticity questions, you need to determine whether demand is elastic or inelastic at the current price, then apply the revenue rule: if demand is elastic, lower prices increase revenue; if demand is inelastic, higher prices increase revenue. Given the demand function

q(p) = 1600 - p^2

, you first calculate the price elasticity of demand using

E_d = \frac{dq/dp \cdot p}{q}

. Taking the derivative:

\frac{dq}{dp} = -2p

. At

p = 20

, the quantity is

q(20) = 1600 - 400 = 1200

. The elasticity becomes:

E_d = \frac{-2(20) \cdot 20}{1200} = \frac{-800}{1200} = -\frac{2}{3}

Since

|E_d| = \frac{2}{3} < 1

, demand is inelastic at this price point. When demand is inelastic, consumers are relatively unresponsive to price changes, so the theater can increase prices to boost total revenue. Answer D correctly identifies both that demand is inelastic and that the theater should increase prices. Answer A incorrectly states demand is elastic when it's actually inelastic. Answer B correctly identifies the inelastic nature but wrongly suggests decreasing prices, which would reduce revenue when demand is inelastic. Answer C suggests the right action (increase prices) but incorrectly claims demand is elastic. Remember this key relationship: elastic demand means lower prices increase revenue, while inelastic demand means higher prices increase revenue. Always calculate the elasticity first, then apply the appropriate pricing strategy.