In this problem, $$ p $$ is in dollars and $$ q $$ is the number of units. (a) Find the elasticity of the demand function $$ 2 p+3 q=180 $$ at the price $$ p=18 $$. (b) How will a price increase affect total revenue? Since the demand is unitary, there will be no change in the revenue with a price increase. Since the demand is elastic, an Increase in price will decrease the total revenue. Since the demand is inelastic, an Increase in price will decrease the total revenue. Since the demand is inelastic, an Increase in price will increase the total revenue. Since the demand is elastic, an increase in price will increase the total revenue.

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UpStudy AI · Accepted Answer

**(a)**  
The demand function is  
$$
2p + 3q = 180.
$$  
Solve for $$ q $$:  
$$
q = \frac{180 - 2p}{3}.
$$  
Differentiate $$ q $$ with respect to $$ p $$:  
$$
\frac{dq}{dp} = -\frac{2}{3}.
$$  
At $$ p = 18 $$, compute $$ q $$:  
$$
q = \frac{180 - 2(18)}{3} = \frac{180 - 36}{3} = \frac{144}{3} = 48.
$$  
Now, the price elasticity of demand is given by  
$$
\varepsilon = \frac{dq}{dp} \times \frac{p}{q}.
$$  
Substitute the values:  
$$
\varepsilon = \left(-\frac{2}{3}\right) \times \frac{18}{48} = -\frac{2}{3} \times \frac{3}{8} = -\frac{1}{4}.
$$

**(b)**  
Since the absolute value of the elasticity is  
$$
\left|\varepsilon\right| = \frac{1}{4} < 1,
$$  
the demand is inelastic. For inelastic demand, an increase in $$ p $$ causes a proportionally smaller decrease in $$ q $$. As a result, the total revenue  
$$
R = p \times q
$$  
will increase when the price is raised.

Thus, the correct answer is:  
Since the demand is inelastic, an increase in price will increase the total revenue.
**(a)** The elasticity of the demand function at $$ p = 18 $$ is $$ -0.25 $$. **(b)** An increase in price will increase the total revenue because the demand is inelastic.

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