Question

The demand equation for the Roland portable hair dryer is given as follows where x (measured in units of a hundred) is the quantity demanded per week and p is the unit price in dollars.
x = 15 ( 256 − p 2)(0 ≤ p ≤ 15)
(a) Is the demand elastic or inelastic when p = 5?elastic or inelastic is the demand elastic or inelastic when p = 10?elastic or inelastic
(b) When is the demand unitary? (Round your answer to two decimal places.) (Hint: Solve E(p) = 1 for p.)
(c) If the unit price is lowered slightly from $10, will the revenue increase or decrease?
increase or decrease
(d) If the unit price is increased slightly from $5, will the revenue increase or decrease?
Increase or decrease

Answer 1

Answer:

(a) When p = 5, the demand is inelastic and when p = 10, then the demand is elastic.

(b) The demand will be unitary when the price will be approximately equal to 9.24.

(c) If the unit price is lowered slightly from $10, then the revenue will increase.

(d) If the unit price is increased slightly from $5, then the revenue will decrease.

Step-by-step explanation:

We are given the demand equation for the Roland portable hairdryer where x (measured in units of a hundred) is the quantity demanded per week and p is the unit price in dollars.

$x=15(256-p^{2} )$  where; (0 ≤ p ≤ 15)

Firstly, we will find the elasticity of demand as only then we can identify that demand is elastic or not.

So, the formula for the elasticity of demand is given by;

          E(p)  =  $-\frac{p \times f'(p)}{f(p)}$

                   =  $-\frac{p\times \frac{dx}{dp}[15(256-p^{2})] }{15(256-p^{2})}$  

                   =  $-\frac{p\times 15[0-2p] }{15(256-p^{2})}$

                   =  $\frac{30p^{2} }{15(256-p^{2})}$

           E(p)  =  $\frac{2p^{2} }{256-p^{2}}$    

(a) When p = 5, then E(p) is given by;

            E(5)  =  $\frac{2\times 5^{2} }{256-5^{2}}$

                    =  $\frac{50 }{231}$  =  0.22 < 1

Since the elasticity at p = 5 is less than 1, so we can say that the demand is inelastic.

When p = 10, then E(p) is given by;

            E(10)  =  $\frac{2\times 10^{2} }{256-10^{2}}$

                    =  $\frac{200 }{156}$  =  1.28 > 1

Since the elasticity at p = 10 is more than 1, so we can say that the demand is elastic.

(b) The demand is said to be unitary when the value of price elasticity would be equal to 1, i.e.;

when E(p) = 1

          $\frac{2p^{2} }{256-p^{2}}=1$

          $2p^{2} =256-p^{2}$

          $3p^{2} =256$

           $p^{2} =\frac{256}{3}$

            $p=\sqrt{\frac{256}{3} }$  = 9.24

So, the demand will be unitary when the price will be approximately equal to 9.24.

(c) As we know that the price elasticity at $10 is elastic, i.e. E(10) > 1, this means that when unit price is lowered slightly from $10, then the revenue will increase because due to fall in price the demand will increase.

(d) As we know that the price elasticity at $5 is inelastic, i.e. E(5) < 1, this means that when unit price is increased slightly from $5, then the revenue will decrease because due to increase in price the demand will fall.