Question

A garden shop determines the demand function q = D(x) = 5x + 200 / 30x + 11 during early summer for tomato plants where q is the number of plants sold per day when the price is x dollars per plant.
(a) Find the elasticity.
(b) Find the elasticity when x = 2.
(c) At $2 per plant, will a small increase in price cause the total revenue to increase or decrease?

Answer 1

Answer: a) $E(x)=\dfrac{-5945}{(30x+11)(5x+200)}$, b) 0.7975, demand is inelastic, c) increase.

Step-by-step explanation:

Since we have given that

$D(x)=\dfrac{5x+200}{30+11}$

So, derivative w.r.t x would be

$D'(x)=\dfrac{5(30x+11)-30(5x+200)}{(30x+11)^2}\\\\D'(x)=\dfrac{150x+55-150x-6000}{(30x+11)^2}\\\\D'(x)=\dfrac{5945}{(30x+11)^2}$

As we know that

$E(x)=\dfrac{-xD'(x)}{D(x)}\\\\\\E(x)=\dfrac{\dfrac{-(-)5945x}{(30x+11)^2}}{\dfrac{5x+200}{30x+11}}\\\\\\E(x)=\dfrac{5945x}{(30x+11)(5x+200)}$

(b) Find the elasticity when x = 2.

So, we put x = 2, we get that

$E(2)=\dfrac{5945\times 2}{(30(2)+11)((5(2)+200))}\\\\E(2)=\dfrac{11890}{(60+11)(10+200)}\\\\E(2)=\dfrac{11890}{71\times 210}\\\\E(2)=\dfrac{11890}{14910}\\\\E(2)=0.7975$

Since, 0.7975 < 1, so the demand is inelastic.

(c) At $2 per plant, will a small increase in price cause the total revenue to increase or decrease?

The total revenue will also increase with increase in price.

As total revenue = $price\times quantity$

Hence, a) $E(x)=\dfrac{-5945}{(30x+11)(5x+200)}$, b) 0.7975, demand is inelastic, c) increase.