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Crank
3 years ago
13

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?
Mathematics
1 answer:
cricket20 [7]3 years ago
4 0

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.

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