A garden shop determines the demand function $q=D(x)=\frac{2 x+400}{20 x+7}$ 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 $x=3$.
(c) At $\$ 3$ per plant, will a small increase in price cause the total revenue to increase or decrease?
(a) The elasticity is $\frac{3993 x}{20 x^{2}+4007 x+1400}$.
(b) When $x=3$, the elasticity is $\frac{11979}{13601}$.
(Simplify your answer. Type an integer or a fraction)
(c) Fill in the blank below.
At $\$ 3$ per plant, a small increase in price will cause the total revenue to

Step 1 ：The elasticity of demand is a measure of the responsiveness of the quantity demanded to a change in price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. In this case, we are given the demand function and we need to find the derivative of this function with respect to price, which will give us the elasticity.

Step 2 ：The elasticity is calculated as \(\frac{3993 x}{20 x^{2}+4007 x+1400}\).

Step 3 ：For part (b), we simply substitute \(x=3\) into the elasticity function we found in part (a). The elasticity when \(x=3\) is \(\frac{-11979}{13601}\).

Step 4 ：For part (c), if the elasticity is greater than 1, a small increase in price will cause the total revenue to decrease. If the elasticity is less than 1, a small increase in price will cause the total revenue to increase. If the elasticity is equal to 1, a small increase in price will not change the total revenue.

Step 5 ：At \(\$ 3\) per plant, a small increase in price will cause the total revenue to decrease.

Step 6 ：Final Answer: \(\boxed{\text{(a) The elasticity is } \frac{3993 x}{20 x^{2}+4007 x+1400}}\)

Step 7 ：Final Answer: \(\boxed{\text{(b) When } x=3, \text{ the elasticity is } \frac{-11979}{13601}}\)

Step 8 ：Final Answer: \(\boxed{\text{(c) At } \$ 3 \text{ per plant, a small increase in price will cause the total revenue to decrease}}\)