Assume that the demand function for tuna in a small coastal town is given by $p = \frac { 50,000 } { q ^ { 1.5 } }$
Where q is the number of pounds of tuna that can be sold in 1 month at the price of p dollars per pound. The town's fishery finds that the demand for tuna is currently 900 pounds per month and is increasing at a rate of 200 pounds per month. How fast is the price changing Round your answer to the three decimal places.

A)-$0.206 per pound per month
B) -$5.556 per pound per month
C) -$0.617 per pound per month
D) -$0.412 per pound per month
E) -$0.274 per pound per month

Question

Assume that the demand function for tuna in a small coastal town is given by   $p = \frac { 50,000 } { q ^ { 1.5 } }$  
Where q is the number of pounds of tuna that can be sold in 1 month at the price of p dollars per pound. The town's fishery finds that the demand for tuna is currently 900 pounds per month and is increasing at a rate of 200 pounds per month. How fast is the price changing  Round your answer to the three decimal places.
 
A)-$0.206 per pound per month
B) -$5.556 per pound per month
C) -$0.617 per pound per month
D) -$0.412 per pound per month
E) -$0.274 per pound per month

Quizplus · Accepted Answer

To find how fast the price is changing, we need to differentiate the demand function with respect to time (t). Given $p = \frac{50,000}{q^{1.5}}$, we first find $\frac{dp}{dq}$ and then use the chain rule to find $\frac{dp}{dt}$ by multiplying $\frac{dp}{dq}$ by $\frac{dq}{dt}$ (which is given as 200 pounds per month). Differentiating $p = \frac{50,000}{q^{1.5}}$ with respect to q gives $\frac{dp}{dq} = -\frac{75,000}{q^{2.5}}$. At $q = 900$, $\frac{dp}{dq} = -\frac{75,000}{900^{2.5}}$. Plugging in the values, we get $\frac{dp}{dq} \approx -0.00308642$.Then, $\frac{dp}{dt} = \frac{dp}{dq} \times \frac{dq}{dt} = -0.00308642 \times 200 \approx -0.617$.Therefore, the price is decreasing at a rate of approximately -$0.617 per pound per month.

Assume That the Demand Function for Tuna in a Small p=50,000q1.5p = \frac { 50,000 } { q ^ { 1.5 } }p=q1.550,000​

Assume That the Demand Function for Tuna in a Small $p = \frac { 50,000 } { q ^ { 1.5 } }$