Suppose that the price p (in dollars) and the weekly sales x (in thousands of units) of a certain commodity satisfy the demand equation 8p cubedplusx squaredequals104 comma 000. Determine the rate at which sales are changing at a time when xequals200 , pequals20 , and the price is falling at the rate of $.50 per week.

dx/dt = 12 (in thousands of units / week)

Step-by-step explanation:

We have from problem statement that:

8*p³ + x² = 104 (1)

Where p is price in $, and "x" is the weekly sales in thousands of units

All variables x and p change in relation to time then

Differentiating on both sides of the equation, we get:

24*p²*dp/dt + 2*x*dx/dt = 0 (1)

We need to find dx/dt and we know

p = 20

x = 200

And price is falling at the rate of 0,5 $/week

Then plugging these values in equation (1)

24*p²*dp/dt = - 2*x*dx/dt

24 * (20) ² (-0,5) = - 400*dx/dt

-4800 = - 400*dx/dt

dx/dt = 4800/400

dx/dt = 12 (in thousands of units / week)