Suppose the quantity x of Super Titan radial tires made available each week in the marketplace is related to the unit-selling price by the following equation where x is measured in units of a thousand and p is in dollars.
p-(1)/(2)x^(2)=48
How fast is the weekly supply of Super Titan radial tires being introduced into the marketplace when x = 9, p = 88.5, and the price/tire is decreasing at the rate of $4/week? (Round your answer to the nearest whole number.)

By taking differentiation of equation (1); dp/dt represents the decreasing rate of change in the tire price and dx/dt represents rate of change of the tires being introduced.

x = 9

p = 88.5

Tire price decreasing rate = dp/dt = 4$ per week

Solution:

Differentiation is performed implicitly.

dp/dt – [(1/2) d(x^2)/dt ]= d(48) /dt

Putting value of dp/dt;

– 4 – [(1/2) * 2 x (dx/dt)] = 0

Where “– “is due to decreasing rate of change of tires price and derivate of a constant “48” is equal to “0”

x dx/dt = -4

dx/dt = -4/x

Where x = 9

So dx/dt = -4/9

Hence the tire supply rate is decreasing the at rate of -4/9.