Question

Suppose the quantity demanded weekly of the Super Titan radial tires is related to its unit price by the equation

p + x2 = 324

where p is measured in dollars and x is measured in units of a thousand. How fast is the quantity demanded weekly changing when

x = 6,

p = 288,

and the price per tire is increasing at the rate of $2/week? (Round your answer to the nearest tire.)

Answer 1

Answer:

The rate of change of weekly demand is -1/6.

Step-by-step explanation:

It is given that the quantity demanded weekly of the Super Titan radial tires is related to its unit price by the equation

$p+x^2=324$

where p is measured in dollars and x is measured in units of a thousand.

We need to find the rate of change in weekly demand, when x=6, p=288 and $\frac{dp}{dt}=2$.

Subtract p from both sides.

$x^2=324-p$

Differential with respect to t.

$\frac{d}{dt}x^2=\frac{d}{dt}(324-p)$

$2x\frac{dx}{dt}=-\frac{dP}{dt}$

Divide both sides by 2x.

$\frac{dx}{dt}=-\dfrac{\frac{dP}{dt}}{2x}$

Substitute x=6 and $\frac{dp}{dt}=2$ in the above equation.

$\frac{dx}{dt}=-\dfrac{2}{2(6)}$

$\frac{dx}{dt}=-\dfrac{1}{6}$

$\frac{dx}{dt}=-0.16667$

Rate of change in weekly demand is 0 (approximate to the nearest tire). It dose not make any sense.

Therefore, the rate of change of weekly demand is -1/6.