Question

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 - \dfrac{1}{2}x^2 = 48 $ How fast is the weekly supply of Super Titan radial tires being introduced into the marketplace when x = 5, p = 60.5, and the price/tire is decreasing at the rate of $3/week? (Round your answer to the nearest whole number.)

Answer 1

600 tires/week OR dx/dt=-3/5 thousand tires/week

Step-by-step explanation:

where x is measured in units of a thousand and p is in dollars. $ p - \dfrac{1}{2}x^2 = 48 $ How fast is the weekly supply of Super Titan radial tires being introduced into the marketplace when x = 5, p = 60.5, and the price/tire is decreasing at the rate of $3/week?

p-1/2x^2=48

differentiating both sides with respect to time

d/dt(p-1/2x^2)=d/dt(48)

`p^(') +(1*2x*x^(') )/(2) =0`

dp/dt+x dx/dt=0

-3=5dx/dt

dx/dt=-3/5

recall that x is measured in units of thousands

dx/dt=-0.6*1000

dx/dt=600 tires/week

OR dx/dt=-3/5 thousand tires/week

the 600 quantity supply of Super Titan radial tires decreases per each week

Answer 2

dx/dt = -3/5 time/week.

Step-by-step explanation:

`p-x^2/2=48\\x^2 = 2(p-48)\\x^2=2p-96`

differentiating both sides w.r.t t time.

`2x(dx/dt)=2dp/dt\\dp/dt=xdx/dt\\`

given
`dp/dt=-3\\x=5\\\\`

`dx/dt =-3/5` times / week