Revenue-28x-.4x^2, Cost=4x+10, Find dr/dt when profit = 25 unitsand dr/dt when production=6 units/day?

Question

Given the revenue and cost functions R=28x-0.4x^2 and C=4x+10, where x is the daily production, find the rate of change of profit with respect to time when 25 units are produced and the rate of change of production is 6 units per day per day.

Answer 1

Oh, that Gopal.

dP/dt requires multiplication, not division.

dP/dt = (dP/dx) * (dx/dt)

dP/dx is as Gopal gives it,

dP/dx = -0.8x+24

dx/dt is given as 6. When x = 25, we have

dP/dx = -.8(25) + 24 = 4

Then

dP/dt = 4 * 6 = $24 per day, or $1 per hour

Notice that profit per unit declines as production increases. Each unit yields .8 less profit than the previous. (The slope of dP/dx is negative.)

As a rough check, the profit per day when 25 units per day are produced is

P = -.4x^2 + 24x - 10
= -.4(625) + 24(25) - 10
= -250 + 500 -10
= 340

Four hours later, the rate of production is 26 units per day and the profit per day is

P = -.4(26)^2 + 24(26) - 10
= 343.6

The profit is not the full 344 per day because the profit per unit falls off as the number of units increases. Profit is increasing at the rate of $1 per hour only at the instant given in the problem, and then falls off as production increases.

Answer 2

P=R-C
=-0.4x^2+24x-10
dP/dt=dP/dx/dt/dx
dP/dx=-0.8x+24
dP/dt=[-0.8(25)+24]/6
=4*6
=$24
you have given R=-28x-0.4x^2 in one place and 28x-0.4x^2 in another place
i have taken the second value
thank you gary for pointingout the typo