A company is selling laptop computers. It determines that its total profit, in dollars, is given by

$P(x) = 0.08x^2 + 80x$ where $$x$$ is the number of units produced and sold. Suppose that $$x$$ is a function of time, in months, where $$x = 5t + 1$$. (a) Find the total profit as a function of time $$t$$. (b) Find the rate of change of total profit when $$t = 48$$ months.

from sympy import *

x, t = symbols("x, t")

def profit(x):
return 0.08 * x**2 + 80 * x

def units(t):
return 5 * t + 1

P = simplify(profit(units(Symbol("t"))))
P
## 2.0*t**2 + 400.8*t + 80.08
rate = diff(P, t)
rate
## 4.0*t + 400.8
rate.evalf(subs = {t : 48})
## 592.800000000000

A company determines that its total cost, in thousands of dollars, for producing $$x$$ items is

$C(x) = \sqrt{5x^2 + 60}$ and it plans to boost production $$t$$ months from now according to the function $$x(t) = 20t + 40$$. How fast will costs be rising 4 months from now?

def cost(x):
return sqrt(5 * x**2 + 60)

def units(t):
return 20 * t + 40

C = simplify(cost(units(Symbol("t"))))
C
## 2*sqrt(500*(t + 2)**2 + 15)
rate = diff(C, t)
rate
## 2*(500*t + 1000)/sqrt(500*(t + 2)**2 + 15)
rate.evalf(subs = {t : 4}) * 1000
## 44702.7372882889