suppose a company wants to introduce a new machine that will produce a rate of annual savings in dollars given by the function S'(x), where ex is the number of years of operation of the machine, while producing a rate of annual costs in dollars given by the function C'(x)
S’(x)=228-x^2, C’(x)= x^2+14/5x
a. For how many years will it be profitable to use this new machine?
b. What are the net total savings during the first year of use of the machine?
c. what are the net total savings over the entire period of use of the machine?
a. To determine for how many years it will be profitable to use the machine, we need to find the point at which the savings equal the costs, S'(x) = C'(x). This can be written as:
228 - x^2 = x^2 + 14/5x
228 = 2x^2 + 14/5x
0 = 2x^2 + 14/5x - 228
Solving this quadratic equation, we find x ≈ 9.19 years. Therefore, the machine will be profitable to use for approximately 9 years.
b. To find the net total savings during the first year of use of the machine, we need to find the difference between the savings and costs at x = 1:
S'(1) = 228 - 1^2 = 227
C'(1) = 1^2 + 14/5(1) = 1 + 14/5 = 19/5
Net total savings = S'(1) - C'(1) = 227 - 19/5 = 226.2
c. To find the net total savings over the entire period of use of the machine (9 years), we need to calculate the net total savings for each year and sum them up:
Net total savings = Σ(S'(x) - C'(x)) for x from 1 to 9
Net total savings = (227 - 19/5) + (226 - 39/5) + ... + (214 - 167/5)
This calculation will give you the net total savings over the entire period of use of the machine.