Here is the given information: The owner of a commercial fishing operation decides to upgrade the hydraulic pumps, hydraulic lines, and the drum drive to a new system that is supposed to save costs over time. The projected savings rate for the new equipment is given by S′(x)=225−x2S′(x)=225−x2 where xx is the number of years the machinery will be used. The rate of additional costs to run and maintain the new equipment is expected to be C′(x)=x2+25x+150C′(x)=x2+25x+150 Both functions have units of thousands of dollars per year. Recall, from the Fundamental Theorem of Calculus (see June 9th presentation and Section 13.5 in the text): f(b)−f(a)=∫baf′(x)dxf(b)−f(a)=∫abf′(x)dx which yields f(b)=f(a)+∫baf′(x)dxf(b)=f(a)+∫abf′(x)dx . Calculate the value of f(10)f(10) when a=0a=0. Find the maximum value of f(x)=f(x)= net savings xx years after the installation of the hydraulic system. In what year does this maximum occur? What geometric quantity (area or distance) does this maximum correspond to? What does the maximum mean in the context of this problem?
Here is the given information:
The owner of a commercial fishing operation decides to upgrade the hydraulic pumps, hydraulic lines, and the drum drive to a new system that is supposed to save costs over time. The projected savings rate for the new equipment is given by
S′(x)=225−x2S′(x)=225−x2
where xx is the number of years the machinery will be used.
The rate of additional costs to run and maintain the new equipment is expected to be
C′(x)=x2+25x+150C′(x)=x2+25x+150
Both functions have units of thousands of dollars per year.
Recall, from the Fundamental Theorem of Calculus (see June 9th presentation and Section 13.5 in the text):
f(b)−f(a)=∫baf′(x)dxf(b)−f(a)=∫abf′(x)dx
which yields
f(b)=f(a)+∫baf′(x)dxf(b)=f(a)+∫abf′(x)dx .
Calculate the value of f(10)f(10) when a=0a=0.
Find the maximum value of f(x)=f(x)= net savings xx years after the installation of the hydraulic system.
In what year does this maximum occur?
What geometric quantity (area or distance) does this maximum correspond to?
What does the maximum mean in the context of this problem?
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