Question 1096406: The number of pounds of apples a cannery can produce and the processing cost are P(h) = 375h and C(n) = 0.35n + 1000 where P(h) is the number of pounds of apples that can be processed in h hours and C(n) is the cost of processing n pounds of apples. Use composition of functions to find the cost of operating the cannery 32 hours
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Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
The equation P(h) = 375h represents the number of pounds of apples that the cannery can produce.
Saying 375h is the same as 375*h which means "375 times h"
The basic idea is that h represents the placeholder for any positive real number. For example, if h = 2, then 2 hours pass by and 375*h = 375*2 = 750 pounds of apples are processed.
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We can replace the P in P(h) = 375h with n to get n(h) = 375h. The n matches with the n in the other equation C(n) = 0.35n + 1000
Saying n(h) = 375h is the same as n = 375h
Since n = 375h, we can replace every copy of n in the second equation with 375h
C(n) = 0.35n + 1000
C(n) = 0.35*n + 1000
C(n) = 0.35*(n) + 1000
C(n) = 0.35*(n) + 1000
C(375h) = 0.35*(375h) + 1000 ... note how the n terms have been replaced with 375h
C(P(h)) = 0.35*(375*h) + 1000 ... note how the 375h on the left side is replaced with P(h)
C(P(h)) = 0.35*(375*h) + 1000
C(P(h)) = (0.35*375)*h + 1000
C(P(h)) = 131.25*h + 1000
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Now plug in h = 32
C(P(h)) = 131.25*h + 1000
C(P(h)) = 131.25*h + 1000
C(P(32)) = 131.25*32 + 1000 ... every h is replaced with 32
C(P(32)) = 131.25*32 + 1000
C(P(32)) = 4200 + 1000
C(P(32)) = 5200
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Therefore the final answer is 5200 meaning that the total operating cost is $5,200 if you run the cannery for 32 hours.
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