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. Be sure to show all of your work and the composed function you used.
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To find the cost of operating the cannery for 32 hours using composition of functions, we need to substitute the value of 32 hours (h) into the function for pounds of apples (P(h)), and then substitute the result into the function for the processing cost (C(n)).
1. Start with the function for pounds of apples: P(h) = 375h
2. Substitute 32 for h: P(32) = 375 * 32
- Multiply 375 by 32: P(32) = 12,000
- Therefore, the number of pounds of apples that can be processed in 32 hours is 12,000 pounds.
3. Next, we use the result from step 2 in the function for the processing cost: C(n) = 0.35n + 1000
4. Substitute 12,000 for n: C(12,000) = 0.35 * 12,000 + 1000
- Multiply 0.35 by 12,000: C(12,000) = 4,200 + 1000
- Add 4,200 and 1,000: C(12,000) = 5,200
- Therefore, the cost of operating the cannery for 32 hours is $5,200.
The composed function used is C(P(32)), where P(32) represents the number of pounds of apples that can be processed in 32 hours.