The total weight of the package was 20 pounds, given the total shipping cost was $3.45, using the total cost function, C(P) = 1.65 + 0.12(P - 5), where P is the weight of the package in pounds.
In the question, we are given that a package weighs P pounds, P being a whole number. To ship this package by express costs $1.65 for the first five pounds and 12c for each additional pound. The total shipping cost was $3.45.
We are asked for the package weight in pounds.
First, we design a general cost function.
The cost is given to be $1.65 for the first 5 pounds, and 12c for each additional pound.
The weight of the package is given to be P pounds.
We assume the total cost function to be C(P), a function of the weight of the package, P pounds.
Thus, the total cost function can be shown as the sum of $1.65 and 12c or $0.12 multiplied with (P - 5), as it is the cost for additional pounds after 5 pounds.
Thus, the total cost function, C(P) = 1.65 + 0.12(P - 5).
Now, we are given that the total shipping cost was $3.45.
Thus, to find the weight of the package, we equate the total cost function, C(P) to 3.45, to get:
1.65 + 0.12(P - 5) = 3.45,
or, 1.65 + 0.12P - 0.6 = 3.45,
or, 0.12P + 1.05 = 3.45,
or, 0.12P = 3.45 - 1.05,
or, 0.12P = 2.40,
or, P = 2.40/0.12 = 20.
Thus, the total weight of the package was 20 pounds, given the total shipping cost was $3.45, using the total cost function, C(P) = 1.65 + 0.12(P - 5), where P is the weight of the package in pounds.
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