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Chapter 0: Problem 5

Fueled by the promotion of testosterone as an antiaging elixir, use of the hormone by middle-aged and older men grew dramatically. The total number of prescriptions for testosterone from 1999 through 2002 is given by \(N(t)=-35.8 t^{3}+202 t^{2}+87.8 t+648 \quad 0 \leq t \leq 3\) where \(N(t)\) is measured in thousands and \(t\) is measured in years with \(t=0\) corresponding to \(1999 .\) Find the total number of prescriptions for testosterone in 1999,2000 , 2001, and \(2002 .\)

Short Answer

Expert verified
The total number of prescriptions for testosterone in 1999, 2000, 2001, and 2002 are 648,000, 902,000, 1,312,000, and 1,756,000 respectively.

Step by step solution

01

Understand the given function

We are given the function: \[ N(t)=-35.8t^3 +202t^2 +87.8t +648 \quad 0 \leq t \leq 3 \] Where N(t) is the total number of prescriptions in thousands, and t is time in years, with t=0 corresponding to 1999.
02

Find the total number of prescriptions for t=0 (1999)

To find the total number of prescriptions in 1999, substitute t=0 in the given function: \[ N(0)=-35.8(0)^3 +202(0)^2 +87.8(0) +648 \] \[ N(0)=648 \] So, the total number of prescriptions in 1999 is 648,000 (since N(t) is in thousands).
03

Find the total number of prescriptions for t=1 (2000)

To find the total number of prescriptions in 2000, substitute t=1 in the given function: \[ N(1)=-35.8(1)^3 +202(1)^2 +87.8(1) +648 \] \[ N(1)=-35.8 +202 +87.8 +648 \] \[ N(1)=902 \] So, the total number of prescriptions in 2000 is 902,000.
04

Find the total number of prescriptions for t=2 (2001)

To find the total number of prescriptions in 2001, substitute t=2 in the given function: \[ N(2)=-35.8(2)^3 +202(2)^2 +87.8(2) +648 \] \[ N(2)=-35.8(8) +202(4) +87.8(2) +648 \] \[ N(2)=1312 \] So, the total number of prescriptions in 2001 is 1,312,000.
05

Find the total number of prescriptions for t=3 (2002)

To find the total number of prescriptions in 2002, substitute t=3 in the given function: \[ N(3)=-35.8(3)^3 +202(3)^2 +87.8(3) +648 \] \[ N(3)=-35.8(27) +202(9) +87.8(3) +648 \] \[ N(3)=1756 \] So, the total number of prescriptions in 2002 is 1,756,000. In conclusion, the total number of prescriptions for testosterone in 1999, 2000, 2001, and 2002 are 648,000, 902,000, 1,312,000, and 1,756,000 respectively.

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