Chapter 4.8, Problem 36E

Solution

a.

Courtney’s weight at the start and at the end of two years.

Courtney’s weight at the start and at the end of two years is 220 and 184 pounds respectively.

Given:

Given, Courtney tried several different diets over a two-year period in an attempt to lose weight. She found that her weight W followed a fluctuating curve that could be modeled by the function $W=220-1.5t+9.81\sin \left(\pi t/4\right)$ , where t was measured in months after January 1 of the first year and W was measured in pounds.

Calculation:

At the start of the two years, t is 0. And at the end of two years i.e., 24 months, t is 24.

Plugging the values:

  $\begin{array}{l}W\left(0\right)=220-1.5\left(0\right)+9.81\sin \left(\pi \left(0\right)/4\right)\\ W\left(0\right)=220-0+9.81\sin \left(0\right)\\ W\left(0\right)=220-0+9.81\left(0\right)\\ W\left(0\right)=220-0+0\\ W\left(0\right)=220\end{array}$

  $\begin{array}{l}W\left(24\right)=220-1.5\left(24\right)+9.81\sin \left(\pi \left(24\right)/4\right)\\ W\left(24\right)=220-36+9.81\sin \left(6\pi \right)\\ W\left(24\right)=220-36+9.81\left(0\right)\\ W\left(24\right)=220-36+0\\ W\left(24\right)=184\end{array}$

Courtney’s weight at the start and at the end of two years is 220 and 184 pounds respectively.

Conclusion:

Courtney’s weight at the start and at the end of two years is 220 and 184 pounds respectively.

b.

Courtney’s maximum weight during the two-year period.

Courtney’s maximum weight during the two-year period is 227 pounds.

Given:

Given, Courtney tried several different diets over a two-year period in an attempt to lose weight. She found that her weight W followed a fluctuating curve that could be modeled by the function $W=220-1.5t+9.81\sin \left(\pi t/4\right)$ , where t was measured in months after January 1 of the first year and W was measured in pounds.

Calculation:

Graphing the function for t between 0 and 24:

  

From the graph, it is seen that the maximum weight during the two-year period is approximately 227 pounds.

Conclusion:

Courtney’s maximum weight during the two-year period is 227 pounds.

c.

Courtney’s minimum weight during the two-year period.

Courtney’s minimum weight during the two-year period is 177 pounds.

Given:

Given, Courtney tried several different diets over a two-year period in an attempt to lose weight. She found that her weight W followed a fluctuating curve that could be modeled by the function $W=220-1.5t+9.81\sin \left(\pi t/4\right)$ , where t was measured in months after January 1 of the first year and W was measured in pounds.

Calculation:

Graphing the function for t between 0 and 24:

  

From the graph, it is seen that the minimum weight during the two-year period is approximately 177 pounds.

Conclusion:

Courtney’s minimum weight during the two-year period is 177 pounds.