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A woman who has recovered from a serious illness begins a diet regimen designed to get her back to a healthy weight. She currently weighs 107 pounds. She hopes each week to multiply her weight by 1.06.

(a) Find a formula for an exponential function that gives the woman's weight \( w \), in pounds, after \( t \) weeks on the regimen.
\[ w = 107 \times 1.06^t \]

(b) How long will it be before she reaches her normal weight of 125 pounds? (Round your answer to two decimal places.)
\(\text{weeks}\)

Answer :

Let's solve the given problem step-by-step:

Part (a): Finding the Formula

The woman starts with a weight of 107 pounds. She hopes her weight multiplies by 1.06 each week. This describes an exponential growth model.

The general formula for an exponential function can be written as:

[tex]w(t) = w_0 \cdot r^t[/tex]

Where:

  • [tex]w(t)[/tex] is the weight after [tex]t[/tex] weeks.
  • [tex]w_0[/tex] is the initial weight.
  • [tex]r[/tex] is the growth rate.

For this problem, [tex]w_0 = 107[/tex] (her current weight), and [tex]r = 1.06[/tex] (the growth factor per week). Therefore, the exponential function is:

[tex]w(t) = 107 \cdot 1.06^t[/tex]

This formula gives us the woman's weight [tex]w[/tex] in pounds after [tex]t[/tex] weeks.

Part (b): Finding How Long It Will Take to Reach 125 Pounds

We need to find the time [tex]t[/tex] when her weight reaches 125 pounds, which means solving the equation:

[tex]125 = 107 \cdot 1.06^t[/tex]

Divide both sides by 107 to isolate the exponential expression:

[tex]\frac{125}{107} = 1.06^t[/tex]

Calculating the left side gives approximately:

[tex]1.1682 \approx 1.06^t[/tex]

To solve for [tex]t[/tex], take the logarithm of both sides. Using the natural logarithm [tex]\ln[/tex]:

[tex]\ln(1.1682) = \ln(1.06^t)[/tex]

Using the power rule for logarithms, [tex]\ln(a^b) = b \cdot \ln(a)[/tex], this becomes:

[tex]\ln(1.1682) = t \cdot \ln(1.06)[/tex]

Solve for [tex]t[/tex]:

[tex]t = \frac{\ln(1.1682)}{\ln(1.06)}[/tex]

Calculating this gives:

[tex]t \approx \frac{0.1552}{0.0583} \approx 2.66[/tex]

Therefore, it will take approximately 2.66 weeks for the woman to reach her normal weight of 125 pounds.

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Rewritten by : Barada