To calculate the number of weeks it will take for the awareness to grow from a beginning value of 3% to an ending value of 75% with a compounded weekly growth rate of 5%, we will use the compound growth formula:
E = B (1 + R) T
where:
- E is the ending value (0.75, or 75%),
- B is the beginning value (0.03, or 3%),
- R is the rate of growth (0.05, or 5% per week), and
- T is the time in weeks we want to solve for.
To solve for T, we need to rearrange the formula:
E = B (1 + R) T
E/B = (1 + R) T
ln(E/B) = ln((1 + R) T)
ln(E/B) = T * ln(1 + R)
T = ln(E/B) / ln(1 + R)
Now we plug in the values:
E = 0.75
B = 0.03
R = 0.05
T = ln(0.75 / 0.03) / ln(1 + 0.05)
We can use a calculator to find the natural logarithms:
T = ln(25) / ln(1.05)
Calculating the natural logarithms:
T ≈ ln(25) / ln(1.05) ≈ 3.21887582487 / 0.0487901641694 ≈ 65.9738844955 weeks
Since you can't have a fraction of a week in this context, we will round up to the next whole number:
T ≈ 66 weeks
So, it will take approximately 66 weeks to reach 75% awareness. The answer is b) 66 weeks.
