Answered

10: Advertisers want to create
starts with 3% awareness and grows awareness by 5% per week (compounded), how long will it take to reach 75% awareness? The formula
for compound growth is E-Bx (1+R), where E is ending value, B is beginning value, R is rate of growth, and T is the amount of time
.
a) 12 weeks
b) 66 weeks
c)
83 weeks
d)
94 weeks

Verified Answer :

GinnyAnswer

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.

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