Question 418·Hard·Math

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The function

A(t)=8\bigl(1.331\bigr)^{t}

models the number of active users, in thousands, on a new app $t$ years after launch.

According to the model, the number of active users is predicted to increase by $r\%$ every 8 months.

What is the value of $r$ ?

An exponential model multiplies by the same factor over equal time intervals. When $t$ is measured in years, the base is the one-year factor, even if the question asks about months. Convert the requested interval to years, use Desmos to compare the model’s outputs before and after that interval, then turn the factor into a percent increase. Dividing the yearly percent by $12$ ignores compounding.

Hints

Hint 1
The exponent tracks time in years here. Convert 8 months to a fraction of a year before using the model. What value of $t$ represents 8 months after launch?
Hint 2
A multiplier is the later amount divided by the earlier amount. Compare the model’s output after 8 months with its output at launch. That ratio gives the eight-month factor, not the number of users.
Hint 3
A growth factor includes the whole original amount, represented by $1$ . Subtract $1$ to keep only the increase, then multiply by $100$ to express it as a percent.

Step-by-step

Compare model outputs eight months apart
Step 1
Express eight months in the model’s units
The model measures $t$ in years, so 8 months is:
$t=\frac{8}{12}=\frac{2}{3}.$
The exponent rises by $1$ over a full year. So $1.331$ is the one-year factor, not the eight-month factor. Convert the requested interval to the exponent’s units before finding its factor.
Step 2
Find the eight-month multiplier
$A(0)$ is the launch count, and $A(2/3)$ is the count 8 months later. Their ratio is the multiplier for that interval. Type $A(t)=8(1.331)^t$ , then $A(2/3)/A(0)$ . Desmos shows $1.21$ . Dividing cancels the starting count of $8$ thousand, leaving the eight-month factor.
Step 3
Turn the multiplier into the requested increase
The factor $1.21$ includes the original amount, represented by $1$ . To get only the increase, type $100(1.21-1)$ . Desmos prints $21$ , so the model predicts a $21\%$ increase in active users every 8 months. Choice C.

Question 418·Hard·Math

Strategy

Hints

Step-by-step

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Strategy

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