SAT Nonlinear Functions Question 105 | Free SAT Prep | Aniko

00:00

Question 105·Hard·Nonlinear Functions

0 / 234

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 $r$ ?

When an exponential model uses time in one unit (like years) but the question asks for a rate over a different interval (like months), first convert the new time interval into a fraction of the original unit. Raise the base (growth factor per original unit) to that fractional power to get the new growth factor, then subtract 1 and convert to a percent. On the SAT, look for opportunities to simplify exponents by rewriting the base as a convenient power (for example, recognizing $1.331 = (1.1)^3$ ), which can turn a tricky fractional exponent into an easy calculation.

Hints

Click me!

Focus on the time units

The exponent $t$ in $A(t) = 8(1.331)^t$ is in years, but the question asks about growth every 8 months. How many years is 8 months?

Express 8 months as part of a year

Write 8 months as a fraction of a year and think about how to represent the growth over that fraction of a year using the base $1.331$ .

Use exponent rules to simplify

You need $(1.331)^{2/3}$ . Try rewriting $1.331$ as a power of a simpler number (like $1.1$ ) so that raising it to the $2/3$ power becomes easier.

Turn the growth factor into a percent

Once you have the growth factor for 8 months, subtract 1 and convert the result to a percentage to match it to one of the answer choices.

Desmos Guide

Compute the 8-month growth factor

In Desmos, type (1.331)^(2/3) to find the growth factor over 8 months (since 8 months is $2/3$ of a year). Note the decimal output.

Convert the factor to a growth rate

In a new line, type ((1.331)^(2/3) - 1) to get the increase as a decimal, then type ((1.331)^(2/3) - 1)*100 to convert that decimal increase into a percentage.

Match with an answer choice

Look at the percentage value from the last expression and select the answer choice whose percent is equal to (or closest to) that value.

Step-by-step Explanation

Interpret the exponential model

The function is

A(t) = 8(1.331)^t

Here $t$ is measured in years, and $1.331$ is the growth factor per 1 year.

That means: each time $t$ increases by 1 year, the number of users is multiplied by $1.331$ (a $33.1\%$ increase per year).

Relate years to 8-month periods

We want the growth over 8 months, not 1 year.

1 year $=12$ months.
8 months is a fraction of a year:

\frac{8}{12} = \frac{2}{3} \text{ of a year}

So, 8 months corresponds to $\tfrac{2}{3}$ of a year. If the yearly growth factor is $1.331$ , then the 8‑month growth factor will be $1.331$ raised to the $\tfrac{2}{3}$ power:

\text{8-month factor} = (1.331)^{2/3}.

Simplify the power using a nicer base

Notice that

$1.1^2 = 1.21$
$1.21 \times 1.1 = 1.331$

So $1.331 = (1.1)^3$ .

Substitute this into the expression for the 8‑month factor:

(1.331)^{2/3} = \bigl((1.1)^3\bigr)^{2/3}.

Use the exponent rule $(a^b)^c = a^{bc}$ :

\bigl((1.1)^3\bigr)^{2/3} = (1.1)^{3\cdot(2/3)} = (1.1)^2.

So the growth factor for each 8‑month period is $(1.1)^2$ .

Convert the growth factor to a percent increase

Compute $(1.1)^2$ :

(1.1)^2 = 1.21.

A growth factor of $1.21$ means the new amount is $1.21$ times the old amount: it has increased by $1.21 - 1 = 0.21$ , or $21\%$ .

So $r = 21\%$ , which corresponds to choice C.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation