Super-Hard SAT Math Question 86 | Free SAT Prep | Aniko

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Question 86·200 Super-Hard SAT Math Questions·Advanced Math

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f(t)=a\cdot 3^{\frac{2t-5}{6}}

The given function $f(t)$ models the amount of money in an investment account $t$ years after the account is opened, where $a$ is a positive constant.

How many months will it take for the amount in the account to increase by $700\%$ from its value when the account is opened (that is, from $f(0)$ )? Round to the nearest whole month.

Convert the percent increase into a multiplier and set up the ratio $\frac{f(t)}{f(0)}=\text{multiplier}$ so constants cancel. Simplify the exponent carefully, then solve the resulting exponential equation (often with logarithms when the multiplier is not a perfect power of the base). Finish by converting years to months and rounding only at the end.

Hints

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Start from the initial amount

The comparison is to the opening value, which is $f(0)$ .

Turn 700% increase into a multiplier

A $700\%$ increase means the new amount is $1+7=8$ times the starting amount.

Use a ratio to simplify the exponential

Consider $\frac{f(t)}{f(0)}$ . The constant $a$ will cancel, and the exponent offset will subtract out.

Expect to use logarithms

If you get an equation like $3^{t/3}=8$ , you can solve it with $\log_3$ or with change of base using calculator logs.

Desmos Guide

Graph the growth ratio

In Desmos, enter

$y=3^{x/3}$

This equals $\dfrac{f(x)}{f(0)}$ because the constant $a$ and the $-5/6$ exponent shift cancel in the ratio.

Graph the target multiplier

Enter the horizontal line $y=8$ .

Read the time in years

Find the intersection point and read its $x$ -value (years).

Convert to months and round

Multiply the intersection $x$ -value by $12$ to convert to months, then round to the nearest whole month and match an answer choice.

Step-by-step Explanation

Translate the percent increase into a multiplier

A $700\%$ increase means the final amount is

f(0)+7f(0)=8f(0).

So we want $f(t)=8f(0)$ .

Use a ratio so the constant cancels

Compute the ratio:

\frac{f(t)}{f(0)}=\frac{a\cdot 3^{\frac{2t-5}{6}}}{a\cdot 3^{\frac{2(0)-5}{6}}} =3^{\frac{2t-5}{6}-\left(-\frac{5}{6}\right)} =3^{\frac{2t}{6}}=3^{t/3}.

Since $f(t)=8f(0)$ , we have $\frac{f(t)}{f(0)}=8$ , so $3^{t/3}=8$ .

Solve for $t$ using logarithms

Take log base $3$ of both sides:

\frac{t}{3}=\log_3(8) \quad\Rightarrow\quad t=3\log_3(8).

Using change of base,

\log_3(8)=\frac{\log(8)}{\log(3)}\approx \frac{0.9031}{0.4771}\approx 1.8928,

t\approx 3(1.8928)=5.6784\text{ years}.

Convert years to months and round

Convert to months:

5.6784\times 12\approx 68.14\text{ months}.

Rounded to the nearest whole month, it will take $68$ months.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation