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.

In an exponential growth problem, the unknown time sits in the exponent. First turn the stated percent increase into a target multiple of the opening amount, then compare the function with its value at time zero. That comparison cancels an unknown scale factor. Use a Desmos regression to find the time, and convert units before rounding.

Hints

Hint 1
A percent increase keeps the original and adds a percent of it. A $700\%$ increase adds seven copies of the opening balance. How many copies are in the new balance?
Hint 2
The initial value is $f(0)$ , not automatically $a$ : the exponent isn't zero when $t=0$ . Compare $f(t)$ with $f(0)$ so the unknown positive factor $a$ cancels.
Hint 3
An exponential equation has its unknown in an exponent. Desmos can find the year by using $t_1$ as the unknown and $\sim$ in place of $=$ . Convert that year to months before rounding.

Step-by-step

Compare with the opening balance, then solve in Desmos
Step 1
Turn the increase into a target balance
A $700\%$ increase adds seven copies of the opening balance on top of the original copy. So the target is $8$ times $f(0)$ , not $7$ times:
$f(t)=f(0)+7f(0)=8f(0).$
Step 2
Compare the balance with its starting value
Substitute $0$ for $t$ in the denominator. Since $a$ is positive, it cancels when you divide. For powers of the same base, subtract the exponents:
$\frac{f(t)}{f(0)}=\frac{a3^{(2t-5)/6}}{a3^{(2(0)-5)/6}}$ $\frac{f(t)}{f(0)}=\frac{a3^{(2t-5)/6}}{a3^{-5/6}}$ $\frac{f(t)}{f(0)}=3^{(2t-5)/6-(-5/6)}$ $\frac{f(t)}{f(0)}=3^{t/3}.$
The $-5$ affects both balances, so it disappears from their ratio, meaning one balance divided by the other.
Step 3
Enter a sample account in Desmos
Because the ratio doesn't depend on $a$ , set $a=1$ for the calculator. Type $a=1$ , define $f(t)=a\cdot3^{(2t-5)/6}$ , then type $f(0)$ . Desmos shows $f(0)\approx0.4003$ for this sample account. The actual opening balance needn't be $0.4003$ ; the time to grow eightfold is the same.
Step 4
Find the time in years
Type $f(t_1)\sim8f(0)$ . The $\sim$ tells Desmos to find the input $t_1$ that makes the two sides match. Under PARAMETERS, it shows $t_1\approx5.67837$ . Since the problem measures $t$ in years, this result is years, not months.
Step 5
Convert to months and round
Type $12t_1$ because a year has $12$ months. Desmos prints about $68.1404$ . Round the months, not the years: the account takes $68$ months to increase by $700\%$ from its opening value. Choice B.

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

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