Question 72·200 Super-Hard SAT Math Questions·Advanced Math

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The function $g$ is defined by $g(t)=500(b)^{t/3}$ , where $b>1$ .

If $g(3)$ is $72.8\%$ greater than $g(0)$ , then when $t$ increases by 1, $g(t)$ increases by $p\%$ . Which choice gives the value of $p$ ?

In an exponential model, the exponent tells you how many intervals the base covers. If a percent increase is given across several input units, turn it into a multiplier first. Use a Desmos regression to find an unknown base, then evaluate the model to get the one-unit multiplier. Don't divide the overall percent increase by the interval length: growth compounds.

Hints

Hint 1
“Greater than” adds to the whole starting amount. A $72.8\%$ increase makes the new value $100\%+72.8\%$ of the old value. What multiplier does that give?
Hint 2
The exponent $t/3$ rises by $1$ when $t$ rises by $3$ . So the base $b$ is the multiplier for three units, not one. How many one-unit multipliers must combine to make $b$ ?
Hint 3
If $r$ is the one-unit multiplier, three equal steps give $r^3=b$ . After finding $r$ , subtract $1$ to get only the increase, rather than the whole new amount.

Step-by-step

Find the three-unit factor, then the one-unit percent
Step 1
Turn “greater than” into a multiplier
The $72.8\%$ increase is added to the original $100\%$ of $g(0)$ , so write:
$g(3)=(1+0.728)g(0).$
Combine the parts:
$g(3)=1.728g(0).$
Using $0.728$ alone would give only the amount added, not the new value.
Step 2
Find what the base multiplies
The exponent $t/3$ goes from $0$ to $1$ as $t$ goes from $0$ to $3$ , so $b$ is the three-unit multiplier. Type $500b^{3/3}\sim1.728(500b^{0/3})$ : the left side is $g(3)$ , and the right side is $1.728g(0)$ . Desmos uses $\sim$ to find the value of $b$ that makes them match. Under PARAMETERS, it shows $b=1.728$ .
Step 3
Find the multiplier for one unit
Three one-unit steps must multiply to $b$ , so each step has multiplier $b^{1/3}$ . Take the cube root of the factor, not one-third of the percent increase. Type $g(t)=500(1.728)^{t/3}$ , then $g(1)/g(0)$ . Desmos prints $1.2$ . This ratio compares the output after one unit with the starting output, so $1.2$ is the one-unit multiplier.
Step 4
Convert the multiplier to a percent increase
The multiplier $1.2$ includes the original amount. To find only the increase, type $100\left(g(1)/g(0)-1\right)$ . Desmos prints $20$ , so $g(t)$ increases by $20\%$ whenever $t$ increases by $1$ . Choice B.

Question 72·200 Super-Hard SAT Math Questions·Advanced Math

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