The function is defined by , where . If is greater than , then when increases by 1, increases by . Which choice gives the value of ?

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Super-Hard SAT Math Question 72 | Free SAT Prep | Aniko

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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 $p$ ?

When an exponential has a fractional exponent like $t/3$ , focus on how much the exponent changes for the stated input change. Use ratios like $\frac{g(t+a)}{g(t)}$ to cancel constants and isolate the base’s power. Convert percent statements to multipliers (e.g., $72.8\%$ greater $\rightarrow 1.728$ ), use roots to move from a larger step (like $+3$ in $t$ ) to a $+1$ step, then convert the final multiplier back to a percent with $(m-1)\times 100$ .

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Convert “percent greater” to a ratio

If a value is $72.8\%$ greater, then the new value is $1+0.728$ times the old value.

Use a ratio to eliminate constants

Form $\frac{g(3)}{g(0)}$ so the 500 cancels and you can solve for $b$ .

Go from a 3-step change to a 1-step change

Once you know $\frac{g(3)}{g(0)}$ , take the cube root to get the multiplier for increasing $t$ by 1.

Multiplier to percent

If the multiplier is $m$ , then the percent increase is $(m-1)\times 100$ .

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Compute the 3-step multiplier from the percent statement

Enter M = 1 + 0.728 to represent the multiplier from $t=0$ to $t=3$ .

Convert the 3-step multiplier to the 1-step multiplier

Enter m = M^(1/3) to compute the multiplier for increasing $t$ by 1.

Convert the multiplier to a percent

Enter p = (m - 1) * 100 and read the value of p (this is the correct choice).

Step-by-step Explanation

Translate the given percent increase into a multiplier

“ $72.8\%$ greater than” means

\frac{g(3)}{g(0)} = 1 + 0.728 = 1.728.

Use the function to relate that ratio to $b$

Compute the ratio using the definition of $g$ :

\frac{g(3)}{g(0)}=\frac{500\,b^{3/3}}{500\,b^{0/3}}=\frac{500\,b}{500}=b.

So $b=1.728$ .

Find the multiplier for increasing $t$ by 1

A 1-unit increase in $t$ changes the exponent by $\frac{1}{3}$ , so the growth factor is

\frac{g(t+1)}{g(t)} = b^{1/3}.

With $b=1.728$ ,

b^{1/3}=\sqrt[3]{1.728}=1.2

(since $1.2^3=1.728$ ).

Convert the multiplier to a percent increase

A multiplier of $1.2$ corresponds to a percent increase of

(1.2-1)\times 100 = 20\%.

So $p=20\%$ .

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Step-by-step Explanation

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