Question 184·200 Super-Hard SAT Math Questions·Advanced Math

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After a valve is adjusted, the water level in a tank begins to approach a stable height. The equation

h(t)=40-15(0.97)^{t-5}

gives the water level $h(t)$ , in centimeters, as a function of time $t$ , in minutes, where $t \ge 5$ . Which choice is the best interpretation of 0.97 in this context?

In a shifted exponential model, the constant outside the power sets the level the output approaches. To interpret the base, identify the gap from that level and check how it changes when the exponent increases by one. Desmos can compare the gap at two neighboring times. The trap is applying the base to the entire output instead of its changing part.

Hints

Hint 1
A baseline is the height the model approaches. The equation has a fixed $40$ outside the exponential term. What would you subtract from $40$ to measure the water’s distance below it?
Hint 2
In an exponential model, increasing the exponent by $1$ multiplies the changing quantity by the base. How much does $t$ have to increase for the exponent $t-5$ to increase by $1$ ?

Step-by-step

Compare the gap at neighboring minutes
Step 1
Identify the changing quantity
The model subtracts a positive amount from $40$ , so the water level is below that stable height. Call its distance below $40$ the gap:
$g(t)=40-h(t)=15(0.97)^{t-5}.$
Step 2
Find the gap at minute 5
In Desmos, use $x$ for the problem’s time $t$ . Type $h(x)=40-15(0.97)^{x-5}$ , then $g(x)=40-h(x)$ , then $g(5)$ . Desmos shows $15$ : at minute $5$ , the water is 15 centimeters below $40$ .
Step 3
Find the gap one minute later
Add $g(6)$ in Desmos. It shows $14.55$ , the gap one minute later. The gap got smaller, even though the water level rose toward $40$ .
Step 4
Interpret the multiplier
Add $\frac{g(6)}{g(5)}$ . Desmos shows $0.97$ . Each one-minute increase in $t$ adds $1$ to the exponent, so this multiplication repeats every minute: $g(t+1)=0.97g(t)$ . Each minute, $0.97$ multiplies the gap below $40$ centimeters, not the whole water level. Choice B.

Question 184·200 Super-Hard SAT Math Questions·Advanced Math

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