SAT Nonlinear Functions Question 154 | Free SAT Prep | Aniko

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Question 154·Easy·Nonlinear Functions

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A sample of a chemical has an initial mass of 180 grams. Every 8 years, the mass decreases by 25%. Which equation models the mass $M$ , in grams, remaining $t$ years after the initial measurement?

For percent growth or decay over fixed time intervals, immediately think exponential: use $M = a \cdot (1 + r)^{t/p}$ , where $a$ is the initial amount, $r$ is the rate as a decimal (negative for decay), and $p$ is the length of one interval. Compute the decay factor $1 - r$ , make the exponent $t$ divided by the interval length, and quickly eliminate choices with the wrong base (greater than 1 for decay, or far too small) or the wrong exponent (like $8t$ instead of $t/8$ ).

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Percent decrease to multiplier

If the mass decreases by 25% each period, what percent remains after each period? Convert that remaining percent to a decimal.

Form of the equation

Because the same percentage change happens every 8 years, think of an equation of the form $M = a \cdot b^{\text{number of periods}}$ . Identify $a$ and $b$ from the context.

What should the exponent represent?

The exponent should be how many times the 8-year change happens. In $t$ years, how many 8-year intervals is that?

Check with a simple time value

After 8 years, the mass should be $180$ multiplied by your decay factor once. Which equation gives that value when $t = 8$ ?

Desmos Guide

Enter the four options as functions

In Desmos, type each option as a separate function, for example:

$f(x) = 180(1.25)^{x/8}$
$g(x) = 180(0.75)^{8x}$
$h(x) = 180(0.25)^{x/8}$
$k(x) = 180(0.75)^{x/8}$ Use $x$ in place of $t$ .

Check the value after one 8-year period

Use the table feature or click on $x = 8$ for each graph. The correct model is the one whose $y$ -value at $x = 8$ equals $180$ multiplied by the remaining fraction after one 25% decrease (that is, $75\%$ of 180).

Confirm long-term behavior

Optional: Look at larger $x$ -values (like $x = 16, 24$ ). The correct graph should keep getting smaller smoothly, always positive, and its values at multiples of 8 years should match repeated multiplication by the same factor each time.

Step-by-step Explanation

Recognize exponential decay

Each equal time interval (every 8 years) the mass changes by the same percentage (decreases by 25%). That means this is an exponential situation, not linear.

A general exponential model looks like

$M = a \cdot b^{\text{number of periods}}$

where:

$a$ is the initial amount,
$b$ is the growth/decay factor per period,
"number of periods" is how many times that change happens.

Find the decay factor per 8-year period

"Decreases by 25%" means the sample loses 25% of its mass but keeps 75% each period.

100% $-\,$ 25% $= 75\%$
As a decimal, $75\% = 0.75$

So the mass is multiplied by $0.75$ every 8 years. That makes $b = 0.75$ in the exponential model.

Express the number of 8-year periods in terms of t

The exponent should be the number of times the 8-year change happens.

In $t$ years, the number of 8-year periods is $\dfrac{t}{8}$ .

So the exponent in the model should be $t/8$ , not $8t$ or just $t$ .

Write the full exponential model

Now put everything into the exponential form $M = a \cdot b^{\text{number of periods}}$ :

Initial mass $a = 180$ grams,
Decay factor per 8 years $b = 0.75$ ,
Number of 8-year periods $= t/8$ .

So the model is

M = 180\,(0.75)^{t/8}

This matches the correct answer choice.

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation