Question 162·Medium·Math

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In a laboratory experiment, the mass of a radioactive substance is modeled by the equation

m = 56(0.92)^t

where $m$ is the mass, in milligrams, and $t$ is the time, in days, after the initial measurement. Which number represents the initial mass of the substance, in milligrams?

For exponential models on the SAT, recognize the standard form $y = a b^t$ , where $a$ is the initial amount and $b$ is the growth or decay factor. When a question asks for the initial value, either identify the coefficient $a$ directly or, if unsure, plug in $t=0$ and simplify; both methods are quick and avoid unnecessary calculation.

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Focus on the meaning of "initial"

"Initial" refers to the amount at the very beginning of the experiment, before any time has passed.

Relate "initial" to the variable $t$

At the very beginning of the experiment, what is the value of $t$ in days?

Use the model to find the starting amount

Substitute that value of $t$ into $m = 56(0.92)^t$ and simplify the expression. The result is the initial mass.

Desmos Guide

Enter the exponential model

In Desmos, type m = 56*(0.92)^t to graph the relationship between $m$ and $t$ .

Use a table to evaluate at $t=0$

Click on the gear icon next to the equation and choose "Table" (or add a table with the plus button and link it to the equation). In the $t$ column, enter 0. Look at the corresponding $m$ value in the table; this is the initial mass.

Step-by-step Explanation

Understand what "initial mass" means

"Initial" means at the start of the experiment, which is when no time has passed.

So the initial mass is the value of $m$ when $t=0$ days.

Substitute $t=0$ into the model

The mass is modeled by $m = 56(0.92)^t$ .

To find the initial mass, plug in $t=0$ :

$m = 56(0.92)^0$ .

Any nonzero number raised to the zero power equals 1, so $(0.92)^0 = 1$ .

So the expression becomes $m = 56 \times 1$ .

Find the initial mass and match the answer choice

Now simplify $56 \times 1$ to get $m = 56$ .

This means the initial mass of the substance is 56 milligrams, which corresponds to choice D) 56.

Question 162·Medium·Math

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation