In a laboratory experiment, the mass of a radioactive substance is modeled by the equation where is the mass, in milligrams, and is the time, in days, after the initial measurement. Which number represents the initial mass of the substance, in milligrams?

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SAT Nonlinear Functions Question 41 | Free SAT Prep | Aniko

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Question 41·Medium·Nonlinear Functions

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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.

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Step-by-step Explanation

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