SAT Nonlinear Functions Question 147 | Free SAT Prep | Aniko

00:00

Question 147·Hard·Nonlinear Functions

0 / 234

A bacteria culture initially has 1,250 cells and doubles every 6 hours.

Which of the following equations can be used to determine the number of hours, $t$ , it will take the culture to reach 40,000 cells?

For exponential growth word problems, first identify three key pieces: the initial amount, the growth factor, and how often that factor is applied. Doubling means a factor of 2, and "every k hours" means the exponent should count how many such periods fit into $t$ , usually $t/k$ . Write a general model like $N(t) = \text{initial} \cdot 2^{t/k}$ , then set this expression equal to the target amount the problem asks about. Finally, match this structure carefully to the answer choices, watching for common traps like using $kt$ instead of $t/k$ in the exponent or using a percent-growth base (like $1.06$ ) when the situation describes doubling.

Hints

Click me!

Identify the type of model

The phrase "doubles every 6 hours" describes exponential growth. Think about equations where a starting amount is multiplied by a constant base raised to a power involving time.

Match the base and the exponent to the situation

If something doubles, what should the base of the exponent be? And if it doubles every 6 hours, how can you write the exponent so it counts how many 6-hour periods fit into $t$ hours?

Check which side has the starting amount and which has the target

At time $t=0$ , the amount is 1,250 cells, and you want to know when the amount reaches 40,000. Which number should be multiplied by the exponential expression, and which number should be alone on the other side of the equation?

Eliminate options that don't fit the story

Look for choices that: (1) start with 1,250 cells, (2) use a base of 2 (for doubling), and (3) use an exponent that correctly represents the number of 6-hour periods. Discard any choice that describes a different kind of growth, like a small percent increase each hour.

Desmos Guide

Graph the growth model for the bacteria

In Desmos, enter the function modeling the bacteria count over time (using $x$ for hours):

y = 1250*2^(x/6)

This shows how the number of cells changes as time increases.

Mark the target population of 40,000 cells

Add a second expression:

y = 40000

This is a horizontal line representing when the culture has 40,000 cells.

Use the graph to connect to the equation choices

Look at where the curve y = 1250*2^(x/6) intersects the line y = 40000 and note that this comes from setting the model $1250\cdot 2^{t/6}$ equal to 40,000. Among the choices, select the equation that represents "future amount = 1250 times $2^{t/6}$ " with the future amount being 40,000.

Step-by-step Explanation

Recognize the exponential growth pattern

The culture doubles every 6 hours, which is exponential growth. A standard form for this kind of situation is

$\text{amount} = \text{initial} \times (\text{growth factor})^{\text{number of doubling periods}}$ .

Here, the growth factor is 2 (because it doubles).

Express the number of doubling periods in terms of t

The bacteria doubles once every 6 hours.

In $t$ hours, the number of 6-hour periods is $t/6$ .

So the exponent should be the number of doubling periods, which is $t/6$ , not $6t$ .

Write a function for the population after t hours

The initial amount is 1,250 cells, and it doubles every 6 hours. Using the pattern from Step 1, the number of cells after $t$ hours can be modeled by

$N(t) = 1250 \cdot 2^{t/6}$ .

This gives the number of cells at any time $t$ .

Set the function equal to the target population

We want to know when the culture reaches 40,000 cells, so we set the expression for $N(t)$ equal to 40,000:

$40{,}000 = 1250 \cdot 2^{t/6}$ .

This is the equation that can be used to find $t$ , and it matches answer choice B.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation