A battery-powered lamp was left on, and its brightness was measured every 30 minutes. The results are shown. | Time elapsed (minutes) | 0 | 30 | 60 | 90 | 120 | 150 | |---|---|---|---|---|---|---| | Brightness (lumens) | 800 | 680 | 578 | 492 | 419 | 357 | Scientists model the brightness (in lumens) at time (in minutes) with an exponential function of the form where is the initial brightness and is a constant decay factor. Which value of is most consistent with the data?

Solution available with step-by-step explanation

SAT Two-Variable Data Question 18 | Free SAT Prep | Aniko

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Question 18·Hard·Two-Variable Data: Models and Scatterplots

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A battery-powered lamp was left on, and its brightness was measured every 30 minutes. The results are shown.

Time elapsed (minutes)	0	30	60	90	120	150
Brightness (lumens)	800	680	578	492	419	357

Scientists model the brightness $B$ (in lumens) at time $t$ (in minutes) with an exponential function of the form

B(t)=B_0\,k^{\,t/30},

where $B_0$ is the initial brightness and $k$ is a constant decay factor.

Which value of $k$ is most consistent with the data?

For exponential models given in table form, first match the time step in the table to the time unit in the exponent; here, each 30-minute increase corresponds to multiplying by the constant factor $k$ . Quickly compute the ratio of a later value to the previous one (for example, $B(30)/B(0)$ ), and that ratio is $k$ . Optionally check with one or two more consecutive ratios to confirm the pattern, then choose the answer choice equal to that factor. This avoids writing full equations for many points and saves time.

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Connect k to the table

Look at how the exponent $t/30$ changes when time increases by 30 minutes. How does that tell you what happens to the brightness every 30 minutes?

Use consecutive data points

Pick two consecutive times in the table, like 0 and 30 minutes. How can you use the brightness values at those times to find the factor $k$ that the brightness is multiplied by?

Form a ratio for one time step

Write an equation using the model for $B(0)$ and $B(30)$ , then solve for $k$ . After you find a value for $k$ from that pair, check if the same factor appears between other consecutive brightness values in the table.

Match your value to an answer choice

Once you have the numerical value of $k$ , compare it to the answer choices and pick the one that is equal (or closest, if there is rounding).

Desmos Guide

Compute the decay factor from the first two points

In a Desmos expression line, type 680/800 and look at the decimal result. This is the multiplicative factor between 0 and 30 minutes, which corresponds to $k$ in the model.

Confirm the factor with other data pairs

In new lines, type 578/680, 492/578, 419/492, and 357/419. Notice that each result is very close to the number you got from 680/800, confirming that this is the consistent decay factor per 30 minutes.

Match to the answer choices

Compare the decimal value you see in Desmos for 680/800 with the four answer choices and select the choice that matches this value.

Step-by-step Explanation

Understand what k means in the model

The model is

B(t) = B_0 \, k^{t/30}.

Here, $t$ increases in steps of 30 minutes in the table (0, 30, 60, ...). When $t$ increases by 30 minutes, the exponent $t/30$ increases by 1, so the brightness is multiplied by $k$ each 30-minute step.

So $k$ is the multiplicative factor per 30 minutes:

Brightness after 30 minutes = (brightness at 0 minutes) $\times k$ .
Brightness after 60 minutes = (brightness at 30 minutes) $\times k$ , and so on.

Use the first two data points to express k

From the table:

At $t=0$ minutes, brightness is $800$ lumens.
At $t=30$ minutes, brightness is $680$ lumens.

From the model:

$B(0) = B_0 = 800$ .
$B(30) = B_0 \cdot k = 800k$ .

But the table also tells us $B(30) = 680$ , so:

800k = 680.

Solve for $k$ :

k = \frac{680}{800}.

This fraction can be simplified, but we will compare it to other data first.

Check that this factor matches other 30-minute steps

To see if the same factor applies for other 30-minute intervals, divide each brightness by the brightness 30 minutes earlier:

From 30 to 60 minutes: $\dfrac{578}{680}$
From 60 to 90 minutes: $\dfrac{492}{578}$
From 90 to 120 minutes: $\dfrac{419}{492}$
From 120 to 150 minutes: $\dfrac{357}{419}$

If you compute these, each ratio is very close to $\dfrac{680}{800}$ . That confirms that $\dfrac{680}{800}$ is the correct per-30-minute decay factor $k$ for this data.

Simplify the fraction and choose the matching option

Now simplify

\frac{680}{800} = \frac{68}{80} = \frac{17}{20}.

And

\frac{17}{20} = 0.85.

So the decay factor $k$ is $0.85$ , which matches answer choice C) 0.85.

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Step-by-step Explanation

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