Question 18·Hard·Two-Variable Data: Models and Scatterplots
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.
Which choice gives the model that best fits the data?
For an exponential model built from a table, identify the starting value first, then compare consecutive outputs over the stated time step to find the repeated multiplicative factor. On the SAT, a common trap is confusing the amount that remains with the percent decrease, or using the wrong time scale in the exponent, so always match the exponent to the interval used in the data table.
Hints
Start with the value at time 0
In the model , the number is the brightness when . Use the first entry in the table.
Use one 30-minute step
Since the exponent is , a 30-minute increase changes the exponent by 1. Divide one brightness value by the previous one to estimate .
Watch for common modeling mistakes
Check whether an answer choice uses the correct starting value, the correct remaining factor rather than the percent decrease, and the correct exponent structure for 30-minute intervals.
Desmos Guide
Enter the table values
In Desmos, enter the time values in one table column and the brightness values in the other so you can compare the pattern of change between consecutive rows.
Compute the repeated factor
Check the ratios of consecutive brightness values, such as 680/800 and 578/680. These are both about 0.85, so the model should multiply by 0.85 every 30 minutes.
Match the equation to the pattern
Use the first table entry to set the initial value to 800, then choose the equation with factor 0.85 and exponent . That gives .
Step-by-step Explanation
Find the initial value
At , the table shows a brightness of 800 lumens. In the model
that means .
Determine the decay factor
Because the exponent is , the value of represents the factor applied every 30 minutes. Use consecutive entries in the table:
The ratios stay close to 0.85, so .
Assemble the model
Substitute the initial value and decay factor into the model:
So the correct answer is .