Question 842·Hard·Math

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The scatterplot shows the time $x$ (in hours) that a crystal was left in a solution and the mass $y$ (in milligrams) of the crystal at the end of the time for 9 trials. The line of best fit for the data is also shown.

Using the slope of the line of best fit, which choice is closest to the crystal’s average growth rate, in micrograms per minute?

(Use $1$ milligram $=1{,}000$ micrograms and $1$ hour $=60$ minutes.)

For a drawn line of best fit, the slope is the model’s change in output per unit of input. Pick two clear points on the line, then use Desmos to divide the vertical change by the horizontal change. Check the requested units against both axes: converting only the mass or only the time leaves the rate in the wrong units.

Hints

Hint 1
A best-fit line shows the trend across the trials. The trial dots might not lie exactly on it. Look for grid crossings on the drawn line, then read their coordinates from the axis labels.
Hint 2
The slope is the change in mass divided by the matching change in time. Keep both subtractions in the same order: if you subtract the earlier mass first, subtract the earlier time first too.
Hint 3
A rate in milligrams per hour has mass on top and time on the bottom. Use the given conversions to change both units. Which way must you place each conversion so milligrams and hours cancel?

Step-by-step

Read the line, then convert its slope
Step 1
Choose two points on the fitted line
The solid best-fit line crosses grid intersections at $(2,15)$ and $(8,55)$ . Read these as hours first, milligrams second. Use points on the fitted line, not trial dots, because the question asks for the line’s slope.
Step 2
Find the growth rate per hour
The slope is the change in mass divided by the change in time. Type $m=\frac{55-15}{8-2}$ into Desmos. It shows $m=6.666\ldots$ . Since the vertical axis measures milligrams and the horizontal axis measures hours, this is about $6.67$ milligrams per hour, not the requested rate yet.
Step 3
Convert both units of the rate
A rate has mass on top and time on the bottom. Use the given conversions so milligrams cancel and hours go on top to cancel the hours in the denominator:
$m\frac{\text{milligrams}}{\text{hour}}\left(\frac{1{,}000\text{ micrograms}}{1\text{ milligram}}\right)\left(\frac{1\text{ hour}}{60\text{ minutes}}\right).$
Type $\frac{m\cdot1{,}000}{60}$ on the next Desmos line. It shows $111.111\ldots$ , so the crystal’s average growth rate is closest to $111$ micrograms per minute. Choice C.

Question 842·Hard·Math

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