A student measured several samples of the element tellurium at STP (standard temperature and pressure).
The equation below models the relationship between the mass \(m\), in grams, and the volume
\(v\), in cubic centimeters, of the samples.
$$ m=5.97v+0.24 $$
The student then measured several samples of the element polonium at STP.
The data revealed that pollonium has a mass that is 1.5 times greater than a sample of tellurium of the same volume. Which of the following equations could model the relationship
between the mass \(m\), in grams, and the volume \(v\), in cubic centimeters, for the element selenium at this temperature?
We want to find an equation that would result in a mass that is \(1.5\) times greater when the volume is the same. We can test the models by using an arbitrary number, such as \(v=1\).
\( m=3.85v-2.15\)
$$ m=3.85(1)-2.15 $$
$$ m=3.85-2.15 $$
$$ m=1.70 $$
\( m=6.15v+0.15 \)
$$ m=6.15(1)+0.15 $$
$$ m=6.15-0.15 $$
$$ m= 6.00 $$
\( m=8.88v-2.32\)
$$ m=8.88(1)-2.32$$
$$m=8.88-2.32 $$
$$m=6.56$$
\( m=9.05v+0.02\)
$$m=9.05(1)+0.02$$
$$ m=9.05+0.02$$
$$m=9.07$$
Comparing this with the initial model,
$$ m=5.97v+0.24 $$
$$ m=5.97(1)+0.24$$
$$m=5.97+0.24 $$
$$ m=6.21 $$
Only the last option seems reasonable. We want a number that is \(1.5\) times greater.
$$ 6.21 \approx 6 $$
$$ 1.5 \text{ times } 6 = 9 $$
$$ 9 \approx 9.07 $$
According to the initial model, the mass of the element scales at approximately \(6\) times the volume.
$$ m=\colorbox{aqua}{$5.97$}v+0.24 $$
If we want a new model where the mass is \(1.5\) times greater, we would want a model that scales at \(6 \times 1.5 = 9 \) times the volume.
$$ m=\colorbox{yellow}{$9.05$}v+0.02 \checkmark$$
Although another option fits this criteria, the constant is significantly different than the initial model and therefore significantly changes the resultant mass.
$$ m=\colorbox{yellow}{$8.88$}v-\colorbox{red}{$2.32$} ✖ $$