Question

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practice: density calculations
answer the following questions. make sure to show all work to receive credit. you may need a separate sheet of paper.

1. find the volume of a box measuring 2 cm by 7 cm by 3 cm.
2. an object placed in a graduated cylinder raises the volume from 12.2 ml to 14.5 ml. find the volume of the object.
3. find the volume of a cube measuring 5 cm on each side.
4. find the density of the box in #1 if it has a mass of 20 g.
5. find the density of the object in #2 if it has a mass of 4 g.
6. find the density of the cube in #3 if it has a mass of 100 g.
7. find the mass of an object that has a density of 1.5 g/cm³ and has a volume of 8 cm³.
8. find the volume of an object with a density of 3.1 g/ml and a mass of 12 g.
9. find the mass of a cube that has a density of 2.7 g/ml and measures 3 cm on each side.
10. challenge: find the density of a soda can with a radius of 3.25 cm, a height of 12.2 cm, and a mass of 40 g.

Explanation:

Step1: Recall volume formula for rectangular - box

The volume $V$ of a rectangular - box with length $l$, width $w$, and height $h$ is $V=l\times w\times h$. For a box with $l = 2$ cm, $w = 7$ cm, and $h = 3$ cm, we have $V=2\times7\times3$.
$V = 42$ $cm^{3}$

Step2: Find volume of object in graduated cylinder

The volume of the object is equal to the change in volume of the liquid in the graduated cylinder. So $V=14.5 - 12.2$.
$V = 2.3$ mL

Step3: Recall volume formula for cube

The volume $V$ of a cube with side length $s$ is $V=s^{3}$. For a cube with $s = 5$ cm, we have $V=5^{3}=5\times5\times5$.
$V = 125$ $cm^{3}$

Step4: Recall density formula

The density $
ho$ is given by $
ho=\frac{m}{V}$, where $m$ is mass and $V$ is volume. For the box in #1 with $m = 20$ g and $V = 42$ $cm^{3}$, $
ho=\frac{20}{42}\approx0.48$ g/$cm^{3}$

Step5: Calculate density of object in #2

Using $
ho=\frac{m}{V}$, with $m = 4$ g and $V = 2.3$ mL, $
ho=\frac{4}{2.3}\approx1.74$ g/mL

Step6: Calculate density of cube in #3

Using $
ho=\frac{m}{V}$, with $m = 100$ g and $V = 125$ $cm^{3}$, $
ho=\frac{100}{125}=0.8$ g/$cm^{3}$

Step7: Recall mass - density - volume relation

Since $
ho=\frac{m}{V}$, then $m=
ho\times V$. With $
ho = 1.5$ g/$cm^{3}$ and $V = 8$ $cm^{3}$, $m=1.5\times8 = 12$ g

Step8: Rearrange density formula for volume

Since $
ho=\frac{m}{V}$, then $V=\frac{m}{
ho}$. With $m = 12$ g and $
ho = 3.1$ g/mL, $V=\frac{12}{3.1}\approx3.87$ mL

Step9: First find volume of cube

The volume of a cube with $s = 3$ cm is $V=s^{3}=3^{3}=27$ $cm^{3}$ (since 1 $cm^{3}=1$ mL). Using $m=
ho\times V$ with $
ho = 2.7$ g/mL and $V = 27$ mL, $m=2.7\times27 = 72.9$ g

Step10: Recall volume formula for cylinder

The volume of a cylinder $V=\pi r^{2}h$. With $r = 3.25$ cm and $h = 12.2$ cm, $V=\pi\times(3.25)^{2}\times12.2=\pi\times10.5625\times12.2\approx399.77$ $cm^{3}$. Using $
ho=\frac{m}{V}$ with $m = 40$ g and $V\approx399.77$ $cm^{3}$, $
ho=\frac{40}{399.77}\approx0.10$ g/$cm^{3}$

Answer:

1. $42$ $cm^{3}$
2. $2.3$ mL
3. $125$ $cm^{3}$
4. $\approx0.48$ g/$cm^{3}$
5. $\approx1.74$ g/mL
6. $0.8$ g/$cm^{3}$
7. $12$ g
8. $\approx3.87$ mL
9. $72.9$ g
10. $\approx0.10$ g/$cm^{3}$