Volume and Surface Area: Additional Practice (Set 2)

Each cube has volume $s^3 = 4^3 = 64 \text{ cm}^3$. Volume is additive, so add the two.

$V = 64 + 64 = 128 \text{ cm}^3$

Multiply the three dimensions.

$V = 9 \times 5 \times 2 = 90 \text{ cm}^3$

Volume is $l \times w \times h$, a product of the three. Multiplying any one factor by $2$ multiplies the whole product by $2$.

$V = l \times w \times h \;\to\; (2l) \times w \times h = 2(lwh)$

So doubling any one dimension doubles the volume.

The edge is the number that cubes to give the volume.

$s^3 = 1000 \;\Rightarrow\; s = 10 \text{ cm}$

because $10 \times 10 \times 10 = 1000$.

Divide the volume by the base area.

$\text{base area} = 10 \times 10 = 100 \text{ cm}^2$

$h = \frac{1000}{100} = 10 \text{ cm}$

Use $V = \pi r^2 h$.

$V = 3.14 \times 1^2 \times 100 = 3.14 \times 100 = 314 \text{ cm}^3$

Use $V = \pi r^2 h$, even though the height is small.

$V = 3.14 \times 10^2 \times 1 = 3.14 \times 100 = 314 \text{ cm}^3$

A wide, thin disk can still hold a sizable volume because its base area is large.

Find each box's volume and add, since volume is additive.

$V_1 = 10 \times 4 \times 3 = 120 \text{ cm}^3, \qquad V_2 = 6 \times 4 \times 3 = 72 \text{ cm}^3$

$V = 120 + 72 = 192 \text{ cm}^3$

Volume is $s^3$, so tripling the edge multiplies the volume by $3^3$.

$\frac{(3s)^3}{s^3} = \frac{27 s^3}{s^3} = 27$

The new volume is $27$ times the old one. (Check: $10^3 = 1000$ and $30^3 = 27000$.)

Volume is $s^3$, so doubling the edge multiplies the volume by $2^3 = 8$.

$\frac{64}{8} = 8$

The volume grew by a factor of $8$.

Subtract the cylinder's volume from the box's volume.

$V_{\text{box}} = 14 \times 14 \times 10 = 1960 \text{ cm}^3$

$V_{\text{cyl}} = 3.14 \times 7^2 \times 10 = 3.14 \times 49 \times 10 = 1538.6 \text{ cm}^3$

$1960 - 1538.6 = 421.4 \text{ cm}^3$

That leftover space is the four corners the round cylinder cannot reach.

Wrapping paper covers the surface area. Find one face of each kind, then double the total.

$lw = 5 \times 5 = 25, \quad lh = 5 \times 8 = 40, \quad wh = 5 \times 8 = 40$

$S = 2(25 + 40 + 40) = 2 \times 105 = 210 \text{ cm}^2$