Question

exercises 23 - 26
in exercises 23 - 26, determine (a) the volume and (b) the surface area of the figure. when appropriate, use the π key on your calculator and round your answer to the nearest hundredth.

1. 11 cm 5 cm 3 cm
2. 9 ft 3 ft
3. 12 mm 16 mm
4. 5 yd

Explanation:

23. Rectangular - prism

(a) Volume calculation

Step1: Recall volume formula

The volume formula for a rectangular - prism is $V = l\times w\times h$, where $l = 11$ cm, $w = 5$ cm and $h = 3$ cm.
$V=11\times5\times3$

Step2: Calculate the volume

$V = 165$ $cm^{3}$
(b) Surface - area calculation

Step1: Recall surface - area formula

The surface - area formula for a rectangular - prism is $SA=2(lw + lh+wh)$.
$SA = 2(11\times5+11\times3 + 5\times3)$

Step2: Calculate inner values

First, calculate $11\times5 = 55$, $11\times3=33$, $5\times3 = 15$. Then $11\times5+11\times3 + 5\times3=55 + 33+15=103$.

Step3: Calculate surface area

$SA=2\times103 = 206$ $cm^{2}$

24. Cylinder

(a) Volume calculation

Step1: Recall volume formula

The volume formula for a cylinder is $V=\pi r^{2}h$, where $r = 3$ ft and $h = 9$ ft.
$V=\pi\times3^{2}\times9$

Step2: Calculate the volume

$V=\pi\times9\times9=81\pi\approx254.47$ $ft^{3}$
(b) Surface - area calculation

Step1: Recall surface - area formula

The surface - area formula for a cylinder is $SA = 2\pi r^{2}+2\pi rh$.
$SA=2\pi\times3^{2}+2\pi\times3\times9$

Step2: Calculate each part

$2\pi\times3^{2}=2\pi\times9 = 18\pi$ and $2\pi\times3\times9=54\pi$.

Step3: Calculate surface area

$SA=18\pi + 54\pi=72\pi\approx226.19$ $ft^{2}$

25. Cone

(a) Volume calculation

Step1: Find the radius

Given the diameter $d = 12$ mm, so the radius $r=\frac{d}{2}=6$ mm and $h = 16$ mm. The volume formula for a cone is $V=\frac{1}{3}\pi r^{2}h$.
$V=\frac{1}{3}\pi\times6^{2}\times16$

Step2: Calculate the volume

$V=\frac{1}{3}\pi\times36\times16 = 192\pi\approx603.19$ $mm^{3}$
(b) Surface - area calculation

Step1: Find the slant height $l$

Use the Pythagorean theorem $l=\sqrt{r^{2}+h^{2}}$, where $r = 6$ mm and $h = 16$ mm. So $l=\sqrt{6^{2}+16^{2}}=\sqrt{36 + 256}=\sqrt{292}=2\sqrt{73}$ mm.
The surface - area formula for a cone is $SA=\pi r^{2}+\pi rl$.
$SA=\pi\times6^{2}+\pi\times6\times2\sqrt{73}$

Step2: Calculate each part

$\pi\times6^{2}=36\pi$ and $\pi\times6\times2\sqrt{73}=12\pi\sqrt{73}$.
$SA = 36\pi+12\pi\sqrt{73}\approx36\pi+12\pi\times8.544\approx36\pi + 102.528\pi\approx434.08$ $mm^{2}$

26. Sphere

(a) Volume calculation

Step1: Recall volume formula

The volume formula for a sphere is $V=\frac{4}{3}\pi r^{3}$, where $r = 5$ yd.
$V=\frac{4}{3}\pi\times5^{3}$

Step2: Calculate the volume

$V=\frac{4}{3}\pi\times125=\frac{500}{3}\pi\approx523.60$ $yd^{3}$
(b) Surface - area calculation

Step1: Recall surface - area formula

The surface - area formula for a sphere is $SA = 4\pi r^{2}$, where $r = 5$ yd.
$SA=4\pi\times5^{2}$

Step2: Calculate the surface area

$SA=4\pi\times25 = 100\pi\approx314.16$ $yd^{2}$

Answer:

23. (a) $165$ $cm^{3}$
23. (b) $206$ $cm^{2}$
24. (a) $\approx254.47$ $ft^{3}$
24. (b) $\approx226.19$ $ft^{2}$
25. (a) $\approx603.19$ $mm^{3}$
25. (b) $\approx434.08$ $mm^{2}$
26. (a) $\approx523.60$ $yd^{3}$
26. (b) $\approx314.16$ $yd^{2}$