Question

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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.
23.
24.
25.
26.

Explanation:

for Problem 23 (Rectangular Prism):

Step1: Calculate volume

The volume formula for a rectangular - prism is $V = l\times w\times h$. Here, $l = 11$ cm, $w = 5$ cm, and $h = 3$ cm. So, $V=11\times5\times3 = 165$ $cm^{3}$.

Step2: Calculate surface area

The surface - area formula for a rectangular prism is $SA = 2(lw+lh + wh)$. Substitute $l = 11$ cm, $w = 5$ cm, and $h = 3$ cm.
$SA=2(11\times5 + 11\times3+5\times3)$
$=2(55 + 33+15)$
$=2\times103$
$ = 206$ $cm^{2}$

for Problem 24 (Cylinder):

Step1: Calculate volume

The volume formula for a cylinder is $V=\pi r^{2}h$. Given $r = 3$ ft and $h = 9$ ft. So, $V=\pi\times3^{2}\times9=\pi\times9\times9 = 81\pi\approx254.47$ $ft^{3}$.

Step2: Calculate surface area

The surface - area formula for a cylinder is $SA = 2\pi r^{2}+2\pi rh$.
$SA=2\pi\times3^{2}+2\pi\times3\times9$
$=2\pi\times9 + 2\pi\times27$
$=18\pi+54\pi$
$=72\pi\approx226.19$ $ft^{2}$

for Problem 25 (Cone):

Step1: Calculate volume

The volume formula for a cone is $V=\frac{1}{3}\pi r^{2}h$. Given $d = 12$ mm, so $r = 6$ mm and $h = 16$ mm. Then $V=\frac{1}{3}\pi\times6^{2}\times16=\frac{1}{3}\pi\times36\times16 = 192\pi\approx603.19$ $mm^{3}$.

Step2: Calculate slant height

First, find the slant height $l$ using the Pythagorean theorem $l=\sqrt{r^{2}+h^{2}}$. Here, $r = 6$ mm and $h = 16$ mm, so $l=\sqrt{6^{2}+16^{2}}=\sqrt{36 + 256}=\sqrt{292}=2\sqrt{73}$ mm.

Step3: Calculate surface area

The surface - area formula for a cone is $SA=\pi r^{2}+\pi rl$.
$SA=\pi\times6^{2}+\pi\times6\times2\sqrt{73}$
$=36\pi+12\pi\sqrt{73}\approx36\pi+12\pi\times8.544$
$=36\pi+102.528\pi$
$=138.528\pi\approx435.20$ $mm^{2}$

for Problem 26 (Sphere):

Step1: Calculate volume

The volume formula for a sphere is $V=\frac{4}{3}\pi r^{3}$. Given $r = 5$ yd, so $V=\frac{4}{3}\pi\times5^{3}=\frac{4}{3}\pi\times125=\frac{500}{3}\pi\approx523.60$ $yd^{3}$.

Step2: Calculate surface area

The surface - area formula for a sphere is $SA = 4\pi r^{2}$. Substitute $r = 5$ yd, so $SA=4\pi\times5^{2}=4\pi\times25 = 100\pi\approx314.16$ $yd^{2}$

Answer:

Problem 23:
(a) $165$ $cm^{3}$
(b) $206$ $cm^{2}$
Problem 24:
(a) $\approx254.47$ $ft^{3}$
(b) $\approx226.19$ $ft^{2}$
Problem 25:
(a) $\approx603.19$ $mm^{3}$
(b) $\approx435.20$ $mm^{2}$
Problem 26:
(a) $\approx523.60$ $yd^{3}$
(b) $\approx314.16$ $yd^{2}$