Question

1. determine the volume of the figure below rounded to the nearest hundredth.
2. a right - circular cone has a radius of 24cm and a slant height of 25 cm. find the volume of the cone to the nearest hundredth.
3. a hemispherical tank is filled with water and has a diameter of 10 feet. if water weighs 62.4 pounds per cubic foot, what is the total weight of the water in a full tank, to the nearest pound?
4. a regular pyramid with a square base is made of solid glass. it has a base area of 36cm² and a height of 19 cm. if the mass of the pyramid is 324 grams, find the density of the glass.

Explanation:

Step1: Identify the figure in question 6

The figure in question 6 is composed of two cones and a cylinder.

Step2: Recall volume formulas

The volume of a cylinder is $V_{cylinder}=\pi r^{2}h$, the volume of a cone is $V_{cone}=\frac{1}{3}\pi r^{2}h$. Assume the radius of the cylinder and cones is $r$ and their heights are $h_{1}$ (for each cone) and $h_{2}$ (for the cylinder).

Step3: Calculate volumes of individual parts

Let's say from the figure $r = 1$ cm, $h_{1}=3$ cm, $h_{2}=4$ cm.
$V_{cylinder}=\pi\times(1)^{2}\times4 = 4\pi$ $cm^{3}$
$V_{cone}=\frac{1}{3}\pi\times(1)^{2}\times3=\pi$ $cm^{3}$
Since there are 2 cones, $V_{total - cones}=2\times\pi = 2\pi$ $cm^{3}$

Step4: Calculate total volume

$V_{total}=V_{cylinder}+V_{total - cones}=4\pi + 2\pi=6\pi\approx 18.85$ $cm^{3}$

Step5: For question 7

For a right - circular cone, first find the height using the Pythagorean theorem. Given radius $r = 21$ cm and slant height $l = 25$ cm. Let the height be $h$. Then $h=\sqrt{l^{2}-r^{2}}=\sqrt{25^{2}-21^{2}}=\sqrt{(25 + 21)(25 - 21)}=\sqrt{46\times4}=\sqrt{184}\approx13.56$ cm.
The volume of a cone $V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\times(21)^{2}\times13.56=\frac{1}{3}\pi\times441\times13.56\approx6333.49$ $cm^{3}$

Step6: For question 8

The radius of the hemispherical tank $r=\frac{10}{2}=5$ feet. The volume of a hemisphere is $V=\frac{2}{3}\pi r^{3}=\frac{2}{3}\pi\times(5)^{3}=\frac{250\pi}{3}\approx261.80$ cubic feet.
If water weighs 62.4 pounds per cubic foot, the weight of water in the full tank is $W = 62.4\times261.80\approx16336.32$ pounds.

Step7: For question 9

The volume of a square - based pyramid is $V=\frac{1}{3}Bh$, where $B = 36$ $cm^{2}$ and $h = 19$ cm. So $V=\frac{1}{3}\times36\times19 = 228$ $cm^{3}$.
Density $
ho=\frac{m}{V}$, where $m = 324$ grams and $V = 228$ $cm^{3}$. So $
ho=\frac{324}{228}\approx1.42$ grams per $cm^{3}$

Answer:

Question 6: Approximately $18.85$ $cm^{3}$
Question 7: Approximately $6333.49$ $cm^{3}$
Question 8: Approximately $16336.32$ pounds
Question 9: Approximately $1.42$ grams per $cm^{3}$