Chapter 12.4, Problem 2CK

a.

To determine

To find: The volume to the nearest cubic centimetre.

Answer to Problem 2CK

Side of the pyramid is 18.61 mm and volume is $759.59{\text{ mm}}^3$ .

Explanation of Solution

Given information:

The total surface area of the pyramid is 600 mm².

Calculation:

Since the total surface area of the pyramid is given as

  $\text{T}\text{.A}.=4\left(\frac{s^2\sqrt{3}}{4}\right)=600{\text{ mm}}^2$

First solve for s:

  $\begin{array}{l}4\left(\frac{s^2\sqrt{3}}{4}\right)=600\\ s^2\sqrt{3}=600\\ s^2=\frac{600}{\sqrt{3}}\\ s^2=200\sqrt{3}\\ s=18.61\text{ mm}\end{array}$

Now calculate volume

  $\begin{array}{l}V=\frac{s^3\sqrt{2}}{12}\\ V=\frac{{\left(18.61\right)}^3\sqrt{2}}{12}\\ V=759.59{\text{ mm}}^3\end{array}$

Hence,

Side of the pyramid is 18.61 mm and volume is $759.59{\text{ mm}}^3$ .

b.

To determine

To find: The volume to the nearest cubic centimetre.

Answer to Problem 2CK

Side of the cube is 10 mm and volume is $1000{\text{ mm}}^3$ .

Explanation of Solution

Given information:

The total surface area of the cube is 600 mm².

Calculation:

Since the total surface area of the cube is given as

  $\text{T}\text{.A}.=6x^2=600{\text{ mm}}^2$

First solve for s:

  $\begin{array}{l}6x^2=600\\ x^2=\frac{600}{6}\\ x^2=100\\ x=10\text{ mm}\end{array}$

Now calculate volume

  $\begin{array}{l}V=x^3\\ V={\left(10\right)}^3\\ V=1000{\text{ mm}}^3\end{array}$

Hence,

Side of the cube is 10 mm and volume is $1000{\text{ mm}}^3$ .

c.

To determine

To find: The volume to the nearest cubic centimetre.

Answer to Problem 2CK

Radius of the sphere is 6.91 mm and volume is $1382{\text{ mm}}^3$ .

Explanation of Solution

Given information:

The total surface area of the sphere is 600 mm².

Calculation:

Since the total surface area of the sphere is given as

  $\text{T}\text{.A}.=4\pi r^2=600{\text{ mm}}^2$

First solve for r:

  $\begin{array}{l}4\pi r^2=600\\ r^2=\frac{150}{\pi }\\ r=\sqrt{\frac{150}{\pi }}\\ r=6.91\text{ mm}\end{array}$

Now calculate volume

  $\begin{array}{l}V=\frac{4}{3}\pi r^3\\ V=\frac{4}{3}\pi {\left(6.91\right)}^3\\ V=1382{\text{ mm}}^3\end{array}$

Hence,

Radius of the sphere is 6.91 mm and volume is $1382{\text{ mm}}^3$ .

d.

To determine

To complete: The given statement.

Answer to Problem 2CK

Of all solid figures with a fixed surface area, the sphere has the largest volume.

Explanation of Solution

Given information:

Of all solid figures with a fixed _____, the ________ has the _______?

Calculation:

From part (a), (b) and (c) observe that the volume of sphere is largest among all though the area of all the solids are fixed.

Therefore, the complete statement will be

Of all solid figures with a fixed surface area, the sphere has the largest volume.

Hence,

Of all solid figures with a fixed surface area, the sphere has the largest volume.