Chapter 12.4, Problem 4CK

a.

To determine

To find: The total surface area to the nearest square centimetre.

Answer to Problem 4CK

Side of the pyramid is 20.4 mm and total surface area is $720.56{\text{ cm}}^2$ .

Explanation of Solution

Given information:

The volume of the pyramid is 1000 cm²

Calculation:

Since the volume of the pyramid is given as

  $V=\frac{s^3\sqrt{2}}{12}=1000{\text{ cm}}^3$

First solve for s:

  $\begin{array}{l}\frac{s^3\sqrt{2}}{12}=1000\\ s^3=\frac{1000\times 12}{\sqrt{2}}\\ s=20.4\text{ cm}\end{array}$

Now calculate total surface area

  $\begin{array}{l}T.A=s^2\sqrt{2}\\ T.A={\left(20.4\right)}^2\sqrt{2}\\ T.A=720.56{\text{ cm}}^2\end{array}$

Hence,

Side of the pyramid is 20.4 mm and total surface area is $720.56{\text{ cm}}^2$ .

b.

To determine

To find: The total surface area to the nearest square centimetre.

Answer to Problem 4CK

Side of the cube is 10 mm and total surface area is $600{\text{ cm}}^2$ .

Explanation of Solution

Given information:

The total surface area of the cube is 600 mm².

Calculation:

Since the volume of the cube is given as

  $V=x^3=1000{\text{ cm}}^3$

First solve for x:

  $\begin{array}{l}{x}^3=1000\\ x=10\text{ cm}\end{array}$

Now calculate total surface area

  $\begin{array}{l}\text{T}\text{.A}.=6x^2\\ \text{T}\text{.A}.=6{\left(10\right)}^2\\ \text{T}\text{.A}.=600{\text{ cm}}^2\end{array}$

Hence,

Side of the cube is 10 mm and total surface area is $600{\text{ cm}}^2$ .

c.

To determine

To find: The total surface area to the nearest square centimetre.

Answer to Problem 4CK

Radius of the sphere is 6.2 cm and total surface area is $483.6{\text{ cm}}^2$ and among all the solid wih fixed volume sphere ha the least total surface area.

Explanation of Solution

Given information:

The volume of the sphere is 1000 cm³.

Calculation:

Since the volume of the sphere is given as

  $V=\frac{4}{3}\pi r^3=1000{\text{ cm}}^3$

First solve for r:

  $\begin{array}{l}\frac{4}{3}\pi r^3=1000\\ r^3=\frac{1000\times 3}{4\pi }\\ r=6.2\text{ cm}\end{array}$

Now calculate total surface area

  $\begin{array}{l}\text{T}\text{.A}.=4\pi r^2\\ \text{T}\text{.A}.=4\pi {\left(6.2\right)}^2\\ \text{T}\text{.A}.=483.6{\text{ cm}}^2\end{array}$

Hence,

Radius of the sphere is 6.2 cm and total surface area is $483.6{\text{ cm}}^2$ and among all the solid wih fixed volume sphere ha the least total surface area.