Chapter 12.3, Problem 36SR

a.

To determine

To explain that the expressions for volume and surface area are monomials

Answer to Problem 36SR

Yes, the expressions for volume and surface area are monomials because each is the product of variables and a real number.

Explanation of Solution

Given information:

The side length of the cube is represented as $\text{'}s\text{'}$

The volume of the cube is represented as $\text{'}{s}^3\text{'}$

The surface area of the cube is represented as $\text{'}6s^2\text{'}$

Yes, the expression for volume is monomial because this expression is the product of variables. And the expression for surface area is also monomial because it is a non-linear expression since the exponent is a positive number other than 1.

Therefore, both the expression, volume and surface area of cube are monomial because each is the product of variables and a real number.

b.

To determine

To calculate the volume and surface area of the cube

Answer to Problem 36SR

The volume of the cube is $27{\text{ ft}}^3$ and surface area of the cube is $54{\text{ ft}}^2$

Explanation of Solution

Given information:

The side of the cube measures 3 feet

The volume of the cube is calculated as,

  $\begin{array}{l}{\left(\text{side}\right)}^3\\ ={\left(3\right)}^3\\ =27{\text{ ft}}^3\end{array}$

The surface area of the cube is calculated as,

  $\begin{array}{l}6\times {\left(\text{side}\right)}^2\\ =6\times {\left(3\right)}^2\\ =6\times 9\\ =54{\text{ ft}}^2\end{array}$

Therefore, the volume of the cube is $27{\text{ ft}}^3$ and surface area of the cube is $54{\text{ ft}}^2$

c.

To determine

To calculate the side length $s$ of the cube

Answer to Problem 36SR

The side length of the cube is $6\text{ units}$

Explanation of Solution

Given information:

The volume and surface area of the cube have the same measure

Calculation:

Since the volume and the surface area of the cube have same measure so its side length is calculated as,

  $s=6\text{ units}$

Therefore, the side length of the cube is $6\text{ units}$

d.

To determine

To calculate the ratio of volume of two cylinders

Answer to Problem 36SR

The ratio of volume of two cylinders is $1:8$

Explanation of Solution

Given information:

The volume of the first cylinder is $V=\pi r^{2}h$

The volume of the second cylinder is $V=\pi {\left(2r\right)}^2\left(2h\right)$

Calculation:

The ratio of the first and second cylinder can be obtained as,

  $\begin{array}{l}\text{Volume of first cylinder : Volume of second cylinder}\\ =\pi r^{2}h\text{ : }\pi {\left(2r\right)}^2\left(2h\right)\\ =\pi r^{2}h\text{ : 8 }\pi r^{2}h\\ =\pi r^{2}h\text{ : 8}\left(\pi r^{2}h\right)\\ \text{= 1 : 8}\end{array}$

Therefore, the ratio of volume of two cylinders is $1:8$