Q16. - Algebra 1

Question

The formula for the surface area of a cube is $SA=6s^2$ where SA is the surface area and s is the length of any side.

 

a. Express the surface area of the cube as a monomial.

 

b. What is the surface area of the cube if $a=3$ and $b=4$?

Step-by-Step Solution

Answer

a. The monomial expression of the surface area of the given cube is $6a^6{b}^2$.

 

b. The surface area of the cube when $a=3$ and $b=4$ is 69984. 

Part a. Step 1. State the concept of ‘surface area of cube’.

The total surface area of a cube is the area covered by all six faces of a cube. If ‘s’ is the length of the sides of a cube, then the total surface area of a cube is given as:

$\text{Surface\hspace{0.17em}\hspace{0.17em}area\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}cube\hspace{0.17em}}\left(SA\right)=6s^2\dots \left(1\right)$

Part a. Step 2. State the concept of ‘monomial’.

A polynomial which contains only one non-zero single term is called a monomial. A monomial consists either one variable or a constant or products of more than one variable with a coefficient along with the exponents as a whole numbers.

 

Note: A monomial cannot have a variable in the denominator.

Part a. Step 3. Calculate the surface area of the given figure.

Observe the figure given below.

From the figure the length of each side is $a^{3}b$.

Substitute $s=a^{3}b$ in (1) and find the surface area of the given cube.

$$\begin{gathered} SA=6s^2 \\ =6{\left(a^{3}b\right)}^2 \\ =6{\left(a^3\right)}^2\left(b^2\right) \\ =6\left(a^{3\times 2}\right)\left(b^2\right) \\ =6\left(a^6\right)\left(b^2\right) \\ =6a^6{b}^2 \end{gathered}$$

Part a. Step 4. State the conclusion.

The surface area of the cube is $6a^6{b}^2$.

 

From the definition, a monomial expression must contain only one non-zero term.

 

The expression $6a^6{b}^2$ contains a constant term 6 and variables a,b. By the definition, they are considered as one single term.

 

Hence, the monomial expression of the surface area of the given cube is $6a^6{b}^2$.

Part b. Step 1. State the concept of ‘surface area of cube’.

The total surface area of a cube is the area covered by all six faces of a cube. If ‘s’ is the length of the sides of a cube, then the total surface area of a cube is given as:

 

$\text{Surface\hspace{0.17em}\hspace{0.17em}area\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}cube\hspace{0.17em}}\left(SA\right)=6s^2\dots \left(1\right)$

Part b. Step 2. Find the surface area of the given figure.

Observe the figure given below.

From the figure see that the length of each side is $a^{3}b$.

Substitute $s=a^{3}b$ in (1) and find the surface area of the given cube.

$$\begin{gathered} SA=6s^2 \\ =6{\left(a^{3}b\right)}^2 \\ =6{\left(a^3\right)}^2\left(b^2\right) \\ =6\left(a^{3\times 2}\right)\left(b^2\right) \\ =6\left(a^6\right)\left(b^2\right) \\ =6a^6{b}^2 \end{gathered}$$

Part b. Step 3. Calculate the surface area of the cube when a = 3 and b = 4 .

From step 2 the surface area of the given cube is $6a^6{b}^2$.

 

Substitute $a=3$ and $b=3$ in $6a^6{b}^2$ to get the required value.

 

$$\begin{gathered} SA=6a^6{b}^2 \\ =6\left(3^6\right)\left(4^2\right) \\ =6\left(729\right)\left(16\right) \\ =69984 \end{gathered}$$

 

Hence, the surface area of the cube is 69984.