Chapter 8.7, Problem 19CYU

a.

To determine

To write: The expression that represents the area of the shaded region.

Answer to Problem 19CYU

The expression that represents the area of the shaded region is $8\left(n-1\right)\left(2n+3\right)$ .

Explanation of Solution

Given:

The shaded region is shown in Figure-1.

  

Formula used:

The area of a square equals (side) 2.

Calculations:

The area A (say) of shaded region is $A=$ Area of larger square of side $\left(4n+1\right)\text{cm}$ - area of smaller and clear- square of side 5 cm.

  $A=\left\{{\left(4n+1\right)}^2-5^2\right\}\text{c}{\text{m}}^{\text{2}}$ .

  $\Rightarrow A=\left\{\left(4n+1\right)-5\right\}\left\{\left(4n+1\right)+5\right\}\text{c}{\text{m}}^{\text{2}}$       Difference of two squares

  $\Rightarrow A=\left(4n-4\right)\left(4n+6\right)\text{c}{\text{m}}^{\text{2}}$

  $\Rightarrow A=4\cdot \left(n-1\right)\cdot 2\left(2n+3\right)\text{c}{\text{m}}^{\text{2}}$

  $\Rightarrow A=8\left(n-1\right)\left(2n+3\right)\text{c}{\text{m}}^{\text{2}}$

Thus, the expression that represents the area of the shaded region is $8\left(n-1\right)\left(2n+3\right)$ .

Conclusion:

The expression that represents the area of the shaded region is $8\left(n-1\right)\left(2n+3\right)$ .

b.

To determine

To find: The dimensions of a rectangle with the same area as the shaded region in part (a)

Answer to Problem 19CYU

The dimensions of a rectangle(s) with the same area as the shaded region in part (a) are $\left(8n-8\right)\times \left(2n+3\right)$ , $\left(n-1\right)\times \left(16n+24\right)$ , $\left(4n-4\right)\times \left(4n+6\right)$ , and $\left(2n-2\right)\times \left(8n+12\right)$ .

Explanation of Solution

Given:

The shaded region is shown in Figure-1.

Method used:

Expression for the area of shaded region in part (a).

Calculations:

The expression for the area of shaded region derived in part (a) is $8\left(n-1\right)\left(2n+3\right)$ .

To determine the dimensions of the rectangle of area $8\left(n-1\right)\left(2n+3\right)$ expression for the area must be written having two factors only; one factor will represent the length and the other the width of the rectangle.

The possible combinations are decided by the factors of $8$ that are $8=8\times 1=4\times 2$ .

Thus, dimensions of the possible rectangles are:

  $\left(8n-8\right)\times \left(2n+3\right)$ , $\left(n-1\right)\times \left(16n+24\right)$ , $\left(4n-4\right)\times \left(4n+6\right)$ , and $\left(2n-2\right)\times \left(8n+12\right)$ .

Conclusion:

The dimensions of a rectangle(s) with the same area as the shaded region in part (a) are $\left(8n-8\right)\times \left(2n+3\right)$ , $\left(n-1\right)\times \left(16n+24\right)$ , $\left(4n-4\right)\times \left(4n+6\right)$ , and $\left(2n-2\right)\times \left(8n+12\right)$ .