Chapter 4.3, Problem 35WE

To determine

To Find: The perimeter and area of the below shaded region

  

Answer to Problem 35WE

Area of a rectangle $=6a^2{b}^2$

Perimeter of a rectangle $=10ab$

Explanation of Solution

Given information:

The diagram:

  

Concept Used:

Rule of exponents for product of powers states that to multiply two powers with same base, the exponents are added.

For example $a^m\cdot a^n=a^{m+n}$

Commutative property of multiplication states that the product of any numbers remains the same irrespective of the order in which we multiply.

For example:

  $2\cdot 5=5\cdot 2=10$

Associative property of multiplication states that the product of the any numbers remains the same regardless of the grouping of factors.

For example:

  $2\left(5\cdot 3\right)=\left(2\cdot 5\right)\cdot 3=30$

Area of a rectangle $=length\cdot width$

Perimeter of a rectangle $=2\left(length+width\right)$

Calculation:

In the given diagram

Length $=3ab$

Width $=2ab$

Area of a rectangle $=length\cdot width$

Substituting the known value

Area of a rectangle $=3ab\cdot 2ab$

Commutative and Associative property of multiplication.

  $=\left(3\cdot 2\right)\left(a\cdot a\right)\left(b\cdot b\right)$

Using rule of exponents for product of powers

  $\begin{array}{l}=6a^{1+1}{b}^{1+1}\\ =6a^2{b}^2\end{array}$

Perimeter of a rectangle $=2\left(length+width\right)$

Substituting the known value

  $\begin{array}{l}=2\left(3ab+2ab\right)\\ =2\cdot 5ab\\ =10ab\end{array}$