Example 2 - Length and width of a rectangle
The diagram shows a rectangle.
The length of the rectangle is \(2x + 5\).
The width of the rectangle is \(3x - 2\).
The The distance around a shape. of the rectangle is 46 cm.
Find the length and width of the rectangle.
1. What do I have to do?
Read the question through twice. Highlight or underline the important pieces of information in the question.
2. What information do I need?
The highlighted words are the most important ones. The key word in the question is perimeter.
The question asks to find the length and width of the rectangle, and to do this you have to find the value of \(x\). The answer might be a whole number or a fraction or decimal.
3. What information don鈥檛 I need?
Everything in this question is relevant to working out the answer.
4. What maths can I do?
Step A
Opposite sides in a rectangle are equal.
Write on the lengths (Numbers, symbols and operators grouped together - one half of an equation, eg 2bf + 2f + 3k.) of the other sides on your diagram.
Step B
The perimeter of a shape is found by adding up all the sides. Find an expression for the perimeter:
\(2x + 5 + 3x - 2 + 2x + 5 + 3x - 2 = 10x + 6\)
Step C
The perimeter of the rectangle is 46 cm, therefore:
\(10x + 6 = 46\)
Step D
Solve this equation to find \(x\):
\(10x + 6 = 46\)
Start by subtracting 6 from both sides:
\(10x = 40\)
Divide both sides by 10:
\(x = 4\)
Find the length and width of the rectangle by substituting the value of \(x\) into the expressions for the length and width.
The length of the rectangle is: \(2x + 5\)
If \(x = 4\) then \(2 \times 4 + 5 = 13\:\text{cm}\).
The width of the rectangle is: \(3x - 2\)
If \(x = 4\) then \(3 \times 4 - 2 = 10\:\text{cm}\).
So the length of the rectangle is 13 cm and the width is 10 cm.
5. Is my solution correct?
Check the answer by calculating the perimeter of the rectangle.
\(13 + 10 + 13 + 10 = 46\)
The perimeter is 46 cm which matches the information given in the question.
6. Have I completed everything?
The question asks for the length and width of the rectangle, these have been found and checked.
Nothing else was asked for.