Question

The area of a rectangle measured in cm2, is numerically equal to its perimeter, measured in cm.

The length of the rectangle is 5 times its width.

Calculate the width and length of the rectangle. Give your answers in centimetres.

Answer 1

Width: 2.4 cm

Length: 12 cm

Explanation:

Let the width of the rectangle be w cm, and the length be 5w cm. The area A and perimeter P of the rectangle can be expressed as follows:

Area (A) = length × width

A = 5w × w = 5w^2 cm^2

Perimeter (P) = 2 × (length + width)

P = 2 × (5w + w) = 2 × 6w = 12w cm

According to the problem, the area is numerically equal to the perimeter:

A = P

5w^2 = 12w

To solve for w, we can rearrange the equation:

5w^2 - 12w = 0

w(5w - 12) = 0

This equation has two possible solutions:

w = 0

In this case, the width would be 0 cm, which doesn't form a valid rectangle.

5w - 12 = 0

5w = 12

w = 12/5 = 2.4 cm

For the second solution, the width is 2.4 cm. Now, we can calculate the length:

length = 5w

length = 5 × 2.4 = 12 cm

So, the width of the rectangle is 2.4 cm, and the length is 12 cm.

Answer 2

Let's use "w" to represent the width of the rectangle and "l" to represent the length.

According to the problem statement, the area of the rectangle is numerically equal to its perimeter, so we can write:

2(l + w) = lw

Simplifying this equation, we get:

2l + 2w = lw

Dividing both sides by 2, we get:

l + w = 0.5lw

Multiplying both sides by 2, we get:

2l + 2w = lw

Since the length of the rectangle is 5 times its width, we can write:

l = 5w

Substituting this into the previous equation, we get:

2(5w) + 2w = 5w^2

Simplifying this equation, we get:

10w + 2w = 5w^2

12w = 5w^2

Dividing both sides by w, we get:

12 = 5w

So the width of the rectangle is:

w = 12/5 = 2.4 cm

And the length of the rectangle is:

l = 5w = 12 cm

Therefore, the width of the rectangle is 2.4 cm and the length of the rectangle is 12 cm.