The Diagonal Of A Rectangle Is 8m Longer Than Its Shorter Side. If The Area Of The Rectangle Is 60 Sq.M, Find Its Dimensions.

The diagonal of a rectangle is 8m longer than its shorter side. If the area of the rectangle is 60 sq.m, find its dimensions.

 

Answer:

The dimensions of the rectangle are 5 and 12 meters.

Step-by-step explanation:

Shorter side (Width): x

Diagonal: x + 8

Longer side: a

Use the Pythagorean theorem to find the longer side a:

c² = a² + b²

Where:

c = diagonal (x+8)

b = shorter side/width (x)

a = longer side/length

(x+8)² = a² + (x)²

a² = (x+8)² - x²

a² = x² + 16x + 64 - x²

a² = 16x + 64

a² = 16(x + 4)

√a² = √16 (x+4)

a = 4 √(x+4)

The longer side is  4√(x+4) m.

Find the dimension of the rectangle when the area is 60 m²:

Area = (length) (Width)

60 = (4√(x+4) (x)

60 = 4x√(x+4)

4x√(x+4)² = (60)²

16x²(x+4) = 3,600

16x³ + 64x² - 3600 = 0

16x³ + 64x² - 3600/16 = 0

x³ + 4x²- 225 = 0

Find the roots or x:

Rational theorem for polynomial roots:

Factors of 225:

(±)1,3,5,9,15,25,45,75,225

Trial: 5

Divide by x - 5:

(x³ + 4x²- 225) / (x-5) = x² + 9x + 45

Therefore, x = 5

Find the roots of x² + 9x + 45:

Completing the square method of solving quadratic equation:

x² + 9x + (9/2)² = -45 + (9/2)²

(x + 9/2)² = -45 + 81/4

(x + 9/2)² = (-180 + 81)/4

√((x + 9/2)² = ±√(-99/4)

x + 9/2 = ± (3i√11)/2

x = (9 ± 3i √11)/4

The roots are:

x = 5 a real solution

x = (9 ± 3i √11)/4  not a real solution

Choose x = 5:

Shorter side/Width: x = 5 meters

Longer side/Length: 4√(x+4) ⇒ 4√(5+4) ⇒ 4√9 ⇒ 4(3) = 12 meters

The dimensions are 5 and 12 meters.

Check:

Area = Length × Width

60 = 12 × 5

60 = 60  (True)