The width of a rectangle is 4cm less than the length. If the area of the rectangle is 192vm², find the:

Question

1) perimeter of the rectangle
2) angle which the diagonal makes with the length correct to the nearest 0.1⁰

Answer 1

Let's assume that the length of the rectangle is x cm.
According to the given information, the width of the rectangle is 4 cm less than the length. So, the width would be (x - 4) cm.

1) Perimeter of the rectangle:
The perimeter of a rectangle is given by the formula: P = 2(l + w), where P is the perimeter, l is the length, and w is the width.

Plugging in the values, we get:
P = 2(x + (x - 4))
P = 2(2x - 4)
P = 4x - 8 cm

2) Angle which the diagonal makes with the length:
Using the Pythagorean theorem, we can find the length of the diagonal (d) of the rectangle. The diagonal, length, and width form a right-angled triangle.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (d^2) is equal to the sum of the squares of the other two sides (l^2 + w^2).

In this case, the hypotenuse is the diagonal (d) and the other two sides are the length (x) and width (x - 4).

So, we have: d^2 = x^2 + (x - 4)^2

Simplifying, we get: d^2 = x^2 + x^2 - 8x + 16

Since we know the area of the rectangle is 192 cm², we can write:
l * w = 192
x * (x - 4) = 192
x^2 - 4x - 192 = 0

Using the quadratic formula, we can solve for x (length):
x = (-(-4) ± √((-4)^2 - 4 * 1 * -192)) / (2 * 1)
x = (4 ± √(16 + 768)) / 2
x = (4 ± √784) / 2
x = (4 ± 28) / 2
x = 16 or x = -12

Since the length cannot be negative, we take x = 16 cm.

Now, we can find the width (w):
w = x - 4
w = 16 - 4
w = 12 cm

Therefore, the length is 16 cm and the width is 12 cm.

To find the diagonal (d), we use the Pythagorean theorem:
d^2 = 16^2 + 12^2
d^2 = 256 + 144
d^2 = 400
d = √400
d = 20 cm

To find the angle (θ) which the diagonal makes with the length, we can use trigonometry.

θ = arctan(w / l)
θ = arctan(12 / 16)
θ ≈ 36.87° (to the nearest 0.1°)