The width of a rectangle is 4cm less than the length. If the area of the rectangle is 192vm², find the:
1) perimeter of the rectangle
2) angle which the diagonal makes with the length correct to the nearest 0.1⁰
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°)
To solve this problem, we'll need to find both the length and width of the rectangle.
Let's first set up the equation based on the given information:
Let the length of the rectangle be "L" cm.
The width of the rectangle is then "L - 4" cm.
The area of the rectangle is 192 cm².
1) Perimeter of the rectangle:
The formula for the perimeter of a rectangle is P = 2(L + W), where P is the perimeter, L is the length, and W is the width.
Substituting the values into the formula, we get:
P = 2(L + (L - 4))
P = 2(2L - 4)
P = 4L - 8
2) Angle which the diagonal makes with the length:
To find this angle, we first need to calculate the length of the diagonal using the Pythagorean theorem.
In a rectangle, the diagonal, length (L), and width (W) form a right-angled triangle.
The diagonal (D) can be found using the formula D² = L² + W².
Substituting the values into the formula, we get:
D² = L² + (L - 4)²
D² = L² + L² - 8L + 16
D² = 2L² - 8L + 16
Now, calculate the value of D by taking the square root of both sides of the equation:
D = √(2L² - 8L + 16)
To find the angle (θ), we can use the following formula:
θ = arctan(W/L)
Substituting the values into the formula and rounding to the nearest 0.1⁰, we get:
θ = arctan((L - 4) / L)
Now, you can solve for L by equating the area of the rectangle to 192 cm²:
L(L - 4) = 192
L² - 4L - 192 = 0
The quadratic equation can be solved to find the value of L.
I hope this helps you to calculate the perimeter and the angle of the rectangle.
To solve this problem, we need to use the given information and apply relevant formulas for finding the perimeter and the angle made by the diagonal with the length. Let's break down the problem step by step:
1) Finding the perimeter of the rectangle:
The perimeter of a rectangle is defined as the sum of all its sides. Since we know the length and the width of a rectangle, we can calculate the perimeter using the formula:
Perimeter = 2 * (Length + Width)
Given that the width is 4cm less than the length, we can express the width in terms of the length:
Width = Length - 4cm
Now, we can substitute the value of the width into the perimeter formula:
Perimeter = 2 * (Length + (Length - 4cm))
Simplifying further gives:
Perimeter = 2 * (2 * Length - 4cm)
Therefore, the perimeter is twice the sum of the length and 4cm.
2) Finding the angle the diagonal makes with the length:
To calculate the angle between the diagonal and the length, we can use trigonometry. Let's assume the length is the base of a right-angle triangle and the diagonal is the hypotenuse. We want to find the angle that the hypotenuse makes with the base.
Using the trigonometric function "inverse tangent" (tan⁻¹), we can find the angle using the following formula:
Angle = tan⁻¹(Height/Base)
In this case, the height represents the width of the rectangle, which is given as Length - 4cm. The base is the length.
Therefore, the angle can be calculated as:
Angle = tan⁻¹((Length - 4cm)/Length)
To find this angle, you can use a calculator or a computer program that has a built-in trigonometric function (e.g., the arctan function).
Remember to round the angle to the nearest 0.1⁰ as mentioned in the question.
By following these steps, you can calculate the perimeter and the angle made by the diagonal accurately.