A rope of length 84cm is made into a circle of which the area exceeds the area of a rectangle by 20 cm square. The length of the diagonal of the rectangle is 13cm. Find its area.
If the circle has radius r and the rectangle's dimensions are w and l, then
2πr = wl+20
w^2+l^2 = 13^2
2πr + 2(w+l) = 84
Solving these, we get
r = 5.843
w = 2.615
l = 12.734
wl = 32.300
Hmmm. With the diagonal a convenient 13, I had expected the rectangle to be 5x12, but not so.
To solve this problem, we need to find the lengths of the sides of the rectangle first, and then we can use those values to calculate its area.
Let's assume that the length of the rectangle is "l" and the width is "w".
We are given that the length of the diagonal of the rectangle is 13cm. Using the Pythagorean theorem, we can find the relationship between the length, width, and diagonal of a rectangle:
diagonal^2 = length^2 + width^2
Substituting the given values, we have:
13^2 = l^2 + w^2
169 = l^2 + w^2 --------(Equation 1)
Now, let's consider the rope. The rope forms a circle, which we can calculate using the formula for the circumference of a circle:
Circumference of the circle = 2 * π * r
Since the rope is made into a circle, the length of the rope will be equal to the circumference of the circle:
Circumference of the circle = 84 cm
2 * π * r = 84
Dividing both sides by 2π, we get:
r = 84 / (2 * π)
r = 42 / π
The area of the circle can be calculated using the formula: Area = π * r^2
Substituting the value of r, we have:
Area of the circle = π * (42/π)^2
Area of the circle = 42^2 / π
Now, we are given that the area of the circle exceeds the area of the rectangle by 20 cm^2:
Area of the circle - Area of the rectangle = 20
(42^2 / π) - (l * w) = 20
To solve for the area of the rectangle, we need to find the values of l and w that satisfy this equation.
Now, we have two equations:
Equation 1: l^2 + w^2 = 169
Equation 2: (42^2 / π) - (l * w) = 20
Using these equations, we can solve for the values of l and w, and then calculate the area of the rectangle using the formula: Area = length * width.