The sum of the area of the six congruent circles is 150 pi square centimeters. What is the area of the rectangle? Exact value
To find the area of the rectangle, we first need to find the area of one of the congruent circles.
Let's assume the radius of each circle is 'r'.
The area of a circle is given by the formula: A = πr^2.
Since we have six congruent circles, the total area of all six circles would be: 6 * πr^2.
We are given that the sum of the area of the six congruent circles is 150π square centimeters. Hence, we can write the equation:
6πr^2 = 150π.
Now, we can divide both sides of the equation by 6π to solve for the value of r^2:
r^2 = 150π / 6π = 25.
Taking the square root of both sides of the equation, we find:
r = √25 = 5.
So, the radius of each circle is 5 centimeters.
Now, let's calculate the area of the rectangle.
We know that the width of the rectangle will be twice the radius of the circle, which is 2 * 5 = 10 centimeters.
To find the length of the rectangle, we need to consider that the circumference of each circle would be equal to one side of the rectangle. The circumference of a circle is given by the formula: C = 2πr.
The sum of the circumferences of all six circles will be equal to the perimeter of the rectangle. So, we can write the equation:
Perimeter of rectangle = 6 * 2πr = 12πr.
We're given that the perimeter of the rectangle is 60 centimeters. Therefore, we can set up the equation:
12πr = 60.
Dividing both sides of the equation by 12π, we find:
r = 60 / (12π) = 5 / π.
Since we know the width of the rectangle is 10 centimeters, we can calculate the length using the formula: Area = length * width.
Area of rectangle = (5 / π) * 10 = 50 / π.
So, the exact area of the rectangle is 50 / π square centimeters.
To find the area of the rectangle, we need to know the dimensions of the rectangle.
Let's assume the radius of each congruent circle is "r". Since there are six congruent circles, the total area of the six circles is given by:
Area of the six circles = 6 * π * r^2
We are given that the sum of the area of the six congruent circles is 150π square centimeters. So we can write the equation as:
6 * π * r^2 = 150π
Dividing both sides of the equation by π, we get:
6 * r^2 = 150
Now, divide both sides of the equation by 6:
r^2 = 25
Taking the square root of both sides, we get:
r = 5
Since the radius of the circle is 5, the diameter would be twice the radius, which is 10.
Now, let's assume the length of the rectangle is "l" and the width is "w". The dimensions of the rectangle are twice the diameter of the circle.
So, l = 2 * 10 = 20 centimeters
And, w = 2 * 10 = 20 centimeters
Now, we can find the area of the rectangle by multiplying the length (l) and the width (w):
Area of the rectangle = l * w = 20 * 20 = 400 square centimeters
Therefore, the area of the rectangle is 400 square centimeters.
What rectangle?
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