A circular copper wire of radius 7cm is bent to form a rectangle. If the breadth and the length of the rectangle are in the ratio 4:7 respectively. What is the breadth of the rectangle?
Given:
radius=7cm
Circumference=perimeter of rectangle
Ratio of breadth:length = 4:7
Let b=breadth,
then length=(7/4)b
circumference=2π(7)=14π
circumference=perimeter
therefore
14π=2(b+(7/4)b)
14&;i;=2(1+7/4)b
(11/2)b=14π
b=14π(5/11)=7.997=8 cm approx.
Check:
perimeter=2(8+(7/4)8)=44 cm
circumference = 14π=43.98=44 cm approx. .......good!
A path that is 2 feet wide surrounds the pond.
What is the combined area of the pond and the path?.
To solve this problem, we need to find the length and breadth of the rectangle.
Let's start by finding the length of the rectangle.
The circular wire is bent into a rectangle, so the circumference of the circle will be equal to the perimeter of the rectangle.
The formula for the circumference of a circle is given by C = 2πr, where r is the radius of the circle.
Given that the radius is 7 cm, we can find the circumference:
C = 2π(7) = 14π cm
Since the circumference is equal to the perimeter of the rectangle, we have:
Perimeter = 2(length + breadth)
14π = 2(length + breadth) ---(1)
Now, let's find the ratio between the breadth and the length of the rectangle.
Given that the ratio between the breadth and length is 4:7,
Let the common ratio between the breadth and length be x.
Therefore, breadth = 4x and length = 7x
Substituting these values into equation (1), we have:
14π = 2(7x + 4x)
14π = 2(11x)
14π = 22x
Dividing both sides by 22, we get:
x = (14π) / 22
Simplifying, we have:
x = 7π / 11
Now, we can find the breadth of the rectangle.
Breadth = 4x = 4 * (7π / 11)
Breadth = 28π / 11
So, the breadth of the rectangle is 28π / 11 cm.
To find the breadth of the rectangle formed by bending a circular copper wire, we need to follow these steps:
Step 1: Calculate the circumference of the circular copper wire.
The circumference of a circle can be found using the formula: C = 2πr, where C is the circumference and r is the radius.
In this case, the radius of the circular wire is 7 cm. Therefore, the circumference of the circular copper wire is:
C = 2π(7) = 14π cm
Step 2: Determine the length of the rectangle.
Given that the breadth and length of the rectangle are in the ratio 4:7, we can set up a proportion:
4/7 = C/length
Cross-multiplying the equation, we get:
4 * length = 7 * C
Step 3: Substitute the value of C into the equation from Step 2.
Substituting the value of C we calculated earlier, we have:
4 * length = 7 * 14π
Step 4: Solve for the length of the rectangle.
Divide both sides of the equation by 4 to isolate the length:
length = (7 * 14π) / 4
Step 5: Calculate the breadth of the rectangle by using the ratio.
Given that the breadth and length are in the ratio 4:7, we know that the breadth is 4 parts and the length is 7 parts. Therefore, we can set up a proportion:
4/7 = breadth/length
Cross-multiplying the equation, we get:
4 * length = 7 * breadth
Step 6: Substitute the value of length we found in Step 4 into the equation from Step 5.
Substituting the value of length we calculated, we have:
4 * ((7 * 14π) / 4) = 7 * breadth
Step 7: Solve for the breadth of the rectangle.
Divide both sides of the equation by 7 to isolate the breadth:
breadth = (4 * ((7 * 14π) / 4)) / 7
Simplifying the equation, we have:
breadth = 2π cm
Therefore, the breadth of the rectangle formed from the circular copper wire is 2π cm.