Solving problems involving area :
Let us recall the formula to find the area of a rectangle.
length × width
To find area of a rectangle, we multiply length and width.
In case we know the area, and only one dimension, we can divide the area by the known dimension to find the other dimension.
The ultimate aim of this section is, how to solve problems using area.
Problem 1 :
The area of a rectangular sandbox is 56 2/3 square feet. The length of the sandbox is 8 1/2 feet. What is the width ?
We know the formula to find area of the rectangle.
Area of the rectangle = length x width
Plug the known values area and length and solve for the unknown value width.
56 2/3 = 8 1/2 x width
170/3 = 17/2 x width
(170/3) x (2/17) = width
20/3 = width
6 2/3 = width
Therefore, the width of the rectangle is 6 2/3 feet.
Problem 2 :
The area of a circle is 154 square cm. Find its perimeter.
To find perimeter of the circle, we have to know the radius.
So, we have to find the radius from the given area.
We know the formula for area of circle.
Area of the circle = ∏r²
Plug ∏ = 22/7, area = 154
154 = 22/7 x r²
154 x 7/22 = r²
49 = r²
√49 = √r²
7 = r
Perimeter of the circle = 2∏r
= 2x (22/7) x 7
= 2 x 22
= 44 cm
Therefore, perimeter of the circle is 44 cm.
Problem 3 :
Mr. Webster is buying carpet for an exercise room where the basement of the room is in the shape of rectangle. The length and width of the room are 18 2/5 feet and 12 1/2 feet respectively. Find the total cost of the carpet, if the price of the carpet is $3 per square feet ?
To know the total cost of the carpet, first we have to know the area of the basement.
Area of the basement = length x width
= 18 2/5 x 12 1/2
= (92/5) x (25/2)
= 46 x 5
= 230 square feet
So, we need 230 square feet of carpet.
The cost each square feet of carpet = $3
Then, the cost of 230 square ft of carpet is
= 3 x 230
Therefore, the total cost of the carpet is $690
Problem 4 :
If the area of a square is 64 square inches, find the length of its diagonal.
Let "a" be the length of one side of the square.
We all know the formula for area of square.
Area of square = side x side = a²
64 = a²
√64 = √a²
8 = a
So, the length of side of the square is 8 inches.
In the above picture, we have to find the length of AC, say "x".
Using Pythagorean theorem in the right triangle ABC, we get
AC² = AB² + BC²
x² = 8² + 8²
x² = 64 + 64
x² = 128
x = √128
x = √(2x64)
x = 8√2 inches
The length of its diagonal is 8√2 inches.
Problem 5 :
If the radii of two circles are in the ratio 2 : 3, then find the ratio between their areas.
From, the given information, we have
Radius of the first circle = 2x
Radius of the second circle = 3x
Area of the first circle = ∏r² = ∏(2x)² = 4∏x²
Area of the second circle = ∏r² = ∏(3x)² = 9∏x²
Ratio between their areas = 4∏x² : 9∏x²
Ratio between their areas = 4 : 9
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