A rectangle has a length 12 feet in width 8 feet. Every dimension of the rectangle has multiplied by 3/4 to form a similar rectangle. How is the ratio of the areas related to the ratio of the corresponding sides
the new area is (3/4)^2 times the old area, so ...
To find the relationship between the ratio of the areas and the ratio of the corresponding sides, we'll start by calculating the dimensions of the new rectangle.
Given that the original rectangle has a length of 12 feet and a width of 8 feet, we can multiply both dimensions by 3/4 to find the dimensions of the new rectangle.
New length = 12 * (3/4) = 9 feet
New width = 8 * (3/4) = 6 feet
Now, let's calculate the areas of both rectangles.
Original area = length * width = 12 * 8 = 96 square feet
New area = new length * new width = 9 * 6 = 54 square feet
The ratio of the areas of the two rectangles is:
Ratio of the areas = New area / Original area = 54 / 96 = 9 / 16
Next, let's calculate the ratio of the corresponding sides.
Ratio of the length = New length / Original length = 9 / 12 = 3 / 4
Ratio of the width = New width / Original width = 6 / 8 = 3 / 4
The ratio of the corresponding sides is the same for both dimensions, which is 3/4.
Therefore, the ratio of the areas is related to the ratio of the corresponding sides squared. In this case, (9/16) = (3/4)^2.
To find out how the ratio of the areas is related to the ratio of the corresponding sides of a rectangle, you can follow these steps:
1. Calculate the area of the original rectangle: Area = length x width = 12 ft x 8 ft = 96 square feet.
2. Multiply each dimension of the original rectangle by 3/4 to find the corresponding sides of the similar rectangle:
Length = 12 ft x (3/4) = 9 ft
Width = 8 ft x (3/4) = 6 ft
3. Calculate the area of the similar rectangle: Area = length x width = 9 ft x 6 ft = 54 square feet.
4. Calculate the ratio of the areas by dividing the area of the similar rectangle by the area of the original rectangle:
Ratio of areas = (Area of similar rectangle) / (Area of original rectangle)
= 54 square feet / 96 square feet
5. Simplify the ratio if possible:
Ratio of areas = (9/2) square feet / (16/3) square feet
= (9/2) x (3/16)
= 27/32
Therefore, the ratio of the areas of the similar rectangle to the original rectangle is 27/32.