We are given that a map is 3 inches wide and 5 inches long. We want to find the perimeter and area of the similar map that is 8 inches long. To do so, we will start by finding the perimeter and area of the original map. Let's start by recalling the formulas for the perimeter and area of a rectangle with length l and width w.
Perimeter of a Rectangle
2 l + 2 w
Area of a Rectangle
l w
In the case of the original map, l= 5 and w= 3. We can substitute these values into each formula to calculate the perimeter and area of the original map. Let's do it!
Perimeter, in.
2( 5) + 2( 3) = 10+6=16
Area, in^2
5 ( 3) = 15
We got that the perimeter of the original map is 16 inches and the area is 15 square inches. Next, we will recall the relationships between perimeters and areas of similar figures.
Perimeter of Similar Figures
If figure B is similar to figure A by a scale factor, then the perimeter of B is equal to the perimeter of A times the scale factor.
Area of Similar Figures
If figure B is similar to figure A by a scale factor, then the area of B is equal to the area of A times the square of the scale factor.
We can find the scale factor between the given maps. To do so, we will calculate the quotient of the two corresponding side lengths.
Length of the new map/Length of the original map substitute ⟶ 8/5
We got that the scale factor is 85. Since we know the perimeter and area of the original map, we can now calculate the perimeter and area of the new map.
Perimeter, in.
16( 8/5) =25.6
Area, in^2
15( 8/5)^2 =15 (64/25)= 38.4
Therefore, the perimeter of the new map is 25.6 inches and the area is 38.4 square inches.
