If the area of two similar decagons are 625 ft² squared and 100 ft² what is the ratio of the perimeters of the decagons
please I desperately need some help!!!!
625:100 = 25^2:10^2
so the perimeters are in the ratio 25:10 = 5/2
omg thanks so much @oobleck
To find the ratio of the perimeters of the decagons, we need to know the relationship between the areas of similar shapes. In this case, since the decagons are similar, their areas will be in proportion to the square of a linear scaling factor.
Let's assume that the linear scaling factor for the two decagons is 'x'. This means that the side length of the larger decagon is multiplied by 'x' to get the side length of the smaller decagon.
We know that the areas of the decagons are in the ratio of 625 ft² to 100 ft². Therefore, we can set up the following equation based on the proportion of their areas:
(x * side length of larger decagon)² : (side length of smaller decagon)² = 625 : 100
Simplifying the equation, we get:
(x² * side length of larger decagon)² : (side length of smaller decagon)² = 625 : 100
(x²² * (10x)²) : (10)² = 625 : 100
(x⁴ * 100x²) : 100 = 625 : 100
x⁴ * x² = 625 / 100
x⁶ = 6.25
Now, to find the value of x, we take the sixth root of both sides of the equation:
x = ∛(6.25)
Calculating ∛(6.25), we get:
x = 1.577
Since we have found the value of x, we can now find the ratio of the perimeters. The perimeter of a decagon is equal to 10 times the side length.
For the larger decagon:
Perimeter = 10 * (1.577 * side length of larger decagon)
For the smaller decagon:
Perimeter = 10 * (side length of smaller decagon)
Therefore, the ratio of the perimeters of the decagons is:
10 * (1.577 * side length of larger decagon) : 10 * (side length of smaller decagon)
Simplifying this expression, we get:
1.577 * side length of larger decagon : side length of smaller decagon