the area of a trapezoid is 360in^2.if the ratio of the height is 6:10:5, find the dimensions of the trapezoid.
To find the dimensions of the trapezoid, we need to use the given information about the area and the ratio of the height. Let's break down the problem step by step:
Step 1: Understand the formula for the area of a trapezoid.
The formula for the area of a trapezoid is given as:
Area = [(sum of the lengths of the bases) ÷ 2] × height
Step 2: Express the given ratio of the height.
The given ratio of the heights is 6:10:5. We can assume that the common factor between them is 'x'. So, the heights can be expressed as 6x, 10x, and 5x respectively.
Step 3: Set up the equation using the formula for the area.
We are given that the area of the trapezoid is 360 square inches. Therefore, we can set up the equation as:
360 = [(sum of the lengths of the bases) ÷ 2] × height
Step 4: Substitute the variables into the equation.
Let's assume the lengths of the bases are 'a' inches and 'b' inches. Substituting the known values into the equation, we get:
360 = [(a + b) ÷ 2] × (6x + 10x + 5x)
Step 5: Simplify the equation.
We can simplify the equation by combining like terms:
360 = [(a + b) ÷ 2] × 21x
Step 6: Solve for the sum of the bases.
Now, we need to isolate the sum of the bases. Divide both sides of the equation by 21x:
(360 ÷ 21x) = [(a + b) ÷ 2]
Step 7: Simplify further.
Simplify the left side of the equation:
17.14 = (a + b) ÷ 2
Step 8: Solve for the sum of the bases.
Multiply both sides of the equation by 2 to isolate (a + b):
2 * 17.14 = a + b
34.28 = a + b
So, the sum of the lengths of the bases is 34.28 inches.
Step 9: Solve for the individual lengths of the bases.
To find the individual lengths of the bases, we need additional information or another equation. Unfortunately, the given information does not provide us with enough details to solve for the individual lengths of the bases.
Therefore, with the given information, we can conclude that the sum of the lengths of the bases is 34.28 inches, but we are unable to determine the individual lengths without more information.