The area of a trapezium is 360 in^2. If the ratio of the bases to the height is 6:10:5, find the dimensions of the tarpezium.
To find the dimensions of the trapezium, we need to use the formula for the area of a trapezium:
Area = (1/2) * (a + b) * h
where a and b are the lengths of the bases, and h is the height.
Given that the area of the trapezium is 360 in², we can write the formula as:
360 = (1/2) * (a + b) * h
Now let's use the information provided in the question: the ratio of the bases to the height is 6:10:5. We can represent this as:
a/b = 6/10/5
To simplify the ratio, we can multiply all the numbers by the least common multiple (LCM) of the denominators, which is 10:
a/b = (6/10) * (10/10) = 60/10
Now that we have the ratio of the bases, we can substitute it into the formula:
360 = (1/2) * (60/10h + h) * h
Simplifying further:
360 = (1/2) * (60h/10 + 10h/10) * h
360 = (1/2) * (70h/10) * h
360 = (1/2) * (7h) * h
360 = 7h²/2
7h² = 2 * 360
7h² = 720
h² = 720 / 7
h² ≈ 102.86
Taking the square root of both sides to solve for h:
h ≈ √(102.86)
h ≈ 10.14
Now that we have the value of h, we can substitute it into the ratio to find the bases.
a/b = 6/10/5
a/b = (6/10) * (10/10) * (5/10)
a/b = 30/100
a/b = 3/10
Substitute the value of h into the area formula:
360 = (1/2) * (a + b) * h
360 = (1/2) * (a + b) * 10.14
Simplify further:
36 = (a + b) * 10.14
Now we can solve for a + b:
36/10.14 = a + b
a + b ≈ 3.55
Since a/b = 3/10, we can assign a value to a and b using a = 3x and b = 10x, where x is a common factor. We can then solve for x:
3x + 10x = 3.55
13x = 3.55
x = 3.55/13
x ≈ 0.27
Substitute x back into a and b:
a = 3 * 0.27 ≈ 0.81
b = 10 * 0.27 ≈ 2.70
So the dimensions of the trapezium are approximately:
a = 0.81 in
b = 2.70 in
h = 10.14 in