The length of the base is 22 feet, and its height is 6 feet, given the area is 66 square feet.
The area of a triangle is given by the formula, area = (1/2)*base*height.
In the question, we are given that the cross-section of a lean-to shelter is in the shape of a triangle. Also, we are told that the base of the triangle is 10 feet more than twice its height.
We are asked to determine the length of the base and the height of the triangle in feet if the area of the triangle is 66 square feet.
We assume the height of the triangle to be x feet.
Thus, the base of the triangle is 10 feet more than twice its height, that is, base = 10 + 2x feet.
Now, the area of the triangle, using the formula, area = (1/2)*base*height can be shown as:
area = (1/2)(10 + 2x)x square feet.
But, the area of the triangle is given to be 66 square feet.
Equating these values, we get:
(1/2)(10 + 2x)x = 66,
or, x(5 + x) - 66 = 0,
or, x² + 5x - 66 = 0,
or, x² + 11x - 6x - 66 = 0,
or, x(x + 11) - 6(x + 11) = 0,
or, (x - 6)(x + 11) = 0, which implies that:
Either x - 6 = 0, or, x = 6,
Or, x + 11 = 0, or, x = -11, which is not possible since x is the height of a triangle which cannot be negative.
Thus, the height of the triangle, x = 6 feet.
The base of the triangle, 10 + 2x = 22 feet.
Thus, the length of the base is 22 feet, and its height is 6 feet, given the area is 66 square feet.
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