Question:

Please help with this tent problem so i can understand fro step to step?

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A two-person tent is to be made so that the height at the center is a = 4 feet (see the figure below). If the sides of the tent are to meet the ground at an angle α = 60°, and the tent is to be b = 10 feet in length, how many square feet of material will be needed to make the tent? (Assume that the tent has a floor and is closed at both ends, and give your answer in exact form.)

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  1. The tent consists of  two isosceles triangles of height 4 ft. with base angles of 60 deg,

    If each half of the base of a triangle measures x ft., then:

    4 / x = tan(60) = sqrt(3)

    x = 4 / sqrt(3)

    x = 4 sqrt(3) / 3.

    The area of each triangular end is:

    A1 = 4x

    = 16 sqrt(3) / 3 ft^2.

    If s ft. is the length of each sloping side of the isosceles triangles, then:

    4 / s = sin(60)

    s = 4 / sin(60)

    8 / sqrt(3)

    = 8 sqrt(3) / 3.

    Each sloping side of the tent has area:

    A2 = 8 sqrt(3) * 10 / 3

    = 80 sqrt(3) / 3 ft^2.

    The base of the tent has area:

    A3 = 2x * 10

    = 80 sqrt(3) / 3 ft^2.

    The total area is:

    2(A1 + A2) + A3

    = [ 2(16 + 80) + 80 ] sqrt(3) / 3

    = 272 sqrt(3) / 3 ft^2.

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