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Math Help - A couple questions (precal)

  1. #1
    Junior Member
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    A couple questions (precal)

    1. The base of an isosceles triangle is half as long as the two equal sides. Write the area of the triangle as a function of the length of the shorter leg.

    I've already gotten pretty far on this one.
    X is the base, H is the height. 2x is the side lengths because its twice the base
    1/2(x)*(h)= area
    h^2+ (1/2x)^2=(2x)^2
    h^2+1/4x^2= 4x^2
    this is where i get lost...
    My teacher told me it goes to
    sqrt of h^2 = sqrt 15/16x^2
    then it goes to h=sqrt15 divided by 4x
    then you just plug that is. Could you tell me how it becomes sqrt of 15/16 x^2?

    2.A sphere inside a cube, tangent to all six faces. Find the surface area of the cube as a function of the radius of the sphere.

    I don't even know where to start here..

    Thanks
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    The base of an isosceles triangle is half as long as the two equal sides. Write the area of the triangle as a function of the length of the shorter leg.

    I've already gotten pretty far on this one.
    X is the base, H is the height. 2x is the side lengths because its twice the base
    1/2(x)*(h)= area
    h^2+ (1/2x)^2=(2x)^2
    h^2+1/4x^2= 4x^2
    this is where i get lost...
    My teacher told me it goes to
    sqrt of h^2 = sqrt 15/16x^2
    then it goes to h=sqrt15 divided by 4x
    then you just plug that is. Could you tell me how it becomes sqrt of 15/16 x^2?


    A = (1/2)b*h
    A = (1/2)x*h

    (2x)^2 = h^2 +(x/2)^2
    h^2 = 4x^2 -(x^2)/4
    h^2 = (4*4 -1)(x^2)/4
    h^2 = (15/4)x^2
    h = sqrt[(15/4)x^2]
    h = (x/2)sqrt(15)

    So,
    A = (1/2)(x)[(x/2)sqrt(15)]
    A = (1/4)(x^2)sqrt(15) -----------------answer.

    ---------------------------------------------------------------

    2.A sphere inside a cube, tangent to all six faces. Find the surface area of the cube as a function of the radius of the sphere.


    Draw the figure.
    It is a circle of radius r inscribed in a square.

    So the side of the square is 2r long.
    Area of the square is (2r)(2r) = 4r^2

    In the cube, the total surface area is 6 of this square, so,
    A(r) = 6(4r^2)
    A(r) = 24r^2 -------------------answer.
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