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Math Help - Variation Worded Problem

  1. #1
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    Exclamation Variation Worded Problem

    The price of painting the outside of a cylindrical tank (the bottom and top are not painted) of radius r and height h varies directly as the total surface area. If r=5 and h=4, the price is $60.
    What is the price when r=4 and h=6
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  2. #2
    Air
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    Quote Originally Posted by Meg.Fondale View Post
    The price of painting the outside of a cylindrical tank (the bottom and top are not painted) of radius r and height h varies directly as the total surface area. If r=5 and h=4, the price is $60.
    What is the price when r=4 and h=6
    If you open up a cylinder into a net then you will observe that it is a rectangle with the top and the buttom being a circle. As the question says the top and the buttom are not painted then this implies that we only consider the side, which is the rectangle.

    The area of a rectangle is width multiplied by length. For the cylinder, the length is the height ( h) and the width is the circumference (because this is the length of the circle) this it is 2\pi r.

    Therefore, the surface area considered is S = (2\pi r)(h) = 2 \pi h r.

    When r=4, h=5, the surface area is S = 2 \times \pi \times 5 \times 4 = 40\pi at which the price is \$ 60.

    Now consider the surface area when r=4, h=6, this would equal S = 2 \times \pi \times 4 \times 6 = 48\pi.

    Now we are able to use general arithmetic (Ratio consideration) to solve for the price. When r=4, h=6, therefore the price ( P) is P= \frac{48 \pi \times 60}{40\pi} = ...


    EDIT: I have attached the cylinder diagram and the net considered.
    Attached Thumbnails Attached Thumbnails Variation Worded Problem-diagram.jpg  
    Last edited by Air; April 18th 2009 at 02:58 AM. Reason: EDIT1: Attaching Diagram, EDIT2: Had Put EDIT1 Reason Into Title.
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  3. #3
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    Quote Originally Posted by Meg.Fondale View Post
    The price of painting the outside of a cylindrical tank (the bottom and top are not painted) of radius r and height h varies directly as the total surface area. If r=5 and h=4, the price is $60.
    What is the price when r=4 and h=6
    To learn how to set this up as a variation equation, try here.

    Once you have learned the basic terms and techniques, the following should make sense:

    i) The variation equation must be of the form P = k(SA), where P is the price, k is the constant of variation, and SA is the surface area.

    ii) You know that SA = 2(pi)rh, so P = 2k(pi)rh.

    iii) You are given that, for r = 5 and h = 4, P = 60. Use this to solve for the constant of variation k.

    iv) Now that you have the value of k, plug this into P = 2k(pi)rh. This is your variation equation.

    v) Plug in 4 for r and 6 for h. Simplify to find the required value for P.

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