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Math Help - pre-calculus urgent help with functions

  1. #1
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    pre-calculus urgent help with functions

    A rectangle is created with two left hand vertices on the y-axis. One of the other vertices is on the line y=10-x and the other is on the line y=2x-5. Assuming the rectangle is bounded by the lines and the y-axis.
    1) write a formula for the area of the rectangle
    2) what is the domain of A(x)?
    3) find the area when x=2
    4) find the maximum area of the rectangle
    PLEASE EXPLAIN HOW YOU FOUND ALL 4 VERTICES
    Please show your work in detail, thanks for taking time to do this!!!
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  2. #2
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    Let the right hand side of the rectangle be x = k. (Left hand side is x = 0).
    Right hand vertices are: (k, y1) and (k, y2) where
    y1 = 10 - k and y2 = 2k - 5.
    1. Area A(k) = (y1 - y2) (k - 0) = 5k(3-k)
    2. Domain of A(k) is bounded by k >= 0 and 3 - k >= 0
    3. Substitute k=2
    4. What is the max of k(3-k) as k goes from 0 to 3?
    (If you don't know derivatives, think of area of a rectangle that has a constant perimeter of 2k + 2(3-k) = 6 units. When is area maximum?)
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  3. #3
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    Hello, shirleychen888!

    Did you make a sketch?


    A rectangle is created with two left hand vertices on the y-axis.
    One of the other vertices is on the line y\:=\:10-x
    . . and the other is on the line y\:=\:2x-5
    Assuming the rectangle is bounded by the lines and the y-axis:

    1) Write a formula for the area of the rectangle
    Code:
          | y = 10-x
       10 *           *
          | *        *
          |   *     *
          +-----*  *
          |:::::| *
          |:::::|*  *
          +-----*     *
          |    *        *
      - - + - * - - - - - * - - -
          |  *           10 *
          | *                 *
          |* y = 2x-5
       -5 *
          |
    The area of a rectangle is: . \text{(Area)} \:=\: \text{(Base)} \times \text{(Height)}

    Our rectangle has: . \text{Base }= x
    and: . \text{Height} \:=\:(10-x) - (2x-5) \:=\:15 - 3x

    Therefore: . A(x) \;=\;x(15-3x)




    2) What is the domain of A(x) ?
    Domain: . 0 \:\leq\: x \:<\:5



    3) Find the area when x=2
    A(2) \;=\;2(15 - 3\cdot 2) \;=\;2\cdot9 \;=\;18



    4) Find the maximum area of the rectangle
    The graph of: A \:=\:15x-3x^2 is a down-opening parabola.
    . . Its maximum is at its vertex.

    The vertex is at: . x \:=\:\frac{\text{-}15}{\text{-}6} \:=\:\frac{5}{2}

    The maximum area is: . A\left(\frac{5}{2}\right)  \:=\:15\left(\frac{5}{2}\right) -3\left(\frac{5}{2}\right)^2 \;=\;\frac{75}{4}

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