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March 25, 2017

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The hypotenuse of a right triangle has one end at the origin and one end on the curve y=x^(7)e^(-7x) , with x greater than or equal to 0. One of the other two sides is on the x-axis, the other side is parallel to the y-axis.

Find the maximum area of such a triangle. Round your answer to 4 decimal places.
Maximum Area =

At what x-value does it occur?

Any help on these two questions would be greatly appreciated.

  • Calculus - ,

    Area of triangle = 1/2(base)(height)
    = 1/2(x)(x^7)(e^(-7x)
    = 1/2(x^8)(e^(-7x)

    Area' = 1/2[x^8(-7e^(-7x)) + 8x^7(e^(-7x))]
    = 0 for a max/min of Area
    (1/2)e^(-7x)x^7[-7x + 8] = 0
    -7x + 8 = 0
    x = 8/7 or x = 1.142857

    so max area = .000488145

    test: take a slighter larger and a slightly smaller value of x

    if x = 1.14
    area = .000488133

    if x = 1.15
    area = .00048807

    so the largest area correct to 4 decimals is .0005 and it occurs when x = 8/7

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