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Math Help - Practical use of Calculus

  1. #1
    NeF
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    Red face Practical use of Calculus

    15.1

    A rectangular container has the following dimensions (cm)

    Length  3x
    Width  90-3x
    depth  \frac{x}{3}

    Determine
    15.1.1 Volume V in terms of  x
    15.1.2 The value for  x for which it has a max volume

    15.2 The sum of two positive numbers x and y is 48
    15.2.1 find y in terms of x
    15.2.2 Prove that the sum of thier squares can be given by  2x^3-96x+2304

    15.2.3 Find the samllest possible value of the sum of thier squares

    I have no idea >< I made a couple of attempts but it doesnt look right. weh!
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  2. #2
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    Quote Originally Posted by NeF
    15.1
    A rectangular container has the following dimensions (cm)
    Length  3x
    Width  90-3x
    depth  \frac{x}{3}

    Determine
    15.1.1 Volume V in terms of  x
    15.1.2 The value for  x for which it has a max volume
    15.2 The sum of two positive numbers x and y is 48
    15.2.1 find y in terms of x
    15.2.2 Prove that the sum of thier squares can be given by  2x^3-96x+2304
    15.2.3 Find the samllest possible value of the sum of thier squares
    I have no idea >< I made a couple of attempts but it doesnt look right. weh!
    Hello,NeF,

    V=l\cdot w\cdot d \Longrightarrow V(x)=3x\cdot (90-x)\cdot \frac{x}{3}=-3x^3+90x^2

    You'll get the maximum value for V if \frac{dV}{dx}=0. Thus:

    \frac{dV}{dx}=-9x^2+180x. Therefore: -9x^2+180x=0.
    Solve this equation for x and you'll get x = 0 or x = 20. x = 0 produces obviously the minimum value of V, so at x = 20 V has its maximum.

    to 15.2:
    x + y = 48. Therefore:
    y = 48-x

    Sum of squares: x^2+(48-x)^2=x^2+2304-96x+x^2=2x^2-96x+2304. I assume that there isa typo in your text.

    As you can see s(x)=2x^2-96x+2304 is the equation of a parabola opening upward. So the smallest value is at the vertex where the derivative of s is zero:

    \frac{ds}{dx}=4x-96. this expression is zero if x = 24. That means if x = y = 24 you get the smallest sum of the squares.

    Greetings

    EB
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