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Math Help - rate/water level word problem

  1. #1
    Junior Member shepherdm1270's Avatar
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    rate/water level word problem

    a trough has a triangular cross section. The trough is 6 ft across the top, 6 ft deep, and 16 ft long. Water is being pumped into the trough at the rate of 4ft^3 per minute. Find the rate at which the height of the water is increasing at the instant that the height is 4 ft.
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    Quote Originally Posted by shepherdm1270 View Post
    a trough has a triangular cross section. The trough is 6 ft across the top, 6 ft deep, and 16 ft long. Water is being pumped into the trough at the rate of 4ft^3 per minute. Find the rate at which the height of the water is increasing at the instant that the height is 4 ft.

    We need to figure out the volume of the trough

    V=A_{base}\cdot l

    To find the area of the base we need to use similar triangles
    3
    *******
    * b *
    6*** *
    * *
    x *
    *
    so \frac{x}{6}=\frac{b}{3} \iff b=\frac{1}{2}x

    The area is 2 \frac{1}{2}x \cdot b=x \cdot \frac{1}{2}x=\frac{x^2}{2}

    So the Volume is V=\frac{x^2}{2} \cdot 16=8x^2

    \frac{dV}{dt}=16x\frac{dx}{dt} \iff \frac{1}{16x}\frac{dV}{dt}=\frac{dx}{dt}

    We want to find \frac{dx}{dt} and we know \frac{dV}{dt}=4 \mbox{ when } h=4

    \frac{dx}{dt}=\frac{1}{16(4)} \cdot 4=\frac{1}{16}
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    Quote Originally Posted by shepherdm1270 View Post
    a trough has a triangular cross section. The trough is 6 ft across the top, 6 ft deep, and 16 ft long. Water is being pumped into the trough at the rate of 4ft^3 per minute. Find the rate at which the height of the water is increasing at the instant that the height is 4 ft.
    A similar problem that might be of interest is discussed at this thread.
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    Junior Member shepherdm1270's Avatar
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    ok, so what is the rate though. 1/16 whats? ft^3?
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    Quote Originally Posted by shepherdm1270 View Post
    ok, so what is the rate though. 1/16 whats? ft^3?

    <br />
\frac{dx}{dt}=\frac{1}{16 \mbox{ft}(4 \mbox{ft})} \cdot \frac{4 ( \mbox{ft})^3}{min}=\frac{1}{16}\frac{\mbox{ft}}{\m  box{min}}<br />
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    An acknowledgement - Thanks for that!, say - of the given solution, followed by asking
    Quote Originally Posted by shepherdm1270 View Post
    ok, so what is the rate though. 1/16 whats? ft^3?
    would not be inappropriate here.

    Also: TheEmptySet did all the work! Seriously - if you've taken the effort to actually understand his/her solution (as opposed to just copying it out) surely you can do that one small thing and work out the unit yourself.

    I ask: How can dx/dt possibly have a unit of volume?
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    May I show you a short cut to these types of related rates problems?.

    By making the observation that the rate of change of volume is equal to the cross-sectional area at that instant times the rate of change of the height.

    \frac{dV}{dt}=A(t)\cdot\frac{dh}{dt}

    \frac{dh}{dt}=\frac{\frac{dV}{dt}}{A(t)}

    We know that dV/dt=4.

    The area of the water surface, A(t), when it is 4 feet deep is just a rectangle with area 16*4=64.

    So, we have \frac{dh}{dt}=\frac{4}{64}=\frac{1}{16}
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    Quote Originally Posted by galactus View Post
    May I show you a short cut to these types of related rates problems?.

    By making the observation that the rate of change of volume is equal to the cross-sectional area at that instant times the rate of change of the height.

    \frac{dV}{dt}=A(t)\cdot\frac{dh}{dt}

    \frac{dh}{dt}=\frac{\frac{dV}{dt}}{A(t)}

    We know that dV/dt=4.

    The area of the water surface, A(t), when it is 4 feet deep is just a rectangle with area 16*4=64.

    So, we have \frac{dh}{dt}=\frac{4}{64}=\frac{1}{16}
    But what's the unit, galactus!
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  9. #9
    Eater of Worlds
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    Oh...I forgot.

    It's \frac{1}{16} \;\ N\cdot{m^{2}}/kg^{2}

    Oops.....that's gravity. I always get that and ft/min mixed up.
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  10. #10
    Junior Member shepherdm1270's Avatar
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    see - this all looks like gibberish to me. My mind can not conceptualize math terms whatsoever. I have a 4.0, and calculus is killin me right now... i've never had something i couldn't figure out, but calculus baffles me. And it's extremely frustrating. So thanks for all your help. I only hope I can recreate it on the final. *crosses fingers*
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