Anonymous asked in Science & MathematicsMathematics · 1 decade ago

Derivatives: Chain Rule - Melting Ice Block?

A cubical block of ice is melting in such a way that each edge decreses steadily by 5.2 cm every hour. At what rate is its volume decreasing when each edge is 5 meters long?

Solution: Let l=l(t) be the length of each edge at time t. Then the volume of the block is given by V = _______ .

Since the length of each edge is changing in time, we conclude that the volume V is also a function of time t. We note that the rate of change of l is constant and that is given as [(dl)/(dt)]= _______cm/h = ________m/h.

The question is to find the rate of change of V when

l= ____________m.

The chain rule gives

dV/dt = (dV/dl)*(dl/dt)

Therfore , when each edge is 5 m long, the rate of change of the volume of the ice block is ________m3/h.

I don't know how to approach this question..

Please answer..

Thank you

3 Answers

  • 1 decade ago
    Favorite Answer

    Volume of a cube with side l is: V = l^3

    dl/dt = 5.2 cm/hr = 0.052 m/hr

    Find rate of change of V when l = 5 m.

    Ok, for this last part, you need to use the chain rule as you have it stated: dV/dt = (dV/dl)*(dl/dt)

    Now dV/dl is simply the derivative of the first equation with respect to l. This gives:

    dV/dl = 3l^2

    Now plug this into the equation for dV/dt.

    dV/dt = 3l^2*(dl/dt)

    Now you can substitute in all the constants, l = 5 m, dl/dt = 0.052 m/hr, to get:

    dV/dt = 3*(5 m)^2*(0.052 m/hr)

    = 75*0.052 m^3/hr

    = 3.9 m^3/hr

    Hope this clears it up for you.

  • 1 decade ago

    V = l^3 = 5*5*5 = 125 cubic meters

    dl/dt = 5.2 cm/hour = 0.052 meters per hour

    Since V = l^3, then dV/dl = 3l^2 (as long as each edge shrinks at the same rate).

    Then dV/dt = (dV/dl)*(dl/dt) = 3l^2*(dl/dt) = (3*5*5*)(0.052) = 3.9 cubic meters per hour.

  • Como
    Lv 7
    1 decade ago

    dl /dt = 5.2 cm / h

    dl /dt = 0.052 m / h

    V = l ³

    dV/dl = 3 l ² m ²

    dV/dt = (dv / dl) x (dl / dt)

    dV/dt = 3 x 5² x 0.052 m³ / h (when l = 5m)

    dV/dt = 3.9 m³ / h

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