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Related Rates volume change

  1. Nov 15, 2015 #1
    1. The problem statement, all variables and given/known data
    All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters?
    2. Relevant equations


    3. The attempt at a solution
    I used the equation for the volume of a cube: V = s3 but I'm not sure if side and edge can be considered the same thing. Anyways, I implicitly differentiated it with respect to time t and got: dV/dt = 3s2(ds/dt) and since they give ds/dt = 3 cm/sec and values for s I just plugged all that in and for (a) I got 9 cm3/sec and for (b) I got 900 cm3/sec but I'm not sure if I did it correctly though.
     
  2. jcsd
  3. Nov 15, 2015 #2

    Mark44

    Staff: Mentor

    Both answers are correct.

    To answer your other question, a side and an edge are the same thing here.
     
  4. Nov 15, 2015 #3
    Thank you for double checking my work
     
  5. Nov 15, 2015 #4

    Mark44

    Staff: Mentor

    You're welcome!
     
  6. Nov 15, 2015 #5

    fresh_42

    Staff: Mentor

    As you mentioned is ##s = s(t) = 3t + s_0## where ##s_0## is the starting length of the cube. Isn't the volume's vvelocity then accelerating quadratic in time?
    And if so when will be measured? At ##3t + s_0 = 1## and ##3t + s_0 = 100## or is ##s_0 = 0##, ##s_0 = 100## resp.?
     
  7. Nov 15, 2015 #6

    Mark44

    Staff: Mentor

    No, you are misinterpreting the problem. The cube isn't moving through space. It is expanding. This is a typical problem in calculus textbooks in the section on Related Rates.
     
  8. Nov 15, 2015 #7

    fresh_42

    Staff: Mentor

    @Mark44 I got that. ##s(t)## has been noted the length of the cube's edges in the OP.

    My misunderstanding was that I first thought the given lengths were those of the original cube when expansion started.
    In that case it would have been just a formula of time and time of measurement needed to be specified.

    But meanwhile I understood it: ##s(t) = 3t + s_0 = 1## or ##100## defines the measurement via the actual edges then.
    I have to admit that I sometimes tend to make things more complicated than they are.
    Thank you for replying.
     
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