Expert: Paul Klarreich - 2/12/2006

Question
Hi Paul,

I'm doing some review problems in preperation for a midterm and was wondering if you could help me with a word problem I'm stuck on.Thanks!!

Sand is leaking from a bag in such a way that after x seconds there are S(x)=50(1-x^2/15)^3 pounds of sand left in the bag.

a. How much sand was originally in the bag?
b. At what rate is sand leaking from the bag after 1 second?
c. How long does it take for all of the sand to leak from the bag? At what rate is the sand leaking from the bag at the time it empties.

Answer
Hi, Tom,

Your Question:

I'm doing some review problems in preperation for a midterm and was wondering if you could help me with a verbal problem I'm stuck on.Thanks!

Sand leaks from a bag in such a way that after t seconds there are S(t)=50(1-t^2/15)^3 pounds of sand left in the bag.

>> I have made a couple of small changes in wording in your problem -- such as 'after t seconds'.  Sounds better.

a. How much sand was originally in the bag?

>> Let's assume this means 'How much ... bag at t = 0?'

S(0) = 50(1)^3 = 50 pounds.  

[I had to carry one of those from Home Depot -- I wished it was leaking.]

b. At what rate is sand leaking from the bag after 1 second?

>> 'At what rate' always means 'what is S'(t)?'

It is time to compute S'(t):  Chain rule (general power rule) of course:

S'(t) = 50(3)(1 - t^2/15)^2(-2t/15) = 20t(1 - t^2/15)^2

'after 1 second' means 'at t=1'

S'(1) = 20(1)(1 - (1)/15)^2 = 20(14/15) = 4(14/3) = 56/3 lb/sec

That is 18 2/3 lb/sec.  Won't take very long.


c. How long does it take for all of the sand to leak from the bag?

This means 'at what time, t, is S(t) = 0?' so we solve S(t) = 0.

50(1-t^2/15)^3 = 0
1-t^2/15 = 0

t^2/15 = 1
t^2 = 15

t = sqrt(15), just under 4 seconds, as we figured.

d. At what rate is the sand leaking from the bag at the time it empties. (new problem of course)

'At what rate' means 'what is S'(t)' and 'at the time it empties' means at  t = sqrt(15), as we found in (c).

S'(sqrt(15)) = 20(sqrt(15))(1 - (sqrt(15))^2/15)^2
            = 20(sqrt(15))(1 - 15/15)^2 = 20(sqrt(15))(0)^2
            = zero!