Due to melting, an ice cube loses 1/2 its weight each hour. after 8 hrs, it weighs 5/16 lb. ?

Question

what is the original weight? I am looking for the rationale that goes into solving this problem. How can I use the melting rate to solve this problem. thanks,

Answer 1

There is an elegant and not-so-elegant way to do this.

The not-so-elegant way realizes that the half-life of the ice cube is 1 hour, so 8 hours will be 8 half-lives.  So if the 8 hour weight is 5/16 lb, it would have weighed 256 times as much originally.    

Answer 2

let original weight = x

after 1 hour weight = x/2

after 2 hours = x/2/2 = x/2^2

after 3           = x/2/2/2 = x/2^3 .......

therefore x/2^8 = 5/16

x=5*2^8/16

x = 80 lbs

Answer 3

Problems of this kind are solved using the exponential formula

W = Wo(e^kt)

where

W = weight of the body at any time "t"

Wo = original weight of the body

k = exponential constant of proportionality

t = time

Substituting the given the data in the problem,

(W/Wo) = 1/2 = e^k(1)

Taking the natural logarithm of both sides,

ln (1/2) = k(ln e)

and since ln e = 1,

k = ln (1/2)

k = - 0.69315

Therefore, the original equation is rewritten as

W = Wo[e^ -(0.69315t)]

If after t = 8 hours, W = 5/16, then the above formula becomes

(5/16) = Wo[e^-0.69315(8)]

(5/16) = Wo[e^-5.54518]

Solving for Wo,

Wo = (5/16)(1/[e^-5.54518]

Wo = 80 pounds