Question 1146030: Calc 2 Problem:
The rate at which sugar dissolves in water is proportional to the amount that remains. Suppose that 10 lb of sugar is placed in a container of water at 1:00pm and one half is dissolved at 4:00pm.
a) Find a formula for the amount of sugar remaining in the container after t hours. How long will it take to have 3 lb of sugar remaining?
b) How much of the 10 lb will remain in the container at 8:00pm?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! a formula that can be used is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the rate per time period
n is the number of time periods.
in this problem:
f = 5
p = 10
r = what you want to find
n = 3 hours
formula becomes 5 = 10 * (1 + r) ^ 3
divide both sides of this formula by 10 to get .5 = (1 + r) ^ 3
take the third root of both sides of this equation to get .5 ^ (1/3) = 1 + r
subtract 1 from both sides of this equation to get .5 ^ (1/3 - 1 = r
solve for r to get r = -.206299474.
replace r in the original equation with that to get:
.5 = (1 - .206299474) ^ 3 = .5
this confirms the value of r is correct.
in t hours, when r = -.206299474 and p = 10 and f = 3, the formula of f = p * (1 + r) ^ n becomes:
3 = 10 * (1 - .206299474) ^ t
divide both sides by 10 to get:
3/10 = (1 - .206299474) ^ t
take the log of both sides to get:
log(3/10) = log(1 - .206299474) ^ t)
by log rules, this becomes:
log(3/10) = t * log(1 - .206299474)
divide both sides by log(1 - .206299474) to get:
log(3/10) / log(1 - .206299474) = t
solve for t to get:
t = 5.2108967682
replce t in the original equation to get:
3 = 10 * (1 - .206299474) ^ 5.2108967683 = 3
this confirms the number of time periods is correct.
the 10 pounds of undissolved sugar will be reduced to 3 in that number of hours.
if the sugar is placed in the container at 1:00 pm, then at 8:00 pm, the number of hours it has been in the container is 7 hours.
the formula becomes f = 10 * (1 - .06299474) ^ 7
solve for f to get:
f = 1.984251315
that's how much sugar has not been dissolved in 7 hours.
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