find years and cost

Question

A gallon of milk costs 1.97 today and the cost has been increasing 2.5% for the last 20 years. how long will it take for a gallon of milk to cost $2.50?

Answer 1

Hi Jacklyn! So first off we have to figure out the rate that the price of milk will increase at in the future. Given that it's been increasing at 2.5% a year for the last 20 years, it's safe to assume it will continue to increase at that price.

 

Next we just have to set up an equation using what we know about exponential rates. 
Let t = number of years it will take a gallon of milk to cost $2.50
Then: $1.97*(1+0.025)^t= $2.50
We can get this equation because because the price of milk increases by 2.5% per year, or 0.025. So milk in one year will cost $1.97+(0.025)*$1.97

We can factor out $1.97 to get: (1 + 0.025) * $1.97 =1.025 *$1.97

Milk in 2 years would cost 1.025 * the price of milk in one year = 1.025*(1.025 *$1.97) = $1.97 * (1.025)²

So milk in t years would cost: $1.97 * (1.025)^t 

We then set this equal to $2.50, to get: $1.97 * (1.025)^t = $2.50

Solving this equation is pretty easy from there! 
1.97 *(1.025)^t = 2.5
1.025^t = 2.5/1.97
1.025^t = 1.269
And then, using logarithms, we can get:

log(1.025^t) = log(1.269)

And using the exponent property of logarithms we can get:

t * log(1.025) = log(1.269)
t = log(1.269)/log(1.025)

t = 9.6478
So it will take a gallon of milk 9.6478 years to cost $2.50. Hope this helps!