Question

2. The price of a gallon of milk has been rising about 1.36% per year since 2000. a. What type of function would be best to model this scenario? Choose one of the types of functions studied in this course. Explain why you chose this answer. b. Write a formula for the function you chose to model this scenario. What does the independent variable in your function represent? c. If milk costs $4.70 now, what will it cost next year? Show how you found the answer. d. If milk costs $4.70 now, how long will it take for the price to top $5? Show how you found the answer.

Answer 1

(a)Exponential

(b)
`P(t)=4.70(1.0136)^t`

(c)The price of milk next year will be: $4.76

(d)5 years

Explanation:

The price of a gallon of milk has been rising about 1.36% per year since 2000.

(a)Since the price grows by a percentage (or constant factor) each year, an exponential function would be best to model the scenario.

(b)The exponential growth model is given as:

`P(t)=P_0(1+r)^t$ where:\\P_0$=Initial Price\\r=Growth factor\\t=time (in years, for this case)`

The independent variable in the function is t. This represents the number of years since 2000.

(c)

`I$nitial Price, P_0=\$4.70\\r=1.36\%=0.0136\\P(t)=4.70(1+0.0136)^t\\\\P(t)=4.70(1.0136)^t`

Therefore, the price of milk next year will be:

`P(1)=4.70(1.0136)^1=\$4.76`

(d)We want to determine how long it will take for the price to top $5.

P(t)=$5

`5=4.70(1.0136)^t\\$Divide both sides by 4.7$\\(1.0136)^t=(5)/(4.7) \\$Change to logarithm form\\t=Log_(1.0136)(5)/(4.7)\\t=4.58`

Therefore, in exactly 4.58 years, the milk price would be $5. Therefore, by the 5th year, the milk price would top $5.