Questions: The price of a gallon of milk is increasing at a rate of 1% per year. If a gallon of milk costs 3 in 2014, how much will a gallon of milk cost in 2018?

The price of a gallon of milk is increasing at a rate of 1% per year. If a gallon of milk costs 3 in 2014, how much will a gallon of milk cost in 2018?

Transcript text: The price of a gallon of milk is increasing at a rate of $1 \%$ per year. If a gallon of milk costs $\$ 3$ in 2014 , how much will a gallon of milk cost in 2018?

Solution

Solution Steps

To find the cost of a gallon of milk in 2018, we need to calculate the compound interest over 4 years (from 2014 to 2018) with an annual increase rate of 1%. The formula for compound interest is used here: $ \text{Final Amount} = \text{Principal} \times (1 + \text{Rate})^{\text{Time}} $.

Step 1: Initial Setup

Let the initial cost of a gallon of milk in 2014 be $ P = 3.0 $ dollars. The annual increase rate is $ r = 0.01 $ (or $ 1\% $). We need to calculate the cost after $ t = 4 $ years (from 2014 to 2018).

Step 2: Apply the Compound Interest Formula

The formula for the future value with compound interest is given by: \[ A = P \times (1 + r)^t \] Substituting the known values: \[ A = 3.0 \times (1 + 0.01)^4 \]

Step 3: Calculate the Final Amount

Calculating the expression: \[ A = 3.0 \times (1.01)^4 \approx 3.0 \times 1.04060401 \approx 3.12181203 \] Rounding to four significant digits, we find: \[ A \approx 3.122 \]