Since the beginning of the year, the price of a bottle of soda at a local discount supermarket
has been rising at a constant rate of 2 cents per month. By November first, the price had
reached $1.56 per bottle. Express the price of the soda as a function of time and determine the
price at the beginning of the

Let the price of the soda bottle be a function of time, $ P(t) $, where $ t $ is the number of months since the beginning of the year. At $ t = 0 $, this represents the price at the beginning of January.

Given the information:

- The price increases at a constant rate of 2 cents per month.
- By November 1st (which is 10 months from the beginning of the year), the price is $1.56.

We can model this with a linear function of the form:

\[
P(t) = P_0 + rt
\]

Where:
- $ P(t) $ is the price at month $ t $.
- $ P_0 $ is the price at the beginning of the year (when $ t = 0 $).
- $ r $ is the rate of price increase per month.

### Step 1: Set up the equation using the known values
We know that by November 1st (at $ t = 10 $), the price is $1.56:

\[
P(10) = P_0 + 10r = 1.56
\]

Since the rate of increase $ r $ is 2 cents per month (which is $0.02 per month), substitute this into the equation:

\[
P_0 + 10(0.02) = 1.56
\]

\[
P_0 + 0.20 = 1.56
\]

\[
P_0 = 1.56 - 0.20 = 1.36
\]

### Step 2: Write the equation for $ P(t) $
Now that we know $ P_0 = 1.36 $ and $ r = 0.02 $, the equation for the price as a function of time is:

\[
P(t) = 1.36 + 0.02t
\]

### Step 3: Determine the price at the beginning of the year
At the beginning of the year ($ t = 0 $):

\[
P(0) = 1.36 + 0.02(0) = 1.36
\]

Thus, the price at the beginning of the year was **$1.36**.

### Summary:
The price of the soda as a function of time is:

\[
P(t) = 1.36 + 0.02t
\]

And the price at the beginning of the year was **$1.36**.

Would you like more details or explanations on this? Here are some related questions you might find interesting:

1. How would the function change if the price increased by 3 cents per month instead of 2 cents?
2. What will the price be at the end of the year?
3. How can you graph the function $ P(t) $ over the course of the year?
4. How would a sudden increase in price mid-year affect the function?
5. What would happen to the function if the price decreased instead of increasing?

**Tip:** When modeling real-world scenarios, linear functions are often used for problems involving constant rates of change over time.

Since the beginning of the year, the price of a bottle of soda at a local discount supermarket has been rising at a constant rate of 2 cents per month. By November first, the price had reached $1.56 per bottle. Express the price of the soda as a function of time and determine the price at the beginning of the year.

Learn how to express the price of a bottle of soda as a linear function over time, with a constant increase of 2 cents per month. By November, the price reached $1.56. Discover how to calculate the price at the beginning of the year and explore the full solution, suitable for students in Grades 7-9.

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