Problem 11 - Algebra 1: Concepts and Skills

Question

Kudzu is a type of Japanese vine that grows at a rate of 1 foot per day during the summer. On August \(1,\) the length of one vine was 50 feet. What was the length on July \(1 ?\) HINT: July has 31 days. Use the verbal phrases to complete the verbal model. Total length Original length Number of days Growth rate

Step-by-Step Solution

Answer

The original length of the vine on July 1 was 19 feet.

Step 1: Identify known quantities

The final length of the vine on August 1 is 50 feet. The vine grows at a rate of 1 foot per day. The number of days from July 1 to August 1 is 31 days.

Step 2: Calculate total growth

Multiply the number of days (31) by the growth rate (1 foot per day) to find the total growth. This equals to 31 feet.

Step 3: Determine the original length

Subtract the total growth from the final length to find the original length. \(50 - 31 = 19\) feet. Therefore, the original length of the vine on July 1 was 19 feet.

Key Concepts

Linear Growth

Kudzu grows at a constant rate, which we describe as linear growth in mathematics. This means that every day, the vine increases by the same amount—exactly 1 foot per day. Linear growth is predictable, which makes it easier to calculate changes over a period of time. When you know the growth rate and the time involved, you can quickly determine how much the only thing that affects the vine’s length is the passing of days.
For example, if you know the growth rate (here it's 1 foot per day), you can calculate the total growth over any number of days by multiplying these values. This process gave us the total growth of 31 feet over 31 days from July 1 to August 1. Understanding this principle is crucial because it applies to many real-world problems where a quantity changes at a consistent rate.

Time Calculation

Calculating time accurately is crucial in solving word problems like determining the length of the Kudzu vine. To identify how many days passed from July 1 to August 1, you simply count up all the days in July. Since July has 31 days, this is our timeframe for calculating the vine's growth.

• Start counting from the first day.
• Make sure you include every day in the month.

For this specific problem, every single day is important because the vine grows at a regular daily rate. Hence, skipping or miscounting even one day could result in an incorrect calculation of the vine's initial length. Understanding basic time calculation is fundamental to tackling problems involving growth over time.

Subtraction

Subtraction is essential in finding the original length of the Kudzu vine. Once you've calculated how much the vine has grown over 31 days, you'll use subtraction to determine its starting length on July 1.
The process involves these steps:

• Find the total length on a particular day—in this case, 50 feet on August 1.
• Subtract the amount of growth over the 31 days, which we calculated as 31 feet.

This calculation: \(50 - 31 = 19\) simplifies to understanding how much the vine had grown during those days. So, its length at the beginning of July was 19 feet. Subtraction enables us to backtrack and accurately determine past values in linear growth scenarios.