Problem 15 - 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. Check that your answer is reasonable.

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

This problem tells that a vine grows at a rate of 1 foot per day. We know the length of the vine on August 1st is 50 feet. We have to find out the length of this vine on July 1st. Since the vine grows equally every day, we can simply subtract the number of days in July from the length of the vine on August 1st to get the answer.

Step 2: Subtract July days from August 1st vine length

There are 31 days in the month of July. Since each day the vine grows 1 foot, the vine would have grown 31 feet in 31 days of July. So, on July 1st, the vine was 50 (Length of the vine on August 1st) - 31 (Growth of the vine in July) = 19 feet long.

Step 3: Check Reasonableness of the Answer

The answer must be smaller than the length on August 1st, which is 50 feet, because the vine is growing every day. Our answer of 19 feet for the vine length on July 1st is less than that of August 1st, so it seems reasonable.

Key Concepts

Rate of Change

In mathematics, the rate of change defines how one quantity changes in relation to another quantity over time. In this exercise, the vital rate of change is the growth of the kudzu vine. The vine grows at a rate of 1 foot per day during the summer.
This means that for each day that passes, the length of the vine increases by 1 foot. The rate of change is often a constant value in scenarios of linear growth, which is what we have here.

• If the rate of growth is constant, like the 1 foot per day for the kudzu vine, the graph representing this growth would be a straight line.
• It is a linear function because the change in the length of the vine is directly proportional to the number of days it grows.
• Understanding this concept is crucial because it allows predictions of future quantities (e.g., the vine's future length) based on past constants and values.

Recognizing and calculating the rate of change gives us a powerful tool to anticipate and backtrack results.

Time Intervals

Time intervals represent the span of time over which changes occur. In this problem, the interval of interest is from July 1st to August 1st, covering the 31 days of July.

• Each day is an individual step where the vine's length changes.
• The total time interval we are looking at is these 31 days, which significantly impacts our calculations for the vine's growth.

Breaking down the problem into these time intervals helps us systematically calculate how much the kudzu vine has grown.
In each daily interval, the vine grows by 1 foot, so across the entire month, it gains 31 feet.
By considering time intervals, learners can compartmentalize complex problems into manageable parts, making the operations simpler and clearer.

Subtraction

Subtraction plays a crucial role in this exercise as it helps determine the initial condition of the vine. Once we know how much the vine grew in July (31 feet), we simply subtract this from the known length on August 1st (50 feet).
With subtraction, we compute:\[\text{Length on July 1st} = \text{Length on August 1st} - \text{Growth in July}\]This subtraction helps us backtrack and find that the vine was 19 feet long on July 1st.

• Subtraction allows comparing current measurements against past or predicted values.
• By taking away the amount of growth in July, we can see what the vine's length was before this growth occurred.

It's a fundamental arithmetic operation that lets us reverse what happens during time intervals of growth, offering a clear view into previous states.