Question

First solve the problem, and then enter your answer on the grid provided on the answer sheet. The instructions for entering your answers follow. A botanist plants a small ivy plant and evaluates its growth function. She finds that 2 months after planting, the plant is 5 inches tall; at 4 months after planting, the plant is 8 inches tall. Additionally, the botanist has noticed that the plant has grown at a constant rate since its initial planting. Given this information, what was the plant's height in inches at the time it was planted?

Short answer

The plant was 2 inches tall when it was planted.

Step-by-step solution

Understand the Problem

We are given that a plant grows at a constant rate. At 2 months, the plant is 5 inches tall and at 4 months, it is 8 inches tall. We are to find the height of the plant at 0 months, i.e., when it was planted.

Identify Key Variables

Let the initial height of the plant at 0 months be represented by \( h_0 \). The plant's height after time \( t \) can be given by a linear equation, which is generally represented as \( h(t) = mt + c \), where \( m \) is the rate of growth (slope) and \( c \) is the initial height, which is \( h_0 \).

Calculate the Growth Rate, m

We have two points: (2, 5) and (4, 8). The growth rate \( m \) can be found by the change in height over the change in time: \[ m = \frac{8 - 5}{4 - 2} = \frac{3}{2} \text{ inches per month.} \]

Use the Linear Growth Equation

Now that we know \( m \), we use the point-slope form to find \( h_0 \). Given the point (2, 5), use the equation: \[ 5 = \frac{3}{2} \times 2 + c \] Solve for \( c \): \[ 5 = 3 + c \] \[ c = 2 \]

Determine Initial Height, h_0

The value of \( c \) we found is indeed \( h_0 \), the initial height at the time it was planted: \[ h_0 = 2 \text{ inches.} \]

Key concepts

Understanding Constant Rate of Change

The concept of constant rate of change is fundamental in understanding linear functions. In any scenario where a rate remains steady over time, we can describe this with a linear equation. Consider the example of our ivy plant. The growth of the plant at a steady, unchanging rate means that the distance (or growth, in this case) covered over a period is predictable.

In a linear equation, the constant rate of change is often referred to as the "slope," represented by the letter \( m \). It's calculated by the change in the dependent variable (plant height) divided by the change in the independent variable (time). For instance:

\[ m = \frac{\text{change in height}}{\text{change in time}} \]

This ensures that for every month that passes, the height increases by the same amount. The linear relationship makes predicting future behavior, like the plant's eventual height, straightforward.

Solving Botany Problems Using Mathematical Models

In botany, especially when studying plant growth, it's important to use mathematical models like linear equations. These help quantify changes and provide insights into growth patterns. Botanists often observe plant changes over time, collecting data at different intervals. With this data, they create models to understand plant behavior and make predictions.

For example, botanists gather data at interval times such as 2 months and 4 months to check a plant's height (5 inches and 8 inches, respectively). Using this, they develop a linear model that demonstrates how plants grow over time. The model can also help foresee how changes in conditions might impact growth.

By following these calculated models, botanists don't just monitor current growth steps; they also extend their understanding to potential future developments. Such models become very useful in agricultural planning and ecosystem management.

Rate of Growth Calculation Techniques

To calculate rate of growth, especially in plants, you would typically look at data points where the growth rate was observed. In our example with the ivy plant, measuring its height at two different time points enabled us to determine its rate of growth.

The process involves several simple steps:

• Identify at least two points in time where you have data.
• Calculate the change in the plant's height between these points.
• Divide the height change by the corresponding change in time.

This computation will yield the rate at which the plant grows per unit of time. For our ivy plant:

\[ m = \frac{8 - 5}{4 - 2} = \frac{3}{2} \text{ inches per month} \]

This consistent rate helps to support planning and timing in farming, gardening, or ecological studies. It's crucial for determining not only current growth trends but also predicting future growth events. Understanding this enables better resource allocation and care strategies for various plant species.