To model the relationship between time (in months) and plant height (in cm) using the data provided, we can start by examining the data points:
| Time (months) | Height (cm) | |---------------|-------------| | 3 | 15 | | 5 | 25 | | 7 | 35 | | 9 | 45 |
a. Model the data with an equation
Looking at the pattern in the data, we can observe that the height of the plant increases linearly with time. The increase in height is consistent:
- From 3 to 5 months: \( 25 - 15 = 10 \) cm increase
- From 5 to 7 months: \( 35 - 25 = 10 \) cm increase
- From 7 to 9 months: \( 45 - 35 = 10 \) cm increase
This consistent increase suggests a linear equation of the form:
\[ h(t) = mt + b \]
where:
- \( h(t) \) is the height at time \( t \),
- \( m \) is the slope (rate of height increase),
- \( b \) is the initial height (height at \( t = 0 \)).
The slope \( m \) can be calculated as follows:
\[ m = \frac{\Delta h}{\Delta t} = \frac{10 \text{ cm}}{2 \text{ months}} = 5 \text{ cm/month} \]
To find \( b \), we can use one of the data points. Let's use the point at \( t = 3 \) months, where \( h(3) = 15 \):
\[ 15 = 5(3) + b \ 15 = 15 + b \ b = 0 \]
Thus, the equation modeling the growth of the plant is:
\[ h(t) = 5t \]
b. Predict the height of the plant at 12 months
Now, substituting \( t = 12 \) months into the model:
\[ h(12) = 5(12) = 60 \text{ cm} \]
Final Results
a. The model of the data is: \[ h(t) = 5t \]
b. The predicted height of the plant at 12 months is: \[ 60 \text{ cm} \]