Question

Once a certain plant begins to grow, its height increases at a linear rate. After six weeks, the plant is 54 centimeters tall. Which of the following functions best models the relationship between \(h(w)\), the height, in centimeters, of the plant, and \(w\), the number of weeks that the plant has been growing? A) \(h(w)=6 w\) B) \(h(w)=9 w\) C) \(h(w)=54 w\) D) \(h(w)=54+w\)

Short answer

The best fitting function for the relationship between the height and the number of weeks is \(h(w) = 9w\) (Option B).

Step-by-step solution

Analyze the given information

We know that the plant's height increases at a linear rate, which means this can be represented by a linear equation in the form of \(h(w) = m w + b\), where \(m\) is the slope (rate of growth) and \(b\) is the initial height of the plant. Since no information is given about the initial height, we'll simply focus on the slope. We are given that after six weeks (\(w = 6\)), the plant is 54 centimeters tall (\(h(w) = 54\)). This can be written as a coordinate point (6, 54).

Calculate the slope

To find the slope, we can use the formula \(m = \frac{h(w) - b}{w}\), but since we don't have information about the initial height, we can use the point (6, 54) to determine the slope by dividing the height by the number of weeks: \(m = \frac{54}{6}\) \(m = 9\) Now we have the slope, which represents the rate of growth per week.

Determine the best fitting function

We found the slope to be 9, and we will compare this with the given functions to see which one is the best fit. A) \(h(w) = 6w\) - The slope in this function is 6, which does not match our calculated slope of 9. B) \(h(w) = 9w\) - The slope in this function is 9, which matches our calculated slope. C) \(h(w) = 54w\) - The slope in this function is 54, which does not match our calculated slope of 9. D) \(h(w) = 54 + w\) - The slope in this function is 1, which does not match our calculated slope of 9. Based on our calculations, the best fitting function for the relationship between the height and the number of weeks is: \(h(w) = 9w\) (Option B)