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Difficulty: MediumLinear Equations and Graphing

A linear model predicts the height of a plant, HH, in inches, based on the number of weeks, ww, since it was planted. According to the model, the height of the plant after 22 weeks is 3123\frac{1}{2} inches, and its height after 44 weeks is 4144\frac{1}{4} inches. What was the initial height of the plant, in inches, when it was planted?

  1. A
    1561\frac{5}{6}
  2. B
    38\frac{3}{8}
  3. C
    2
  4. 2342\frac{3}{4}Answer
  5. E
    916\frac{9}{16}

Answer

The initial height of the plant was 2342\frac{3}{4} inches.
The correct answer is 2342\frac{3}{4} inches. First, the two given data points are (2,72)(2, \frac{7}{2}) and (4,174)(4, \frac{17}{4}). The rate of growth (slope) is calculated as m=174−724−2=38m = \frac{\frac{17}{4} - \frac{7}{2}}{4 - 2} = \frac{3}{8} inches per week. Substituting this into the slope-intercept form equation H=mw+bH = mw + b at w=2w = 2 gives 72=38(2)+b\frac{7}{2} = \frac{3}{8}(2) + b, which simplifies to b=72−34=234b = \frac{7}{2} - \frac{3}{4} = 2\frac{3}{4} inches.

Step-by-Step Solution

1
Convert the mixed numbers for the plant heights into improper fractions.
The height at 22 weeks is 312=723\frac{1}{2} = \frac{7}{2} inches, and the height at 44 weeks is 414=1744\frac{1}{4} = \frac{17}{4} inches.
Improper fractions are easier to use in slope and linear equation calculations.
2
Calculate the slope (weekly growth rate) of the linear function using the formula m=H2−H1w2−w1m = \frac{H_2 - H_1}{w_2 - w_1}.
m=174−724−2=342=38m = \frac{\frac{17}{4} - \frac{7}{2}}{4 - 2} = \frac{\frac{3}{4}}{2} = \frac{3}{8}
The slope represents the constant rate of height increase per week.
3
Use the slope-intercept form H=mw+bH = mw + b and substitute one of the points to solve for the y-intercept, bb.
Using the point (2,72)(2, \frac{7}{2}): 72=38(2)+b⇒72=34+b⇒b=72−34=144−34=114=234\frac{7}{2} = \frac{3}{8}(2) + b \Rightarrow \frac{7}{2} = \frac{3}{4} + b \Rightarrow b = \frac{7}{2} - \frac{3}{4} = \frac{14}{4} - \frac{3}{4} = \frac{11}{4} = 2\frac{3}{4}
The y-intercept, bb, represents the initial height of the plant at w=0w = 0 weeks.

Key Concept

Determining the equation of a line from two points and finding the y-intercept in a real-world context.
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Topics

Linear Equations and Graphing

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