Question

A gardener measured the heights of two plants at the end of every week. The function y = 3x + 8.5 gives the height of plant A in centimeters at the end of x weeks. The function y = 2.5x + 14.5 gives the height of plant B in centimeters at the end of x weeks. Based on this information, which of these statements is true?

1) The height of plant A is always greater than the height of plant B.
2) The height of plant B is always greater than the height of plant A.
3) The height of plant A is initially greater than the height of plant B, but eventually becomes less.
4) The height of plant B is initially greater than the height of plant A, but eventually becomes less.

Answer 1

Initially, Plant B is taller because it has a larger y-intercept (14.5 cm compared to 8.5 cm for Plant A). However, Plant A grows at a faster rate, and by solving the equation 3x + 8.5 = 2.5x + 14.5, we find the exact week when Plant A becomes taller, thus Plant A's height is initially less but eventually becomes greater than Plant B.

Step-by-step explanation:

The heights of two plants, A and B, are described by the linear functions y = 3x + 8.5 and y = 2.5x + 14.5 respectively, where y represents the height in centimeters and x the number of weeks after planting. To determine which plant will be taller and when, we can compare the y-intercepts and slopes of the equations.

Plant A starts with a height of 8.5 cm (y-intercept) and grows 3 cm per week (slope). Plant B starts higher with 14.5 cm but grows at a slower rate of 2.5 cm per week. Initially, Plant B is taller because 14.5 cm > 8.5 cm. However, since Plant A has a steeper growth rate, there will come a point where Plant A's height will surpass Plant B's.

To find when Plant A becomes taller than Plant B, we set their equations equal to each other: 3x + 8.5 = 2.5x + 14.5. Solving this equation for x will give us the week when the height of Plant A becomes equal to that of Plant B, after which Plant A will continue to be taller.