Question

There are 150 marigold plants in a back yard. Each month, the number of marigold plants decreases by 15%. There are 125 sunflower plants in the back yard. Each month, 8 sunflower plants are removed. Part A: Write functions to represent the number of marigold plants and the number of sunflower plants in the back yard throughout the months. (4 points) Part B: How many marigold plants are in the back yard after 3 months? How many sunflower plants are in the back yard after the same number of months? (2 points) Part C: After approximately how many months is the number of marigold plants and the number of sunflower plants the same? Justify your answer mathematically. (4 points)

Answer 1

Part A: There are initially 150 marigold plants and the number at the end of each month is 100% - 15% = 85% of what it was at the beginning. The function representing the number of marigold plants can be written as an exponential function.
m(x) = 150·0.85^x

There are initially 125 sunflower plants and the number at the end of each month is 8 fewer than at the beginning. The decrease is the same every month. The function representing the number of sunflower plants is a linear function.
s(x) = 125 - 8x

Part B:
At the end of 3 months, the number of marigold plants remaining is
m(3) = 150·0.85³ ≈ 92
At the end of 3 months, the number of sunflower plants remaining is
s(3) = 125 - 8·3 = 121

Part C:
There are no algebraic methods for solving an equation like
m(x) = s(x)
150·0.85^x = 125 - 8x
However, it can be solved using a graphing calculator. The graph shows there to be two solutions:
after 1.93 months . . . . about 110 of each plant
after 13.55 months . . . . about 17 of each plant