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)

Let us first assign variables. We say that:

X = number of marigold plants

Y = number of sunflower plants

n = number of months

We can see that in the given problem, X is decreasing by a percentage, this means that we have to set-up a geometric equation while for Y the decrease is linear so we set-up an arithmetic equation.

Part A.

For marigold plants X, a geometric sequence has a general form of:

X = Xo * (1 + r)^n

where r = -15% = -0.15   (negative since it is decreasing)

Xo = the initial amount of marigold plants = 150

X = 150 * (1 – 0.15)^n

X = 150 (0.85)^n

For the sunflower plants Y, an arithmetic sequence has a general form of:

Y = Yo + d * n

where d = -8 and Yo = 125

Y = 125 – 8 n

Part B. For n = 3

X = 150 (0.85)^3 = 92.12 = 92

Y = 125 – 8 (3) = 101

Part C. From Part B we see that the two values are very far from each other when n = 3, therefore they must be similar when n < 3. So we try n = 2

X = 150 (0.85)^2 = 108.38 = 108

Y = 125 – 8 (2) = 109

Therefore the two plants have approximately similar amount after 2 months.

what is the original number? you forgot to add it.