Question 4 (essay worth 10 points) (05.03 hc) 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)

A: Marigold is y = 150*(0.85)^x, Sunflower is y = 125- 8x. x is month here.

B: y3 = 150*(0.85)^3 = 92 for Marigold, y3 = 125-8*3 = 101

C: Let 150*(0.85)^x = 125-8x, you can draw the graph and see the intersection, where I get x = 2 or 13.5.                                                                                                  If a quantity A is decreased by 15%, it means that what is left is 85% of it.

85%A=

Part A.

Consider the 150 marigolds.

After the first month, 0.85*150 are left
After the second month, 0.85*0.85*150=
After the third month, 0.85*0.85*0.85*150 = 
.
.
so After n months,  marigolds are left.

in functional notation:  is the function which gives the number of marigolds after n months

consider the 125 sunflowers.

After 1 month, 125-8 are left
After 2 months, 125-8*2 are left
After 3 months, 125-8*3 are left
.
.
After n months, 125-8*n sunflowers are left.

In functional notation: S(n)=125-8*n is the function which gives the number of sunflowers left after n months

Part B.

 marigolds are left after 3 months.

S(3)=125-8*3=125-24=121 sunflowers are left after 3 months.

Part C.

equalizing M(n) to S(n) produces an equation which is very complicated to solve algebraically.

A much better approach is to graph both functions and see where they intersect.
 
Another approach is by trial, which gives 14 months

which are close numbers to each other?

We need more info then that