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?