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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)

Respuesta :

barnuts

To solve this problem, let us first assign variables. Let us 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.

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