A gardener can increase the number of dahlia plants in an annual garden by either buying new bulbs each year or dividing the existing to create new plants. The table below shows the expected number of bulbs for each method. Need Part A-C

Step-by-step explanation:

Part A

If the gardener increases the number of dahlia plants by buying the new bulbs,

Let the equation representing year (x) and number of bulbs purchased (y) is,

y = mx + b

Here, m = Number of plants purchased each year

b = Initial numbers of plants

For x = 0, and y = 6,

6 = m(0) + b

b = 6

For x = 1, y = 56,

56 = m(1) + 6

m = 50

Therefore, equation for this method of increasing the numbers of plants will be,

y = 50x+ 6

Let the equation for the second method of increasing the number of plants is represented by the equation,

y = a(b)ˣ

From the data given in the table,

For x = 0, y = 6

6 = a(b)⁰

a = 6

For x = 1, y = 12

12 = 6(b)¹

b = 2

Therefore, equation representing the second method (By diving the existing bulbs) will be,

y = 6(2)ˣ

Part B

By substituting the value of x = 10 years,

Method - 1

y = 50(10) + 6

y = 506 plants

Method - 2

y  = 6(2)¹⁰

y = 6144 plants

Part C

After 5 years number of plants by method - 1,

y = 50(5) + 6

  = 256

After 5 years number of plants by method - 2,

y = 6(2)⁵

y = 192

Similarly, After 10 years number of plants by method - 1,

y = 50(10) + 6

  = 506

By method - 2,

y = 6(2)¹⁰

y = 6144

Since, number of plants after 5 years are less by the second method as compared to method 1

But after 10 years second method results the number of plants more than method-1,

Therefore, gardener should adopt method 2 for increasing the numbers of plants.