Answer:
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.
Functions are used to represent tables and equations.
(a) The equation of the functions
The function of "Buy New Bulbs" is a linear function with an initial value of 6, and a rate of 50 (i.e. 56 - 6)
So, the equation is:
[tex]y = 6 + 50x[/tex]
The function of "Divide Existing Bulbs" is an exponential function with an initial value of 6, and a rate of 2 (i.e. 12/6)
So, the equation is:
[tex]y = 6(2)^x[/tex]
Hence, the equations of the functions are [tex]y = 6 + 50x[/tex] and [tex]y = 6(2)^x[/tex]
The expected number of bulbs in 10 years
This means that x = 10.
So, we have:
[tex]y = 6 + 50x =6 + 50*10 = 506[/tex]
[tex]y = 6(2)^x =6 *2^{10} = 6144[/tex]
Hence, the expected number of bulbs in 10 years for both functions are 506 and 6144, respectively
Compare the expected number of bulbs in 5 and 10 years
When x = 5.
We have:
[tex]y = 6 + 50x =6 + 50*5 = 256[/tex]
[tex]y = 6(2)^x =6 *2^5 = 192[/tex]
When x = 15
We have:
[tex]y = 6 + 50x =6 + 50*15 = 756[/tex]
[tex]y = 6(2)^x =6 *2^{15} = 196608[/tex]
By comparison;
In 5 years, the expected number of bulbs for the function of "Buy New Bulbs" is greater than the function of "Divide Existing Bulbs", however the expected number of bulbs for the function of "Buy New Bulbs" is less than the function of "Divide Existing Bulbs" in 15 years
Read more about functions at:
https://brainly.com/question/15602982
