Question

Mr. Samuels planted 40 pepper plants when he began his garden in 2005. Each year the number of pepper plants doubled. The number of plants, f(x), after x years is modeled with an exponential function. Which statement is true?
A.
The growth rate is 200%, and the y-intercept represents the 40 plants he started with.

B.
The growth rate is 100%, and the y-intercept represents the 40 plants he started with.

C.
The growth rate is 200%, and the y-intercept represents the year 2005.

D.
The growth rate is 100%, and the y-intercept represents the year 2005.

Answer 1

Given:

Initital Number of plants = 40

Given each year the number of pepper plants doubled.

f(x) represents the number of plants after x year.

To Find:

The correct growth rate regarding the function and its y - intercept.

Solution:

Given the number of plants doubles each year.

• Initial number of plants = 40
• Number of plants after 1 year = f(1) = 40 x 2
• Number of plants after 2 year = f(2) = 40 x 2 x 2 = 40 x  2²
• Number of plants after x year = f(x) = 40x $2^{x}$

Therefore the function can be represented as f(x) = y = 40x $2^{x}$

• Y intercept is the value of y for which x = 0.
• At x = 0,
• f(0) = 40 x 1 = 40
• y intercept represents the 40 plants he started with.

Lets find the growth rate.

• Growth rate = Number in (x + 1)th year - Number in x th year / Number in x th year.
• Growth rate = {f(x+1) - f(x)} / f(x)  x 100
• Growth rate = 40 ( $2^{x+1}$ - $2^{x}$)/40x$2^{x}$ = 1 x 100  = 100%

Therefore,

The growth rate is 100%, and the y-intercept represents the 40 plants he started with.