f(n)= 10(1.02)^n

PART A) When the scientist concluded his study, the height of the plant was approximately 11.04 cm. What is a resonable domain to plot the growth function?

PART B) What does the y-intercept of the graph of the function f(n) represent?

PART C) What is the average rate of change of the function f(n) from n=1 to n= 5, and what does it represent?

PART A)

11.04 = 10(1.02)^n

1.104 = 1.02^n

ln 1.104 = ln 1.02^n

ln 1.104 = n ln 1.02

n = ln 1.104/ ln 1.02

n = 4.99630409516

4.99 can be rounded to 5. **So a reasonable domain would be 0 ≤ x < 5 **

PART B)

f(0) = 10(1.02)^0

f(0) = 10(1)

f(0) = 10 **The y-intercept represents the height of the plant when they began the experiment. **

PART C)

f(1) = 10(1.02)^1

f(1) = 10(1.02)

f(1) = 10.2

(1, 10.2)

f(5) = 10(1.02)^5

f(5) = 10(1.1040808)

f(5) = 11.040808

(5, 11.040808)

Find the slope between those two points.

\( \text{Slope } = \frac{ y_2 - y_1 } { x_2 - x_1 } \\ \\= \frac{ \frac{1380101}{125000} - \frac{51}{5}}{ 5 - 1} \\\\ =\frac{ \frac{105101}{125000}}{ 4} \\\\= \frac{ \frac{105101}{125000}\cdot125000}{ 4\cdot125000} \\\\=\frac{ 105101}{ 500000}\\\\\\=\boxed{\bf{0.210202}} \)**The slope represents how much the plant is growing every day. **

Let

f(n)-> is the height of the plant in cm

n-> is the number of days

we now that

The equation to show the height of the plant is equal to

\( f(n)= 10*(1.02)^n \)

**Part a)** When the scientist concluded his study, the height of the plant was approximately \( 11.04\ cm \). What is a reasonable domain to plot the growth function?

**Find the value of n for \( f(n)=11.04 \)**

\( f(n)= 10*(1.02)^n \)

\( 11.04= 10*(1.02)^n \)

\( 1.104=(1.02)^n \)

Applying logarithm both sides

\( ln(1.104)=ln(1.02)^n \)

\( ln(1.104)=n*ln(1.02)\\\\ n= \frac{ln(1.104)}{ln(1.02)} \\ \\ n=5\ days \)

So a reasonable domain would be

\( 0 \leq n \leq 5 \)

therefore

**the answer Part a) is**

**\( 0 \leq n \leq 5 \)**

see the attached figure N 1

**Part b)** What does the y-intercept of the graph of the function f(n) represent?

we know that

the y-intercept is when the value of n is equal to zero

so

\( n=0 \)

\( f(0)= 10*(1.02)^0 \)

\( f(0)= 10\ cm \)

The y-intercept represents the height of the plant when they began the experiment (\( 10\ cm \))

**see the attached figure N 1**

therefore

**the answer Part b) is**

**The y-intercept represents the height of the plant when they began the experiment**

**Part c)** What is the average rate of change of the function f(n) from n=1 to n= 5, and what does it represent?

**\( For \ n=1\ find\ f(1) \)**

\( f(1)= 10*(1.02)^1 \)

\( f(1)= 10.2\ cm \)

**\( For \ n=5\ find\ f(5) \)**

\( f(5)= 10*(1.02)^5 \)

\( f(5)= 11.04\ cm \)

**Find the slope m**

\( m=\frac{f(5)-f(1)}{(5-1)} \\ \\ m=\frac{(11.04-10.2)}{4} \\ \\ m=0.21\ \frac{cm}{day} \)

The slope represents how much the plant is growing every day (\( 0.21\ \frac{cm}{day} \))

therefore

**the answer part c) is **

**The slope represents how much the plant is growing every day**