1.A scientist is studying the growth of a particular species of plant. He writes the following equation to show the height of the plant f(n), in cm, after n days:

f(n) = 12(1.03)n

Part A: When the scientist concluded his study, the height of the plant was approximately 16.13 cm. What is a reasonable domain to plot the growth function? (4 points)

Part B: What does the y-intercept of the graph of the function f(n) represent? (2 points)

Part C: What is the average rate of change of the function f(n) from n = 3 to n = 10, and what does it represent? (4 points)

Part A: A reasonable domain to plot the growth function would be the range of values for n that the scientist observed during his study. Let's assume the scientist studied the growth of the plant for a period of 20 days. Therefore, a reasonable domain to plot the growth function would be n = 0 to n = 20.

Part B: The y-intercept of the graph represents the initial height of the plant, or the height of the plant on day 0. In this case, the y-intercept of the function f(n) would be f(0) = 12(1.03)^0 = 12.

Part C: The average rate of change of the function f(n) from n = 3 to n = 10 can be calculated by finding the difference in height between the two days and dividing it by the difference in time (days) between the two days.

f(3) = 12(1.03)^3 ≈ 12.948 cm
f(10) = 12(1.03)^10 ≈ 16.130 cm

The average rate of change would be (16.130 - 12.948) / (10 - 3) = 0.521 cm/day.

This represents the average daily increase in height of the plant from day 3 to day 10.

Part A: To plot the growth function, we need to determine a reasonable domain. In this case, the height of the plant is given by the function f(n) = 12(1.03)^n, where n represents the number of days. Since the height of the plant is already given as 16.13 cm, we can substitute this value into the equation:

16.13 = 12(1.03)^n

To find a reasonable domain, we need to solve for n.

Dividing both sides of the equation by 12, we have:

16.13/12 = (1.03)^n

Simplifying the left side of the equation:

1.344166667 = 1.03^n

To isolate the variable, we take the logarithm of both sides of the equation:

log(1.344166667) = log(1.03^n)

Using the logarithm properties, we can bring the exponent down:

log(1.344166667) = n*log(1.03)

Finally, we can solve for n by dividing both sides of the equation by log(1.03):

n = log(1.344166667) / log(1.03)

Using a calculator, we find that n ≈ 7.15.

Therefore, a reasonable domain to plot the growth function would be from n = 0 to n = 8 (rounded up from 7.15).

Part B: The y-intercept of the graph of the function f(n) represents the initial height of the plant when n = 0.

Substituting n = 0 into the equation f(n) = 12(1.03)^n, we get:

f(0) = 12(1.03)^0

Simplifying, we have:

f(0) = 12(1)

f(0) = 12

Therefore, the y-intercept of the graph represents an initial height of 12 cm for the plant.

Part C: The average rate of change of the function f(n) from n = 3 to n = 10 represents how the height of the plant changes on average per day over that range.

To find the average rate of change, we need to calculate the difference in height between f(10) and f(3), and divide it by the difference in days:

Average rate of change = (f(10) - f(3)) / (10 - 3)

Substituting the values into the equation:

Average rate of change = (12(1.03)^10 - 12(1.03)^3) / (10 - 3)

Using a calculator, we find:

Average rate of change ≈ 1.194 cm/day

Therefore, the average rate of change of the function f(n) from n = 3 to n = 10 is approximately 1.194 cm/day, which represents the average daily growth rate of the plant over that time interval.

Part A: To determine a reasonable domain to plot the growth function, we need to consider the given information. The height of the plant is approximately 16.13 cm when the scientist concluded the study. We can set up the equation and solve for n:

16.13 = 12(1.03)n

Divide both sides by 12:

1.3444 = (1.03)n

Take the logarithm of both sides:

log(1.3444) = log(1.03)n

Using the logarithmic property, we can bring the exponent down:

log(1.3444) = n * log(1.03)

Now, solve for n by dividing both sides by log(1.03):

n ≈ log(1.3444) / log(1.03)

Calculating this value, we find that n ≈ 11.3.

Since the study concluded when the height was approximately 16.13 cm, it is reasonable to plot the growth function in the domain [0, 11.3] or approximately [0, 12] (rounded up to the nearest whole number).

Part B: The y-intercept of the graph of the function f(n) represents the initial height of the plant when the number of days (n) is zero. In this case, plugging in n = 0 into the equation:

f(0) = 12(1.03)^0
f(0) = 12(1)
f(0) = 12

So the y-intercept is 12, which means the initial height of the plant was 12 cm.

Part C: The average rate of change of the function f(n) from n = 3 to n = 10 can be found by calculating the slope of the function over that interval. The average rate of change is the change in the function's value divided by the change in the input (days in this case).

f(10) = 12(1.03)^10 ≈ 18.1908cm
f(3) = 12(1.03)^3 ≈ 13.3212cm

Change in function's value = f(10) - f(3) ≈ 18.1908cm - 13.3212cm ≈ 4.8696cm
Change in input = 10 - 3 = 7

Average rate of change = (Change in function's value) / (Change in input) = 4.8696cm / 7 ≈ 0.6957cm/day

The average rate of change of the function f(n) from n = 3 to n = 10 is approximately 0.6957 cm/day. It represents the average growth rate of the plant during that time interval.