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) = 8(1.05)n
Part A: When the scientist concluded his study, the height of the plant was approximately 11.26 cm. What is a reasonable 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 = 2 to n = 6, and what does it represent?
I genuinely dont know how to do this :-( Please help me understand
PART A:
f(n) = 8 × (1.05)^n
At n = 0 days, the plant was 8 cm high, according to this rule.
(Note: 1.05^0 = 1)
The difference in growth is:
11.26cm - 8cm = 3.26cm
The time taken for this extra growth is found from:
3.26 = 8×(1.05)^n
ln(3.26) = ln(8) + n ln(1.05)
1.18 = 2.08 + 0.05n
This gives a negative value of n. Please check the problem statement again and be certain that you have the correct equation!
PART B:
x represents time, y represents height of plant
PART C:
At n = 2, the height of the plant is 8×(1.05)^2 = 8.82 cm
At n = 6, the height of the plant is 10.72 cm
The average rate of growth is:
(10.72 - 8.82)/(6 - 2) = 1.9cm/4days
But again, I am using the equation you supplied. This equation needs to be checked very carefully!
The amount of money in an account may increase due to rising stock prices and decrease due to falling stock prices. Emerson is studying the change in the amount of money in two accounts, A and B, over time.
The amount f(x), in dollars, in account A after x years is represented by the function below:
f(x) = 1,264(1.09)x
Part A: Is the amount of money in account A increasing or decreasing and by what percentage per year? Justify your answer. (5 points)
Part B: The table below shows the amount g(r), in dollars, of money in account B after r years.
r (number of years) 1 2 3 4
g(r) (amount in dollars) 1,375 1,512.50 1,663.75 1,830.13
The amount of money in an account may increase due to rising stock prices and decrease due to falling stock prices. Emerson is studying the change in the amount of money in two accounts, A and B, over time.
The amount f(x), in dollars, in account A after x years is represented by the function below:
f(x) = 1,264(1.09)x
Part A: Is the amount of money in account A increasing or decreasing and by what percentage per year? Justify your answer. (5 points)
Part B: The table below shows the amount g(r), in dollars, of money in account B after r years.
r (number of years) 1 2 3 4
g(r) (amount in dollars) 1,375 1,512.50 1,663.75 1,830.13
f(n) = 8(1.05)^n, sorry
Part A: To determine a reasonable domain to plot the growth function, we need to consider the values of n that would make sense in the context of the problem. In this case, since we're studying the growth of a plant over time, it would make sense to have non-negative values for n. We cannot have negative days, and fractional or decimal days might not be meaningful for measuring plant growth. So, a reasonable domain to plot the growth function would be all non-negative integers (n = 0, 1, 2, 3, ...).
Part B: The y-intercept of a graph represents the value of the function when the independent variable (in this case, n) is equal to zero. Substituting n=0 into the equation f(n) = 8(1.05)^n, we get f(0) = 8(1.05)^0 = 8(1) = 8. Therefore, the y-intercept of the graph of the function f(n) represents the initial height of the plant when the study began, which is 8 cm.
Part C: The average rate of change of the function f(n) from n = 2 to n = 6 can be calculated by finding the slope of the line connecting the two points on the graph. The formula for average rate of change is (f(6) - f(2))/(6-2).
f(2) = 8(1.05)^2 = 8(1.1025) = 8.82 cm
f(6) = 8(1.05)^6 = 8(1.340096) ≈ 10.72 cm
The average rate of change is then (10.72 - 8.82)/(6-2) = 1.9/4 = 0.475 cm per day.
The average rate of change of the function f(n) from n = 2 to n = 6 represents the average increase in height per day over the four-day period.
Sure! Let's break down each part of the question step by step.
Part A: To find a reasonable domain to plot the growth function, we need to determine the range of valid input values for "n" (the number of days). In this case, since the plant's height is being measured in days, we would want to consider a non-negative whole number domain (i.e., natural numbers or positive integers) as it would not make sense to have a fraction or negative number of days. Thus, a reasonable domain to plot the growth function would be n ≥ 0.
Part B: The y-intercept of a graph represents the point at which the graph intersects the y-axis (when x = 0). Let's substitute x = 0 into the equation f(n) = 8(1.05)^n to find the y-intercept:
f(0) = 8(1.05)^0
f(0) = 8(1)
f(0) = 8
So, the y-intercept of the graph is at the point (0, 8). In the context of the plant's growth, this means that at the start of the study (n = 0 days), the height of the plant was approximately 8 cm.
Part C: The average rate of change of a function represents how much the output of the function changes on average for each unit increase in the input. To find the average rate of change of f(n) from n = 2 to n = 6, we subtract the initial value of the function at n = 2 from the final value at n = 6 and then divide by the difference in the input values:
Average rate of change = (f(6) - f(2)) / (6 - 2)
Substituting into the equation:
Average rate of change = [8(1.05)^6 - 8(1.05)^2] / (6 - 2)
Calculating this value will give you the average rate of change of the function f(n) from n = 2 to n = 6. In the context of the plant's growth, this value represents the average increase in height per day over the four-day period from day 2 to day 6.