Alex is deciding between two investment options, Option A and Option B. He wants to choose the option that has the highest rate of growth over the first 3 years, because he plans to withdraw his investment at that time. The growth of Option A and Option B can be modeled by their respective functions, A(x) and B(x), which follow.


A(x)=5√2x+7

B(x)=2^x+9

Part A: What is the average rate of growth of A(x) for the first 3 years?

Part B: What is the average rate of growth of B(x) for the first 3 years?

Part C: Which option should Alex invest in? Why?

Select one answer for Part A, one answer for Part B, and one answer for Part C.

To find the average rate of growth for the first 3 years, we need to calculate the difference in the values of the functions at the endpoints of the 3-year interval and divide by the length of the interval.

Part A: Average rate of growth of A(x) for the first 3 years
To find the difference in the values of A(x) at the endpoints of the 3-year interval, we evaluate A(x) at x=3 and x=0:
A(3) = 5√2(3)+7 = 5√6+7
A(0) = 5√2(0)+7 = 0+7 = 7

Now, we calculate the difference in the values:
Difference = A(3) - A(0) = (5√6+7) - 7 = 5√6

The length of the interval is 3 years. Therefore, the average rate of growth of A(x) for the first 3 years is:
Average rate of growth of A(x) = Difference / Interval length = 5√6 / 3

Part B: Average rate of growth of B(x) for the first 3 years
To find the difference in the values of B(x) at the endpoints of the 3-year interval, we evaluate B(x) at x=3 and x=0:
B(3) = 2^3+9 = 8+9 = 17
B(0) = 2^0+9 = 1+9 = 10

Now, we calculate the difference in the values:
Difference = B(3) - B(0) = 17 - 10 = 7

The length of the interval is 3 years. Therefore, the average rate of growth of B(x) for the first 3 years is:
Average rate of growth of B(x) = Difference / Interval length = 7 / 3

Part C: Which option should Alex invest in? Why?
To determine which option Alex should invest in, we need to compare the average rates of growth of A(x) and B(x) for the first 3 years.

Comparing the average rates of growth, we have:
Average rate of growth of A(x) = 5√6 / 3
Average rate of growth of B(x) = 7 / 3

Since 5√6 is greater than 7, Option A has a higher average rate of growth. Therefore, Alex should invest in Option A because it is expected to have a higher rate of growth over the first 3 years.

To find the average rate of growth of a function over a specific interval, you need to calculate the change in the function's values divided by the change in the input variable over that interval.

Part A: To find the average rate of growth of A(x) for the first 3 years, we need to evaluate A(x) at the endpoints of the interval and then calculate the change in the function's values divided by the change in the input variable. In this case, the input variable is x, which represents time in years. So, for the first 3 years, we need to evaluate A(x) at x = 0 and x = 3.

A(0) = 5√2(0) + 7 = 7
A(3) = 5√2(3) + 7 = 5√2(3) + 7

The change in the function's values is A(3) - A(0):
Change in A(x) = [5√2(3) + 7] - 7

The change in the input variable is 3 - 0 = 3.

The average rate of growth of A(x) for the first 3 years is (A(3) - A(0)) / 3.

Part B: Similarly, to find the average rate of growth of B(x) for the first 3 years, we need to evaluate B(x) at the endpoints of the interval x = 0 and x = 3.

B(0) = 2^0 + 9 = 1 + 9 = 10
B(3) = 2^3 + 9 = 8 + 9 = 17

The change in B(x) is B(3) - B(0):
Change in B(x) = 17 - 10

The change in the input variable is 3 - 0 = 3.

The average rate of growth of B(x) for the first 3 years is (B(3) - B(0)) / 3.

Part C: To determine which option Alex should invest in, we need to compare the average rates of growth of A(x) and B(x) for the first 3 years. Whichever option has the higher average rate of growth will provide the better return on investment.

Compare the average rates of growth of A(x) and B(x) calculated in Part A and Part B respectively. Select the option with the higher rate as the answer for Part C.

Part A: (A(3)-A(0))/(3-0)

Part B: (B(3)-B(0))/(3-0)
Since the denominators are the same, just compare the growth A and B over the interval.