An investment grows according to the exponential equation y = 15,000 · 1.07x, where x is the number of years invested. Which of the following statements is true?
A. The investment will continue to grow at a rate of 7% per year compounded quarterly.
B. The investment will increase by $1050 per year.
C. The investment will more than double within 12 years.
D. The investment will triple within 15 years.
I got D is that correct. If incorrect please correct me. Thank you!
It is not exponential the way you typed it.
If you meant y = 15,000(1.07)^x , which is what I suspect ...
A is certainly not the one
B is out since that would be linear
C.
2 = 1.07)^x
log2 = xlog1.07
x = log2/log1.07 = 10.24
so that is a correct choice so far
D
3 = 1.07^x
x = log3/log1.07 = 16.23
so 15 won't do it
correct choice is C
To determine which statement is true, we need to analyze the given exponential equation:
y = 15,000 · 1.07^x
Statement A states that the investment will continue to grow at a rate of 7% per year compounded quarterly.
To check if this statement is true, we can analyze the exponent of 1.07. The base, 1.07, represents an annual interest rate of 7%. If the interest is compounded quarterly, we need to divide the annual interest rate by 4 (since there are 4 quarters in a year). So the quarterly interest rate would be 7% / 4 = 1.75%.
Since the exponent x represents the number of years, there would be 4 times as many compounding periods in a year if the interest is compounded quarterly. Therefore, we need to multiply x by 4 in the exponential equation to calculate the growth rate compounded quarterly:
y = 15,000 · (1 + 0.0175)^(4x)
Since the given equation y = 15,000 · 1.07^x does not match this modified equation, Statement A is false.
Statement B states that the investment will increase by $1050 per year. To determine if this statement is true, we need to find the difference in the investment value between two consecutive years. Let's calculate the investment growth from year 1 to year 2:
y2 = 15,000 · 1.07^2 = 15,000 · 1.1449 ≈ 17,173.50
Investment growth from year 1 to year 2: y2 - y1 = 17,173.50 - 15,000 = 2,173.50
This indicates that the investment increases by $2,173.50 from year 1 to year 2. The same pattern continues in subsequent years. Therefore, Statement B is false since the growth is not a fixed amount of $1050 per year.
Statement C states that the investment will more than double within 12 years. To verify this statement, we can calculate the investment value after 12 years:
y12 = 15,000 · 1.07^12 ≈ 36,175.21
The investment value after 12 years is approximately $36,175.21, which is more than double the initial investment of $15,000. Thus, Statement C is true.
Statement D states that the investment will triple within 15 years. Let's calculate the investment value after 15 years:
y15 = 15,000 · 1.07^15 ≈ 49,093.10
The investment value after 15 years is approximately $49,093.10, which is more than triple the initial investment. Therefore, Statement D is true.
In summary, the correct statements are:
C. The investment will more than double within 12 years.
D. The investment will triple within 15 years.