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 C is that correct or incorrect?
If it grows exponentially you mean
15,000 * 1.07^x
in other words compounding
not 1.07 x
In that case C is correct
1.07^12 = 2.25
A rough estimate to remember is that 7 percent compounded for ten years is about a doubling.
Yes, your answer is correct. The statement "The investment will more than double within 12 years" is true.
To verify this, let's substitute x = 12 into the equation y = 15,000 · 1.07x and calculate the value:
y = 15,000 · 1.07^12
≈ 15,000 · 1.967151
≈ 29,507.27
The investment amount after 12 years is approximately $29,507.27, which is more than double the initial investment of $15,000.
To determine which of the statements is true, let's analyze the given exponential equation y = 15,000 · 1.07^x.
A. The investment will continue to grow at a rate of 7% per year compounded quarterly.
To determine if this statement is true, we need to examine the exponent in the equation. In this case, the exponent is x, which represents the number of years invested. Since the base of the exponent is 1.07, this means that the investment will grow by 7% every year. However, the compounding frequency is not mentioned in the equation, so we cannot conclude if it will be compounded quarterly. Therefore, statement A is incorrect as it contains uncertain information.
B. The investment will increase by $1050 per year.
To determine if the investment will increase by $1050 per year, we need to find the difference between two values of y for consecutive years. Let's calculate the values:
For x = 1: y = 15,000 · 1.07^1 = 15,000 · 1.07 = 16,050
For x = 2: y = 15,000 · 1.07^2 = 15,000 · 1.1449 = 17,173.50
The difference between these two values is: 17,173.50 - 16,050 = 1,123.50, which is not equal to $1050. Therefore, statement B is incorrect as well.
C. The investment will more than double within 12 years.
To determine if the investment will more than double within 12 years, we can calculate the value of y for x = 12 and compare it with twice the initial investment, which is $30,000:
For x = 12: y = 15,000 · 1.07^12 = 15,000 · 2.094045 = 31,410.67
Since the value of y for x = 12 is greater than $30,000, the investment will indeed more than double within 12 years. Therefore, statement C is correct.
D. The investment will triple within 15 years.
To determine if the investment will triple within 15 years, we can calculate the value of y for x = 15 and compare it with three times the initial investment, which is $45,000:
For x = 15: y = 15,000 · 1.07^15 = 15,000 · 3.172055 = 47,580.83
Since the value of y for x = 15 is greater than $45,000, the investment will indeed triple within 15 years. Therefore, statement D is correct.
In summary, both statements C and D are true, so your answer of statement C being correct is correct.