I am confused on how to solve this problem :
$5000 is invested for 4 years at 7% per annum compound interest.
a. what will the amount be at the end of this period?
b. what is the expression for the value of the investment after n full years?
c. the investment will exceed $10,000 after n full years. what is an inequality that represents this information? what is the value of n?
The formula that i have learned for this isn't working for me and i don't know how to do this.
a. The fomula you should be using, for a four year period of deposit, is
Final value = (Initial value)*(1.07)^4
b. Replace "4" with the number of years "n" for other lengths of time
c. 5000*(1.07)^n > 10,000
(1.07)^n > 2
n log 1.07 > log 2
n > 10.24 years
thanks
To solve this problem, you need to understand compound interest and how it is calculated. Compound interest is different from simple interest because it accumulates not only on the original amount invested but also on any interest that has already been earned.
a. To find the amount at the end of the 4-year period, you can use the formula:
Final value = Initial value * (1 + interest rate)^number of years
In this case, the initial value is $5000, and the interest rate is 7% per annum. The number of years is 4. So, the calculation will be:
Final value = $5000 * (1 + 0.07)^4
You can use a calculator or a spreadsheet to evaluate this expression and find the final value.
b. The expression for the value of the investment after n full years can be written as:
Value = Initial value * (1 + interest rate)^n
In this case, the initial value remains $5000, and you replace "4" with the variable "n" to represent the number of years. This allows you to calculate the value of the investment for any number of years.
c. To find the value of n when the investment exceeds $10,000, you need to set up an inequality. Since the value of the investment is given by the expression from part b, the inequality becomes:
5000 * (1 + 0.07)^n > 10,000
Simplifying, you have:
(1.07)^n > 2
To isolate n, you can take the logarithm of both sides of the inequality. So:
n log 1.07 > log 2
Using a calculator, you can evaluate log 2 and find that it is approximately 0.301. Dividing both sides of the inequality by log 1.07, you get:
n > 0.301 / log 1.07
Evaluating this expression, you find that n is approximately 10.24 years.
So, the inequality representing the information is 5000 * (1 + 0.07)^n > 10,000, and the value of n is approximately 10.24 years.