the compound interst forumla is A= P(1+i)^n. consider the function in the form of y= 100(1+x/2)^20, where y is the compound amount.
a) determine the required annual interest rate of investment to the nearest tenth of a percentage of 10 years that will give an amount of $300
b) Graph the function y= 100(1+x/2)^20 and y=300
(a) just plug and chug:
100(1+x/2)^20 = 300
(1+x/2)^20 = 3
1+x/2 = 3^(1/20)
x = 2(3^(1/20)-1) = 0.1129 or 11.3%
(b) See
http://www.wolframalpha.com/input/?i=plot+y%3D100(1%2Bx%2F2)%5E20,+y%3D300+for+0%3C%3Dx%3C%3D0.12
To find the required annual interest rate of the investment, we can rewrite the formula for compound interest:
A = P(1 + i)^n
In this case, we are given that A (the compound amount) needs to be $300, P (the principal amount) is $100, n (the number of years) is 10, and we need to solve for i (the annual interest rate).
a) To find the required annual interest rate, we can rearrange the formula and solve the equation:
A = P(1 + i)^n
300 = 100(1 + i)^10
To isolate (1 + i)^10, divide both sides of the equation by 100:
3 = (1 + i)^10
Now, to solve for i, we need to take the 10th root of both sides of the equation:
(1 + i) = √3
Subtract 1 from both sides to find the value of i:
i = √3 - 1
Calculating this value, we get:
i ≈ 0.7321
Therefore, the required annual interest rate to the nearest tenth of a percentage for 10 years that will give an amount of $300 is approximately 0.7%.
b) To graph the function y = 100(1 + x/2)^20 and y = 300, we can plot multiple points for each equation and then connect them to create the graph.
For y = 100(1 + x/2)^20:
- Choose different values for x, such as -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5.
- Substitute these x-values into the equation to calculate the corresponding y-values.
- Plot the points (x, y) on a graph.
For y = 300:
- Draw a horizontal line at y = 300 on the graph.
Connect all the points corresponding to y = 100(1 + x/2)^20, and also draw the horizontal line at y = 300.