For a fixed rate, a fixed principal amount, and a fixed compounding cycle, the return is an exponential function of time. Using the formula, , let r = 10%, P = 1, and n = 1 and give the coordinates (t, A) for the points where t = 0, 1, 2, 3, 4. Round your answer to the hundredth's place. Show coordinates and how would I show the graph?
Was answered under the post by 'a'
To solve this problem, we will use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the amount of money after time t
P = the principal amount (initial investment)
r = the fixed annual interest rate (expressed as a decimal)
n = the number of times interest compounds per year
t = the time (in years)
Given:
r = 10% = 0.10 (expressed as a decimal)
P = 1
n = 1
We will calculate A for different values of t (0, 1, 2, 3, and 4) using the given values.
For t = 0:
A = 1(1 + 0.10/1)^(1*0)
A = 1(1 + 0.10)^0
A = 1(1)^0
A = 1 * 1
A = 1
So the coordinates for t = 0 are (0, 1).
Now let's calculate A for t = 1:
A = 1(1 + 0.10/1)^(1*1)
A = 1(1 + 0.10)^1
A = 1(1.10)
A = 1.10
The coordinates for t = 1 are (1, 1.10).
Similarly, we can calculate A for t = 2, 3, and 4 as follows:
For t = 2:
A = 1(1 + 0.10/1)^(1*2)
A = 1(1 + 0.10)^2
A = 1(1.10)^2
A = 1.21
The coordinates for t = 2 are (2, 1.21).
For t = 3:
A = 1(1 + 0.10/1)^(1*3)
A = 1(1 + 0.10)^3
A = 1(1.10)^3
A = 1.331
The coordinates for t = 3 are (3, 1.33).
For t = 4:
A = 1(1 + 0.10/1)^(1*4)
A = 1(1 + 0.10)^4
A = 1(1.10)^4
A = 1.4641
The coordinates for t = 4 are (4, 1.46).
To show the graph, you can plot the coordinates (t, A) on a graphing software or a spreadsheet program like Microsoft Excel. The x-axis represents time (t) and the y-axis represents the amount (A). Label the axes accordingly and plot the points (0, 1), (1, 1.10), (2, 1.21), (3, 1.33), and (4, 1.46). Then, connect the points with a smooth curve to represent the exponential function.