If 75100 dollars is deposited in an account for 13 years at 5.35 percent compounded continuously , find the average value of the account during the 13 years period.
The average value in the account is ?
** im alittle confused in how to find the average value please help and show work please and Thank You
To find the average value of the account over the 13-year period, we need to consider the continuous compounding formula and integrate it over the time period.
The continuous compounding formula is given by:
A = P * e^(rt)
Where:
A is the final amount
P is the principal amount (initial deposit)
e is the base of the natural logarithm (approximately 2.71828)
r is the annual interest rate (as a decimal)
t is the time in years
Given:
P = $75100
r = 5.35% = 0.0535 (as a decimal)
t = 13 years
Using these values, we can calculate the average value of the account as follows:
Average Value = (1/t) * ∫(0 to t) A dt
First, let's calculate A.
A = P * e^(rt)
A = $75100 * e^(0.0535 * 13)
Using the approximation e^(0.0535 * 13) = 2.0518:
A ≈ $75100 * 2.0518
A ≈ $154021.918
Now, let's calculate the average value:
Average Value = (1/13) * ∫(0 to 13) A dt
Since A is a constant value, we can simplify the integral:
Average Value = (1/13) * A * ∫(0 to 13) dt
Average Value = (1/13) * $154021.918 * [t] (evaluated from 0 to 13)
Average Value = (1/13) * $154021.918 * (13 - 0)
Average Value = (1/13) * $154021.918 * 13
Average Value = $154021.918
Therefore, the average value of the account during the 13-year period is approximately $154,021.92.
To find the average value of the account during the 13-year period, we need to calculate the final value of the account and divide it by the number of years.
To calculate the final value of the account, we use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = the final value of the account
P = the initial deposit
e = the mathematical constant approximately equal to 2.71828
r = the interest rate
t = the time in years
Plugging in the given values:
P = $75,100
r = 5.35% = 0.0535 (in decimal form)
t = 13 years
A = $75,100 * e^(0.0535 * 13)
Now we can calculate the final value of the account using a calculator or a software that supports the exponential function:
A ≈ $75,100 * e^0.6955 ≈ $75,100 * 2.0049 ≈ $150,633.049
Now that we have the final value of the account, we can calculate the average value by dividing it by the number of years.
Average Value = Final Value / Number of Years
Average Value = $150,633.049 / 13 ≈ $11,585.618
Therefore, the average value in the account during the 13-year period is approximately $11,586.