Final answer:
To calculate the mean annual growth rate for an asset that declines in value from $5000 to $3500 over nine years, the formula for compound interest can be rearranged to solve for the growth rate. The formula is r = (Vf/Vi)^(1/n) - 1, where r is the growth rate, Vf is the final value, Vi is the initial value, and n is the number of years. In this case, the calculated rate will indicate a decrease in value.
Step-by-step explanation:
If an asset declines in value from $5000 to $3500 over nine years, we need to find the mean annual growth rate, which in this case would be a negative growth rate since the asset is declining in value. The mean annual growth rate, often called the average annual growth rate (AAGR), can be found using the formula for compound interest, which is similar to the formula used for calculating the growth of investments. However, because the value is decreasing, the growth rate will be negative.
To find the AAGR, we use the formula:
Vf = Vi (1 + r)^n
Where:
Vf is the final value of the asset,
Vi is the initial value of the asset,
r is the growth rate,
and n is the number of periods.
We can rearrange the formula to solve for the growth rate r:
r = (Vf/Vi)^(1/n) - 1
Plugging in the values, we have:
r = (3500/5000)^(1/9) - 1
Calculating this will give us the mean annual growth rate, which will be a negative percentage indicating a decrease in value each year over the nine-year period.