I got 21 years and it was wrong. Can you help me solve this problem?
sometime ago, julie purchased 11 acres of land costing $15,590. Today, that land is valued at $55,433. How long had she owned this land if the price of the land has been increased at 5% per year?
15,590(1.05)^n = 55,433 , where n is the number of years
1.05^n = 3.55567.. (I stored it in memory)
take logs of both sides
nlog 1.05 = log 3.55567..
n = 26.00002..
I would say she owned it 26 years
Sure! To solve this problem, we need to find out how many years it took for the land's value to increase from $15,590 to $55,433, given a yearly growth rate of 5%.
To start, let's define the initial value of the land as V₀, which is $15,590, and the final value of the land as Vₙ, which is $55,433.
Now, we can use the compound interest formula to find the number of years, which is given by:
Vₙ = V₀ * (1 + r/100)ᵗ
Where:
- Vₙ is the final value of the land ($55,433)
- V₀ is the initial value of the land ($15,590)
- r is the annual growth rate (5% or 0.05)
- t is the number of years we want to find
Plugging in the given values, we get:
$55,433 = $15,590 * (1 + 0.05)ᵗ
To isolate "t" (the number of years), we need to rearrange the equation. Divide both sides by $15,590:
($55,433 / $15,590) = (1 + 0.05)ᵗ
Now, using some basic algebra, we can take the logarithm base (1 + r/100) of both sides:
log(base 1.05) of ($55,433 / $15,590) = t
To calculate this in detail, you can follow these steps:
1. Divide $55,433 by $15,590.
2. Take the logarithm of the result using base 1.05.
3. Round the answer to the nearest whole number to find the number of years.
Alternatively, you can use a calculator or an online logarithm calculator with a custom base to directly find the value of "t".
Remember to round the final answer to the nearest whole number since we are dealing with the number of years.