Question

Bob bought some land costing $15,990. Today, that same land is valued at $46,017. How long has Bob owned this land if the price of land has been increasing at 5 percent per year?

Answer 1

Bob has owed this land for 21.68 years.

Explanation:

The price of the land can be modeled by the following function:

`P(t) = P(0)(1+r)^(t)`

In which P(t) is the price after t years, P(0) is the initial price, and r is the growth rate, as a decimal.

In this problem, we have that:

We want to find t, when
`P = 46017, P(0) = 15990, r = 0.05`. So

`P(t) = P(0)(1+r)^(t)`

`46017 = 15990(1+0.05)^(t)`

`(1.05)^(t) = (46017)/(15990)`

`(1.05)^(t) = 2.88`

We have that:

`\log{a^(t)} = t\log{a}`

So to find t, we apply log to both sides

`\log{(1.05)^(t)} = \log{2.88}`

`t\log{1.05} = \log{2.88}`

`t = \frac{\log{2.88}}{\log{1.05}}`

`t = 21.68`

Bob has owed this land for 21.68 years.