original value of painting is $1,800.value increases 12% each year.The owner would like to sell it once it reaches a value of $6000. How long should the owner wait before selling the painting?

Question

A) 10.6 years

B) 1.5 years

C) 3.3 years

D) 6.6 years

Answer 1

Hello, Fay,

The value of the painting increases by 12% per year. We can calculate the value after x years with (1800)*(1.12)^x. When x = 1, the new value is (1800)*(1.12)¹ = 2016. The x is simply the number of times the value is increased by 12%.

The question we want to solve is to find the value of x (years) when the total new value will be $6000.

(1800)*(1.12)^x = 6000

I should be able to solve this mathematically, but I'm a little lost. A spreadsheet or graph (Desmos) both show around 10.6 years. Wolfram Alpha returns 10.62 years. That matches Option A.

Bob

Answer 2

Since the value of the painting increases by 12%/year, our "growth factor" is 1.12.

The painting is worth $1,800 now, and so the function, v, that gives the value of the paining in terms of the number of elapsed years, t, is given by

v(t) = 1800(1.12)^t.

We want the value to be $6000, and find the corresponding t value, so solve the following equation:

6000 = 1800(1.12)^t

Divide both sides by 1800:

(6000/1800) = (1.12)^t.

Apply a logarithm to both sides, say, the natural log (the base doesn't matter, in terms of a handheld calculator log base 10 would work too)

ln (6000/1800) = t ln(1.12). Divide both sides by ln(1.12):

ln(6000/1800) / ln(1.12) = t ≈10.6237

Hence, it will take about 10.6 years for the value of the painting to be $6,000.

Alternatively, you could evaluate v(t) for each of the given values (since it's multiple choice), but make sure you understand the "complete" approach