I need the answer please

Question

Objective 3.9 - Doubling Time and Half-Life

The hourly wage of some Toyota automobile assembly line workers is being increased annually by the same percentage

If their wage went from $7.60 to $11.81 in 7 years what is the doubling time?

round to (nearest year)

Answer 1

The wage is being increased by a certain percentage every year. The percentage is always the same, and being multiplied to the initial wage. This is exponential growth. The general formula looks something like a=a₀(1+r)^t .

1. We start at $7.60 so a₀ = 7.6
2. They tell us information after 7 years, so t =7
3. They tell us the ending balance, that is a after seven years. So when t=7, a = 11.81
4. r is what we want to find out

11.81 = 7.6(1+r)⁷

1. 11.81/7.6 = (1+r)⁷
2. ⁷√11.81/7.6 = 1+r. You would definitely use a calculator for this part. It might be easier to put in (11.81/7.6)^1/7=1+r
3. r = (11.81/7.6)^1/7-1 → r ≈ .065

We are not done though. The double life is asking for the amount of time to double the initial amount. We want to end up with 7.6*2 = 15.2

We now know r, but we want to know the time t.

15.2 = 7.6(1+.065)^t

1. 15.2/7.6 = (1.065)^t
2. 2 = 1.065^t
3. We want to solve for the exponent, so we need to take the log of both sides, and use the log power rule to bring down the exponent.
4. log(2) = log(1.065^t)
5. log(2) = t*log(1.065)
6. log(2)/log(1.065) = t Use a calculator for this again. t = 11.007

At around 11 years, the wage would double.

Just as a sanity check, calculate 7.6(1.065)¹¹ where we plug in our answer 11. This should equal double of 7.6, if 11 truly is the double life.

You get 15.19, which is just about 15.2, double the initial wage.