Calculus Problem. - Exponential Functions?

Question

Nikki invested $10,000 in the stock market. The investment was a loser, declining in value 10% per year for 10 years. How much was the investment worth after ten years?

- After ten years the stock began to gain value at 10% per year. After how long will the investment regain its initial value?

Steps please?

Thanks :)

Answer 1

Each year, the stock goes down by 10%. That means after each year, the stock is at only 90% of its value from the previous year.

So after the first year, the stock is worth 10000*0.9 = 9000.

After the second year, it's worth 9000*0.9 = 8100.

Or, you could say it's 10000 * 0.9 * 0.9.

After the third year, it's 10000 * 0.9 * 0.9 * 0.9.

And so on. You should see a pattern: at year N, the value is 10000*(0.9)^N. So plug in N=10, and let's call the final value of the stock X.

The second part of the question says that the value then goes up by 10% a year. This means that after a given year, the stock will be worth 1.1 times what it was before. Using the same logic as above, we can say that the value at year K is

V = X * (1.1)^K

We know X from the first part, and we want V=10000 (the original value). So we just solve for K and we're done! (if they want an integer value for the number of years, just round up K).

Answer 2

Following on from the above, after 10 years the investment is worth 10,000 * 0.9^10 = 3486.78

Therefore in terms of the above V/X = 2.867976

1.1^K = 2.867976

Taking logarithms, log(1.1^K) = log( 2.867976)

Now, log( 1.1^K) = K * log(1.1)

Therefore K * log(1.1) = log(2.867976)

So k * 0.04139 = 0.45758

K = 11 years