Suppose human activity has caused a 0.1 Fahrenheit increase in global temperatures so far, and that this sift will grow exponentially with a doubling time of 10 years.

a. How much will temperatures have risen 50 years from now?

b. A politician suggests that a crisis can be averted by simply slowing the doubling time to 20 years. He calculates that after 50 years the shift will be less than 0.6 Fahrenheit, which he can live with. What is the temperature shift after 200 years?

To answer these questions, we need to understand how to calculate exponential growth and apply it to the given information.

Let's start with the basic formula for exponential growth:

A = P * (1 + r)^t

Where:
A = the final amount
P = the initial amount
r = growth rate
t = time in years

In our case, "A" represents the temperature increase, "P" is the initial 0.1 Fahrenheit increase, "r" is the rate of growth, and "t" is the time in years.

a. To find out how much temperatures will have risen 50 years from now, we need to calculate the final temperature increase using the given doubling time of 10 years:

First, we need to find the growth rate "r." Since temperatures are doubling every 10 years, the growth rate can be calculated as the number 2 raised to the power of (1/10) - 1.

r = 2^(1/10) - 1

Now we can plug the values into the exponential growth formula:

A = P * (1 + r)^t
A = 0.1 * (1 + r)^(50/10)

Calculate the value of "A" to find the temperature increase 50 years from now.

b. The politician suggests slowing the doubling time to 20 years. We need to calculate the new growth rate and then use it to find the temperature shift after 200 years.

To find the new growth rate, we use the same formula:

r = 2^(1/20) - 1

Now, use the new growth rate to calculate the temperature shift after 200 years:

A = P * (1 + r)^(t/10)
A = 0.1 * (1 + r)^(200/10)

Calculate the value of "A" to find the temperature shift after 200 years.

By following these steps, we can calculate the temperature increases for both scenarios.