Which equation represents the magnitude of an
earthquake that is 100 times more intense than a
standard earthquake?

Step-by-step explanation:

The Richter scale, the standard measure of earthquake intensity, is a logarithmic scale, specifically logarithmic base 10. This means that every time you go up 1 on the Richter scale, you get an earthquake that's 10 times as powerful (a 2.0 is 10x stronger than a 1.0, a 3.0 is 10x stronger than a 2.0, etc.).

How do we compare two earthquake's intensities then? As a measure of raw intensity, let's call a "standard earthquake" S. What's the magnitude of this earthquake? The magnitude is whatever power of 10 S corresponds to; to write this relationship as an equation, we can say , which we can rewrite in logarithmic form as .

We're looking for the magnitude M of an earthquake 100 times larger than S, so reflect this, we can simply replace S with 100S, giving us the equation

.

To check to see if this equation is right, let's say we have an earthquake measuring a 3.0 on the Richter scale, so . Since taking 100 times some intensity is the same as taking 10 times that intensity twice, we'd expect that more intense earthquake to be a 5.0. We can expand the equation using the product rule for logarithms to get the equation

And using the fact that and our assumption that , we see that as we wanted.

56.6

step-by-step explanation:

the ratio of the perimeters is equal to the scale.

the ratio of the areas is equal to the square of the scale.

34 / 20 = k

a / 19.6 = k²

substituting:

a / 19.6 = (34 / 20)²

a = 19.6 (34 / 20)²

a ≈ 56.6