The energy (in joules) released by an earthquake of magnitude ...

Question

The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 ⋅ 10^1.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)
Compute the energy released by earthquakes of magnitude 1, 2, 3, 4, and 5. Plot the points on a graph and join them with a smooth curve.

Compute the energy released by earthquakes of magnitude 1, 2, 3, 4, and 5. Plot the points on a graph and join them with a smooth curve.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A certain type of star emits energy according to the formula. E equals 150 multiplied by 10 to the power of 0.5 multiplied by M, where E is the energy in joules, and M is the magnitude of the star's brightness. Calculate the energy emitted by stars of magnitude 246, and 8, then plot the points on the graph and connect them through a smooth curve. Awesome, so it appears for this particular problem we're asked to solve for. A several different answers, so we're asked to figure out. Uh, so we're trying to figure out the energy emitted by stars at a magnitude of 2, that's one answer. 4, that's our second answer. 6, that's our third answer, and 8 is our fourth answer. So we're trying to find the magnitude at 4 different. So we're trying to find the energy when the magnitude is at 4 different values, so at 246, and 8. And then our final answer that we're ultimately trying to solve for is we're trying to take our answers that we find for the energy that these stars emit at these specific magnitudes, and we're going to take those values and plot them on a graph and connect them with a smooth curve. So that's our 2nd that's our second part that we're ultimately trying to solve for. We're trying to figure out the energy at these different magnitude values. That's our first part, and then our second part is we're trying to create a graph of our energies that we find at these specific magnitudes. Awesome. So let's take a moment here to look at our blank graph that is provided to us by the problem itself. Our Y axis denotes the energy E. Which is in units of joules, and our X-axis denotes the magnitude M. Awesome. So, and then once again, the magnitude will be the star's brightness. Fantastic. So, with that said, our first step in order to solve this problem is we need to evaluate the function at, and let's make a note of this, we need to evaluate our functions at M equals 246, and 8. Fantastic. So let us do that, shall we? So let us start with 2, so when we the energy or Or 2, so EF X, so we're treating it like a function, so we're gonna be plugging in 2 for M. So essentially, rather than saying EFX we could say EFM, but we're dealing with this function. So EF 2. is equal to 150 multiplied by 10 to the power of 0.5 multiplied by 2. These are plugging in 2 for M. And when we do that, we will find that this is equal to 1500 when we plug that into our calculator. And we're instead of writing out this expression every single time, since it's gonna be the exact same thing, I'm just gonna skip a few steps. So instead of writing 150 times 10 to the power of 0.5 multiplied of value, I'm just gonna write the answer. So following similar steps for EF4, so EF 4 is going to be equal to 15,000 when you plug in a value of 4 in for M, and for EF 6 EF6 is going to be equal to 150,000. When you plug in a value of 6 for M, and finally E of 8 is going to be equal to 1 million. 500,000. Fantastic, and those are our final answers. Hooray, we did it. So these are our final answers for the energy. At the specific magnitude values. So, Essentially now we all we need to do, this is the easy part, is now we're gonna be taking these different energy values and we're gonna be plotting them on our graph, so meaning what we need to do. we need to plot the points 21,500, we also need to plot the points 4. 15,000. Which I guess so we don't get confused. I'm gonna remove the comma. Awesome. And we also need to plot the 0.6,150,000, and finally we need to plot the 0.8. 1,500,000. Fantastic. So we need to take these points which I wrote and read, and we need to plot them on our graph and then connect them with the smooth curve. And when we do that, I've gone ahead and put this into a graph some graphing software to create a You know, digitized graph here. And as you can see when we plot these different points on our graph, so when we plot at 246, and 8, and we connect them with a smooth curve, we could see that there's exponential growth, so it gradually gets Bigger, bigger and bigger and starts to curve upwards. So it starts out slow and then it starts to pick up and then gradually increase. So it's a smooth showing exponential growth it appears. And that's it. That is how you solve for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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