hectictar

Its a little unclear to me what it means by "powerful" so I can understand how this question is a bit confusing.... maybe there is a specific meaning of the word "powerful" here and I just don't know it! But because of the fact that it bothers to tell us that the number on the Richter Scale is the exponent of 10 of the amplitude..... I'm guessing that perhaps the actual question it's asking is this:

How many times greater was the amplitude of the 2011 earthquake than the amplitude of the 2021 earthquake?

amplitude of the 2011 earthquake = 10^5.8

amplitude of the 2021 earthquake = 10^1.9

Now the question is:    10^5.8  is how many times greater is than 10^1.9  ?

Or in other words...  10^5.8  is equal to what times 10^1.9  ?

\(10^{5.8}\ =\ x\cdot10^{1.9}\)

                               Divide both sides of the equation by  10^1.9   to solve for  x

\(\dfrac{10^{5.8}}{10^{1.9}}\ =\ x\)

                               Now we can apply the rule which says  \(\frac{x^a}{x^b}\ =\ x^{a-b}\)

\(10^{5.8-1.9}\ =\ x \\~\\ 10^{3.9}\ =\ x\)

And so the amplitude of the 2011 earthquake is 10^3.9  times greater than the amplitude of the 2021 earthquake.

However, I could be misinterpreting this question too, and your original answer might be correct! Just because they told us that information about the Richter scale doesn't necessarily mean we have to use it to answer the question....maybe they were just trying to throw us off with a red herring

perimeter  =  twice the distance between (1, 4) and (-6, 4)   +   twice the distance between (1, 4) and (5, -5)

perimeter  =  \(2\cdot\sqrt{(4-4)^2+(-6-1)^2}\ +\ 2\cdot\sqrt{(-5-4)^2+(5-1)^2}\)

perimeter  =  \(2\cdot\sqrt{49}\ +\ 2\cdot\sqrt{97}\)

perimeter  =  \(2\cdot7\ +\ 2\cdot\sqrt{97}\)

perimeter  =  \(14\ +\ 2\sqrt{97}\)

If I had to guess right now, I would guess option 4. My next guess would be option 1.

But I am honestly not sure!! I think this is a bad question.

Can you let us know when you find out the "correct" answer according to the answer key?

I do agree Probolobo, but then the confusing thing is that

\(5^{\log y}\ =\ y^{\log 5}\)

Which means option 2 and option 4 are the same function and so are equivalent....

Hmm...actually....I see what you mean.... (maybe I made the question harder than it has to be!)

I take back my original answer!! Now I think the answer is option 4

I'm not sure what you meant by "by the way if i do the same operation and interchange x and y i get 2 and 4 option , that satifies both conditions"....but if this didn't answer your question then please feel free to ask for more clarification!

And I think the answer is option 1:  \(x=y^\frac{1}{\log 5}\)

The inverse function can be graphed by:    \(y=x^{\frac{1}{\log5}}\)

Here's a graph:

(Assuming that  log  means natural log)

y   =   5^{log x}

                                       Take the log of both sides

log y   =   log( 5^{log x} )

                                       Now we can apply the rule that says:  log(x^y)  =  y log x

log y   =   log x log 5

                                       Divide both sides by  log 5

log y / log 5   =   log x

                                                  Do the reverse of natural log to both sides (make both sides the exponent of e )

e^( log y / log 5  )   =   e^log x

                                                  Now the right side simplifies to just  x

e^( log y / log 5  )   =   x

                                                  So we have...

\(x\ =\ e^{\frac{\log y}{\log 5}}\\~\\ x\ =\ (e^{\log y})^{\frac{1}{\log 5}}\\~\\ x\ =\ y^{\frac{1}{\log 5}}\)

Part A:

tan( x )  =  opposite / adjacent

tan( x )  =  16500 / 6000

tan( x )  =  11/4

x   =   arctan( 11/4 )

x   ≈   70°

Part B:

This is the length of the third/unknown side of the triangle, which we can find using the Pythagorean Theorem. Let's call this side  "c". Then...

6000²  +  16500²   =   c²

\(\color{} c\ =\ \sqrt{6000^2+16500^2}\)

We could plug the above into a calculator, or we can first simplify it like this before plugging it in:

\(\color{gray} c\ =\ \sqrt{6000^2+16500^2} \\~\\ \color{gray} c\ =\ \sqrt{(4\cdot1500)^2+(11\cdot1500)^2} \\~\\ \color{gray} c\ =\ \sqrt{4^2\cdot1500^2+11^2\cdot1500^2} \\~\\ \color{gray} c\ =\ \sqrt{1500^2(4^2+11^2)} \\~\\ \color{gray} c\ =\ \sqrt{1500^2}\cdot\sqrt{(4^2+11^2)} \\~\\ \color{gray}c\ =\ 1500\sqrt{137} \)

Now if we put this into a calculator we get:

c   ≈   17557   (And this is in feet)

m∠L  +  m∠P   =   180°

(14x - 22)°  +  (4x + 4)°   =   180°

14x - 22  +  4x + 4   =   180

18x + 18   =   180

18x   =   180 - 18

18x   =   162

x   =   162/18

x   =   9

Now that we know  x  we can find the measure of angle  P

m∠P   =   (4x + 4)°   =   (4(9) + 4)°   =   (36 + 4)°   =   40°