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c = \(\sqrt{3^2 + 4^2} = 5\)

This means b = \(\sqrt{(a + 3)^2 - c^2}\)

but b also equals = \(\sqrt{a^2 + 4^2}\)

We can put these two into an equation:

\(\sqrt{(a + 3)^2 - c^2} = \sqrt{a^2 + 4^2}\)

\(a^2 + 6a + 9 - c^2 = a^2 + 16\)

\(6a - c^2 = 7\)

and since c = 5, 

\(6a - 25 = 7\)

\(6a = 32\)

\(a = 5\frac{1}{3}\)

You are welcome Kes

Perpindicular is oposite slope, (-2/3) to (3/2). 

y = 3/2x+b

Now we just need to find b, and we can plug in the point (-4, 2)

2 = (3/2)(-4)+b

2 = -6 +b

b = 8

So our equation is y = 3/2x+8

I hope this helped. :)))

=^._.^=

that's incorrect, but thank you for taking the time to answer my question!

another question , another superb explanation from mod

Melody you are simply awesome ! thanks for the helping mate

I think you are making hard work of it

the inverse of   

\(y=5^{logx}\)

is simply

\(x=5^{logy}\)

You just have to switch the x and y over.

there would be restrictions on x and on y but the question isn't worrying about that.

Here is the graphs

First, we know that each square in the window has an area of 100 inch². This means the dimensions are 10 x 10. 

To find the area of the window, I'm going to split up the square part and the semicircle. 

This means the area of the square part is (10 x 4) x (10 x 4) = 40 x 40 = 1600 inch². This is because this area is 4 squares by 4 squares. 

Now, the area of the semicircle is \(\frac{1}{2} (10 \cdot 2)^2 π = 200π ≈ 628\)

This means the area of the window is 2228 inch²

The shade is 4 inches beyond, and so the dimensions of the 4 by 4 square would be (40 + 4 + 4) by (40 + 4 + 4). 

This means the area is 48 x 48 = 2304 

Then the semicircle's radius would be (20 + 4). Which means the area is \(\frac{1}{2} (20 + 4)^2π = 288π ≈ 904\)

This means the area of the shade is 3208 inch²

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?

That's correct, then there are actually 2 correct answers, 2 & 4. Didn't see that equivalence on first glance ;)

so whats the final answer so that i can challenge 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....

For computing the inverse function, the plan I know is 
1. interchange x & y

2. solve for y

but actually, after 1. you already have a term for the inverse function. It's just not written in the "usual" way, wich is y=f(x).

Answer for is exactly what you get after interchanging x&y, so the correct answer is answer 4.

brother do you really believe its option 4 or just to keep my heart , please clarify if you still believe the answer is 1 , if yes then please prove the same to me as well

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

brother , if you see your second answer and change it in terms of x , wont you get option 2 and 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:  \(x=y^\frac{1}{\log 5}\)

brother so what you feel (as a math expert) should be the right answer to the given question ,

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

Here's a graph:

brother what you have found is the same function just in terms y , but inverse function is symmetrical about y=x axis ,

also f(f^-1(x))=x which is not satisfied , 

by the way if i do the same operation and interchange x and y i get 2 and 4 option , that satifies both conditions

(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)

It is a stupid question.     (I'd want to cry too)

The percentage is the rate.

I suppose they mean,     1) R,     2) B,    3) P

But it is still a badly worded question.

Joanie takes a  6000 dollar  loan to pay for her car. The annual interest rate on the loan is 12percent . She makes no payments for 4 years, but has to pay back all the money she owes at the end of 4 years. How much more money will she owe if the interest compounds quarterly than if the interest compounds annually? Express your answer as a dollar value to the nearest cent.

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Compounded annually it will become   6000(1+0.12)^4 = 9441.11 dollars,   which means the interest component is 441.11 dollars

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4 years is 4*4=16 quarters,    and     12% per year (nominal) is   12/4 = 3% per quarter

So 6000 dollars will grow to       6000(1+0.03)^16 = 9628.24 dollars (to the nearest cent)  

so that will be  3628.24 dollars interest

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You can finish it :)

I had to get rid of the dollar signs because the program thought it was latex and it didn't print properly.  

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