Alan

1.  Divide both sides by 1000000 to get 211 = 0.9^t

2.  Take logs of both sides:   ln(211) = ln(0.9^t)        where ln is log to the base e (though you could use log to the base 10 if you prefer).

3.  Use a property of logs:  ln(211) = t*ln(0.9)

4.  Rearrange:   t = ln(211)/ln(0.9)

I'll leave you to evaluate (you can use the calculator on the Home page here).

Hi Melody, I make the transformed area 10.

The scaling on length is \(\sqrt{\sqrt3^2+1^2}=2\)  so the scaling on area is 2² = 4. 

(Note: there's no guarantee that I know what I'm talking about either!)

The following should help:

As follows:

What prime is 4 greater than a perfect square and 7 less than the next perfect square?

\(\frac{1}{3}\times(-2\times3.15)=\frac{1}{3}\times(-6.3)=-\frac{6.3}{3}=-2.1\)

Here's a helping hand:

See if you can take it from here.

As follows:

\(\frac{5/\sqrt{80}+\sqrt{845}/9}{\sqrt5}=\frac{5/\sqrt{16\times5}+\sqrt{169\times5}}{\sqrt5}=\frac{5/(4\sqrt5)+13\sqrt5}{\sqrt5} =\frac{\sqrt5}{4}+13\)

Calculate sin(sin(X)).  Plot this point on the graph.  Draw a horizontal line from -10pi to 10pi at this level. 

Now calculate sin(sin(Y)).  All we know is that it must be on the same horizontal line as well as on the curve.

There are 20 possible values of sin(sin(Y)) that are the same as sin(sin(X)).  

Hence the probability that X = Y is 1/20.

(At least, I think that's the interpretation of this question!).

This graph should help