Melody

x^2+x^x2

\(x^2+x^{x^2}\ne x^4\\~\\ x^2\times x^2=x^4\\~\\ x^2+ x^2=1x^2+ 1x^2=2x^2\\~\\\)

Max's equation.

\(\begin{align*}\sqrt[3]{(8^{2x})}&=\sqrt{2^{(4x-1)}}\\ (8^{2x})^{1/3}&=(2^{(4x-1)})^{1/2}\\ ((8^{2x})^{1/3})^6&=((2^{(4x-1)})^{1/2})^6\\ (8^{2x})^2&=(2^{(4x-1)})^3\\ ((2^3)^{2x})^2&=(2^{3(4x-1)})\\ (2^3)^{4x}&=2^{3(4x-1)}\\ (2)^{12x}&=2^{3(4x-1)}\\ 12x&=3(4x-1)\\ 12x&=12x-3\\ 0&=-3\\ &\mbox{No solutions}\end{align*}\)

OR

\(\begin{align*}\sqrt[3]{(8^{2x})}&=\sqrt{2^{(4x-1)}}\\ 8^{2x/3}&=2^{((4x-1)/2)}\\ (2^{2x})&=2^{((4x-1)/2)}\\ 2x&=2x-0.5\\ 0&=-0.5\\ &no \;solution\end{align*}\\ \)

Thanks Alan.  ://

CONGRATULATIONS   MAX !!!

I will assume you mean this ...

3+(csc45)^4

\(3+ {\sqrt2}^4\\ =3+4\\ =7 \)

A spring cannon can shoot its projectile at a maximun of 30m/s. what is the maximum height that the projectile will reach? How long will it take to rech that height?

Hint: consider the object's velocity at its maximum height

It will go the highest if it is shot straight up.  (you better not be underneath when id comes back down though.)

Gavity is downwards acceleration - it is about 9.8m/s^2

\(\ddot y=-9.8\\ \dot y =-9.8t+c\\ \mbox{velocity at t=0 is 30 so c=30}\\ \dot y =-9.8t+30\\ y =-\frac{9.8t^2}{2}+30t+k\\ \mbox{Let the initrial displacement be 0 then k=0}\\ y =-\frac{9.8t^2}{2}+30t\\ \)

the height will  be greatest at the turning point which is where velocity =0

\(\begin{align*}-9.8t+30&=0\\ 9.8t&=30\\ t&=\frac{30}{9.8}\\ t&=3.0612\;sec \end{align*}\)

\(y=-\frac{9.8t^2}{2}+30t\\ y=-4.9t^2+30t\\ y=-4.*3.0612^2+30*3.0612\\\)

-4*3.0612^2+30*3.0612 = 54.35221824

Max height approx 54.35 metres

Maybe this will help :)

ALIGN

This is very interesting Max, thanks for showing me.

I used to use align all the time but it did not work here -

There are many different versions of Latex - each one seems to have its own limitations and peculiarities. :)

You are right    IT DOES WORK!

It seems you cannot add any more latex after you finish the alignment.

That is a pity but most times it would be convenient enough just to close it and start a new latex insert.

You can use this with more than one alignment - you probably know that.  :)

\begin{align*}2x - 5y &=&  8&\qquad \mbox(1)

\\3x + 9y &=&-12&\qquad \mbox(2)\end{align*}

\(\begin{align*}2x - 5y &=& 8&\qquad \mbox(1) \\3x + 9y &=&-12&\qquad \mbox(2)\end{align*}\)

Now I need to remember to use this   

PS   I am sure your Mum is a very fine mathematician :))

Definitely 2ml. :)

I do not understand this statement.     

An interesting feature is created when the blocks are sewn together: each colour forms a shape. The shape and its area are exactly the same for each colour. (The feature appears throughout the quilt, except at the edges.)

Do you understand it?    

Three fair dice is rolled. Calculate the probability that the sum of the numbers on the dice are even, given that the numbers on the dice are all the same.

Could be

2 odd and 1 even   or    3 even

\(3C2*0.5^3+3C3*0.5^3 = 3*0.5^3+0.5^3=4*0.5^3\)

4*0.5^3 = 0.5

Lets think about this answer..

Can't be

2 even and 1 odd   or  3 odd

So 50% makes perfect sense :))