Melody

Why doesn't some smart person show their skill and do this with calculus.   

Thank you Alan.  Yes is usual for two half radii to equal 1 radii.  I guess I had a brain freeze  

3 vowel choices for the first place

3 consonant choices for the second position

2 vowels left for 3rd position

2 consonants left for 4th

rest is set in stone one of each left no more choices :)

30×2

4×15

and there are more :))

3×3×2×2×1×1

Or

3!×3!

Thanks Alan.

I misinterpreted the question 😦

I was just looking at what Heureka was saying.  It took me a while to work it out.

                                                   B

This is the triangle that Heureka was describing

The radius that bisects the chord is perpendicular to the chord so <AHE=90 degrees

which means <AEH=30 degrees (Angle sum of a triangle)

Using trig I know that AH=0.5*AE=2.25  but I am not allowed to use trig.  

Now Heureka says that AH=HB  

Edited: that is because AB is one radius and AH is 1/2 radius so  HB must also be a 1.2 radius 

Join EB forming a new triangle HEB

Now triangle HEB is congruent to triangle HEA because all the angles are congruent and they have a common corresponding side.

therefore triangle AEB is equilateral  and HA=HB

therefore all the sides are  4.5 inches

therefore HA=HB=2.25 inches

 $$\\x^2=4.5^2-2.25^2\\
x^2=20.25-5.0625\\
x^2=15.1875\\
x^2=151875/10000\\
x=\sqrt{151875/10000}\\
x=\sqrt{25*25*81*3/10000}\\
x=(25*9/100)\sqrt{3}\\
x=(2.25)\sqrt{3}\\
side = 4.5\sqrt{3}\;\;inches$$

Thanks Heureka 

You mean that you get 1c on the first day,2c on the second day,4c on the 3rd day,...... for 30days?

This is the sum of a GP    a=1, r=2, n=30

$$\\\boxed{S_n=\frac{a(r^n-1)}{r-1}}\\\\
S_{30}=\frac{1(2^{30}-1)}{2-1}\\\\
S_{30}=(2^{30}-1)$$

$${{\mathtt{2}}}^{{\mathtt{30}}}{\mathtt{\,-\,}}{\mathtt{1}} = {\mathtt{1\,073\,741\,823}}$$  cents

$10,737,418.23     Not bad for a months pocket money  

Your question does not make any sense