CPhill

Using Heron's formula where s = 9 ...... and a = b = c = the side lengths .....we have......

A = √[s(s -a)^3] =  √[9*3^3]   = √[9*27] = √[9*9*3] =   √[81*3)   = 9√3  sq. in. 

Here's one more way using  u-du substitution.......

∫ (arccos(x))^2 / √(1 - x^2)   dx

Let  u   = arccos(x)      du = -1/ √(1 - x^2)   dx    implies that.....-du = 1/ √(1 - x^2)   dx 

So we have......

-∫ u^2 du =     -(1/3)u^3  + C    =  -(1/3)[arccos(x)] + C

The number of blue and white tiles  = total tiles

B +  W   =   T     

And we're told that:    9B = 5W  ...so...   B   = (5/9)W   .....which implies that....

(5/9)W  + W = T         

And if 9 white tiles are replaced with blue tiles, we have.....

[(5/9)W + 9 ]  : [ W - 9 ]  = 13 : 15    ....or.....expressing another way.....

[(5/9)W + 9 ] *15  = 13*[W - 9]  ..... solve for W

(75/9)W + 135 =  13W -  117

75W + 1215  = 117W  - 1053

2268  = 42W

So, W = 54 = the number of original white tiles ....  and B = (5/9)W =  30 = number of original blue tiles....so...... there are  84 tiles total

Check......  before   =   30/54   = 5/9    ...    after =  39/45  =  13/15    √

To get the slope......subtract the y components of each point in any order....then......subtract the x components in the same order.....put the first result over the second.....so we have.....

-7 - (-5)   = -2

Subtract the x components in the same order......

1 - (-2)   = 3

Put the first result over the second........-2/3     .....    and that's the slope....!!!!

Irrational........

Think about this   .......if we took  √17  (an irrational) and added it four times, we would just have 4√17.....but the "√17"  part would remain, and thus, the "irrational" part would remain......the only way to make this "rational"  would be to multiply it by an irrational!!!

As long as this inequality is satisfied, he can make it back home.....

($400)( 600) / x  ≤   $100,000    where x is the mpg of gas that his car gets ......so.....rearranging

240,000/100,000  ≤   x

2.4   ≤    x    

In other words, as long as his car gets at least 2.4 mpg of gas, he can make it back

43 minutes is 43/60 degrees......so.....

tan (126 + 43/60)°   =  -1.340788783609   = -1.3408 to the nearest ten-thousandth

That's very nice, Melody.......!!!

Thanks, Alan......that method seemed a little convoluted for everyday practicality......IMHO.....!!!!