Alan

I don't see how you can avoid doing the integration again, since r is now a function of t.  The result I get when so doing is as follows:

a.) What is the expected value of the number shown when we draw a single slip of paper?

E = (8*2 + 2*4)/(8 + 2) → 2.4

b.) What is the expected value of the number shown if we add one additional 4 to the bag?

E = (8*2 + 3*4)/(8 + 3) → 2.545...

Now try c) and d) yourself.

This graph should help:

cos(A - 90°) = cos(A)cos(90°) + sin(A)sin(90°)  or  cos(A - 90°) = cos(A)*0 + sin(A)*1 or cos(A - 90°) = sin(A)

If A = x + 34, then sin(x + 34) can be written as cos(x + 34 - 90) or cos(x - 56)

So if sin(x+34) = cos(2x + 20) we could also write  cos(x - 56) = cos(2x + 20) which means that x - 56 = 2x + 20 

rearranging, we have: x = -76°  

Check:

sin(-76+34) = -0.6691

cos(2*(-76)+20) = -0.6691

If you want a value of x between 0 and 360, simply add 360 to x (i.e 360 - 76 = 284).

sin R will be the same as cos P, and cos R will be the same as sin P.  This should be enough information for you to find sin R cos R.

There are 6 solutions when y = -3 and when y = -4.  i.e. a horizontal line at y = -3 cuts the curve in 6 places (similarly for y = -4).  Hence c = -3 and c = -4. I'll leave you to add them together!

The expression can be written as \(\frac{(x+1)^2}{(x-1)(x+3)}\)    which is undefined when the denominator is zero.

This occurs when x = 1 and x = -3, so the product is -3.

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This can be rearranged as (x+1)^2 = -2y^2/(y^2+5)

The right hand side is negative unless y is zero, whereas the left hand side is positive unless x = -1.

The only way to achieve equality is for both sides to be zero, which means the only ordered pair of real numbers to satisfy the equation is (x,y) = (-1,0)

Find AI using Pythagoras:

AI² = (AC/2)² + (18+5)²

where AC² = 5² + 7²

To find angle AIE first find length EI.

EI² = (EH/2)² + 5²   and EH = AC

Having found AI and EI, now use the cosine rule to find AIE:

cos(AIE) = (AI² + EI² - 18²)/(2*AI*EI) so AIE = cos^-1((AI² + EI² - 18²)/(2*AI*EI))

I’ll leave you to crunch the numbers!

Using Google translate we have:

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