Alan

tan(x) = 15/8 so x = tan^-1(15/8) or x ≈ 61.9º

Arc length 32π from angle 240 means circumference = 32π*360/240 → 48π

Cirumference is 2πr so r = 48π/(2π) → 24

x=2y  (1)

x^2+2x-y-y^2=36 (2)

Replace x in (2) by 2y

4y^2 +4y -y - y^2 = 36

3y^2 + 3y = 36

y^2 + y = 12 

y^2 + y - 12 = 0

(y + 4)(y - 3) = 0

Hence (i) y = -4, and using (1) x = 2*(-4) or x = -8

and     (ii) y = 3, and using (1) x = 2*3 or x = 6

This requires a numerical solution method.  Draw a graph first to see where the solution(s) might lie.

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The change in momentum is 2(30-0) kg.m/s → 60 kg.m/s

The rate of change of momentum (= force) = 60/0.01 N → 6000 N

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I don't know what Guest #1's answer was, but here's a possible derivation (I've assumed Dx[...] means differentiate [...] with respect to x):

The resistance in the circuit on the right (R_r) is twice that of the circuit on the left (R), so the current (I_r) is half that of the left circuit (I).  Power dissipated is equal to I_r²*R_r , so the power dissipated in the right circuit is given by:  P_r = (I/2)²*(2R)

so P_r = I²*R/2,  but P = I²*R  so P_r = P/2

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For powers of x, we have:

\(\int{x^n}dx=\frac{x^{n+1}}{n+1}\text{ as long as }n\ne-1\)   

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Correct!

The answer sheet has the results as indicated by the red boxes below (you still need to add them together):

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