Alan

If we assume the average resistance acts as a constant force, F, which adds to the gravitational force, then we have a constant acceleration (or deceleration in this case) of a = -(9.81+F/1.07).  Using the constant acceleration formula v² = u² + 2*a*s, where v is final velocity (0), u is initial velocity (29.3) and s is distance travelled (40) we have 

0 = 29.3² - 2*(9.81+F/1.07)*40

It's a straightforward matter to find F from this.

Efficiency = (useful output/total input) multiplied by 100 to turn it into a percentage. 

Represent the complex number z by x + iy where x and y are real numbers. here, the arg function represents the angle between the line from the origin to the point {x+2, y} and the x-axis.  We are told this angle is pi/6. The tangent of this angle is just y/(x+2) so we know that:

tan(pi/6) = y/(x+2)

Now tan(pi/6) or tan(30°) is just 1/sqrt(3) so 1/sqrt(3) = y/(x+2)

Multiply both sides by x+2 to get y = (x+2)/sqrt(3)

$${\mathtt{tanofpiby6}} = \underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{30}}^\circ\right)} = {\mathtt{tanofpiby6}} = {\mathtt{0.577\: \!350\: \!269\: \!19}}$$

$${\mathtt{oneonsqrt3}} = {\frac{{\mathtt{1}}}{{\sqrt{{\mathtt{3}}}}}} = {\mathtt{oneonsqrt3}} = {\mathtt{0.577\: \!350\: \!269\: \!189\: \!625\: \!8}}$$

If you meant what shape has the biggest area for a given perimeter, then the answer is a circle.

Energy required to lift piano to apartment is 397*9.81*19.7 Joules (where acceleration of gravity is 9.81ms^-2)

Power available = 413 Watts = 413 Joules/second, so 

Time taken = 397*9.81*19.7/413 seconds

$${\mathtt{TimeTaken}} = {\frac{{\mathtt{397}}{\mathtt{\,\times\,}}{\mathtt{9.81}}{\mathtt{\,\times\,}}{\mathtt{19.7}}}{{\mathtt{413}}}} = {\mathtt{TimeTaken}} = {\mathtt{185.770\: \!046\: \!004\: \!842\: \!615}}$$seconds

Work (J)= Force (N) x Distance (m) 

214.9 = Force x 6.2

so Force (weight of backpack) = 214.9/6.2 Newtons

 $${\mathtt{Force}} = {\frac{{\mathtt{214.9}}}{{\mathtt{6.2}}}} = {\mathtt{Force}} = {\mathtt{34.661\: \!290\: \!322\: \!580\: \!645\: \!2}}$$ Newtons

e^ix = cos(x) + i*sin(x)

When x = pi we have cos(pi) = -1 and sin(pi) = 0

so e^ipi = -1

Is this what you were looking for?

First let z = e^x. Now the equation can be written as z² - 4z - 5 = 0 This factorises into (z+1)(z-5)=0 so there are two solutions for z; namely z = -1 and z = 5.  Because z is e^x we have

e^x = -1 and e^x = 5

Now, for all real values of x, e^x is positive, so there is no real solution to e^x = -1

For the other possibility we take logs of both sides to get ln(e^x) = ln(5) or just x = ln(5)  (where ln is log to the base e).

If you want the ln(5) in approximate decimal form:

$${\mathtt{x}} = {ln}{\left({\mathtt{5}}\right)} = {\mathtt{x}} = {\mathtt{1.609\: \!437\: \!912\: \!434\: \!100\: \!4}}$$

The expression for radioactive decay is N = N₀e^-ln(2)t/th  where N is the mass, at time t, N₀ is the mass at the start (30grams) and th is the half-life (5.25days), so here: 

$${\mathtt{N}} = {\mathtt{30}}{\mathtt{\,\times\,}}{{\mathtt{e}}}^{{\mathtt{\,-\,}}\left({\frac{{ln}{\left({\mathtt{2}}\right)}{\mathtt{\,\times\,}}{\mathtt{42}}}{{\mathtt{5.25}}}}\right)} = {\mathtt{N}} = {\mathtt{0.117\: \!187\: \!500\: \!000\: \!000\: \!1}}$$ grams

To the nearest thousandth of a gram this is 0.117grams.