Alan

Resistance of wire at 1500°C is  R = 110V/1.6A  or  R = 68.75 ohms 

Electrical resistivity of tungsten at 20°C is  rho₂₀ = 5.5*10^-8 ohm.m 

Electrical resistivity of tungsten at 1500°C is  rho₁₅₀₀ = rho₂₀*(1 + alpha*deltaT)

                                                                       rho₁₅₀₀ = 5.5*10^-8*(1 + 4.5*10^-3*(1500-20)) = 4.213*10^-7 ohm.m

Length of tungsten wire at 1500°C  L = rho₁₅₀₀/R  or  L = 4.213*10^-7 ohm.m/68.75ohms  ≈  6*10^-9 m

Hmm!  Seems very short!  Someone needs to check this.

(1/2)b*h = 124  so b*h = 248 square units

New area = (1/2)*(b/2)*3h → (3/4)b*h → (3/4)*248 → 186 square units

100! = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

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There is no last digit.  It is also known that it doesn't repeat; if it did it would be a rational number, but it is known to be irrational.

Let x = e^-r,  then the equation can be written as:

8x + 8x² + 108x³ - 100 = 0

This factors as  4(27x - 25)(x² + x + 1) = 0

The only real root is x = 25/27

so e^-r = 25/27  or  r = ln(27/25)   or  r ≈ 0.077

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You can enter it in the calculator on the home page here exactly as you've written it above, then press "=" or "Enter"

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Here are a couple of relevant graphs:

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Gradient of the secant from x = a to x = b is given by (f(b) - f(a))/(b - a)

So gradient of secant from x = 2 to x = 3 is   (3^2 - 2^2)/(3 - 2)  →  5

You should now be able to do the others.

Yes, 550mm is the pressure head at B.