Calculus, linear aproximations, and differentials...?

Question

How can I use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with a diamter of 50m?

Answer 1

A,the surface area of a hemisphere=2pir^2
differentiating dA/dr=4pi*r at r=25,delta A=(dA/dr)*delta r
plugging in delta A= increase in area= approx.4pi(25)(.05) cm^2
so from this we know the increase in area due to paint and any paint manufacturer will tell you the quantity of paint needed for this approximated area

Answer 2

This will provide a good approximation:

Diameter is 50m so radius is 25m

Formula for surface area of sphere is 4πr^2=4π(25)^2=2500π m^2
So a hemisphere would be 1250π m^2
you want it 0.05cm=0.0005m thick so 2500π•0.0005= 0.625π m^3≈ 1.963495409 m^3 ≈1964 litres

If you want an exact figure, this is how you do it:

You want to integrate 25≤r≤25.0005, 0≤θ≤2π, 0≤φ≤π/2
of ∫(∫(∫r^2sin(φ)dr)dφ)dθ ≈ 1.963534678 m^3 ≈ 1964 litres

So they are pretty close.

Answer 3

The 1st step is to construct your mathematical model. You will observe that the variables and dependents
of your mathematical model resembles a polynomial equation, such as V = x2(pie)...

There is usually more than one model to a problem. You will know that the answer is correct, when more than one model gets you the "same magnitude".

Cheers !

Answer 4

Eulercrosser has a pretty good answer and that's just geometry mostly. No need for calculus.