Is there something wrong in how I presented the problem
I have got such a problem:
Use the linear approximation to estimate the amount of paint in cubic cm needed to apply a coat of paint 0.020000 cm thick to a hemispherical dome with a diameter 45.000 meters.
From what I understood I am supposed to use this formula in order to find the amount of paint V: f(x)= f(a) + f’(a)(x-a)
I tried to solve this by this way:
let x be 0 cm; Diameter = 45 m = 4500 cm; Radius = 2250cm
Area = 4(pi)r^2 = f(a)
Area = 4(pi)r^2 = 4(pi)(2250)^2 = 4(pi)5062500 cm = 63617251,2351cm^2
Area’ = 8(pi)r = f’(a) = 8(pi) 2250cm = 56548.66776461cm
f(x) = 4(pi)r^2 + 8(pi)r (x - a) = 63617251,2351cm^2 + 56548.66776461cm( 0 – 0.02 ) = ...but something tells me that I made a mistake and I can't figure which one
I checked what answer I'm supposed to obtain by using this formula Volume = 63617251,2351 x 0,02cm^3 = 1272345 cm^3
What do you think? Where am I mistaken?
The problem is that you are trying to use surface area to find the volume of the paint. Instead use so that . Notice that this is just the surface area multiplied by the thickness of the paint. That's because the linear approximation treats the spherical surface as if it were a plane.