Thread: Linear approximation - word problem: something is wrong with my solution, Thanks

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?

Is there something wrong in how I presented the problem

Originally Posted by dokrbb

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?

I don't expect you to tell me the result, I just need a suggestion, like maybe to use x = 0.01, instead of 0 ; dividing the final result by 2 (because it's an hemispherical dome),
Thank you in advance

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.

So, I just need to do the following dV= 4*pi r^2 dr =>(1/2)* 4*pi (2250)^2 0.02 = ?