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

• Mar 8th 2013, 09:45 AM
dokrbb
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?

• Mar 8th 2013, 03:09 PM
dokrbb
Re: Linear approximation - word problem: something is wrong with my solution, Thanks
Is there something wrong in how I presented the problem
• Mar 9th 2013, 06:19 AM
dokrbb
Re: Linear approximation - word problem: something is wrong with my solution, Thanks
Quote:

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),
The problem is that you are trying to use surface area to find the volume of the paint. Instead use $\displaystyle V= \frac{4}{3}\pi r^3$ so that $\displaystyle dV= 4\pi r^2 dr$. 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.