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?

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

Is there something wrong in how I presented the problem

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),

Thank you in advance

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

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.

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

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