# [SOLVED]estimate the amount of paint needed to apply a coat of paint 0.05 cm thick

#### karush

##### Well-known member
Use differentials to estimate the amount of paint needed
to apply a coat of paint $$\displaystyle 0.05 \text{ cm}$$ thick
to a hemispherical dome with a diameter of $$\displaystyle 50\text{ m}$$

$$\displaystyle V_h=\frac{1}{2} \cdot\frac{4}{3}\pi\text{r}^3 \text{ = } \frac{4}{6}\pi\text{r}^3$$

$$dV_h = 2\pi\cdot r^2\cdot \text{dr}$$

so if $$r=25\text{ m} = 2500\text{ cm}\text { and }dr = 0.05\text{ cm}$$ then

$$dV_h = 2\pi \ 2500\text{ cm}^2\cdot 0.05\text{ cm} \approx 1.96\text{ m}^3$$

or should $$r=2500.05\text{ cm}$$

my first try at this so....

#### MarkFL

Staff member
You did it correctly. I always like to, if possible, compare the approximate to the true value:

$$\displaystyle V=\frac{2\pi}{3}\left((r+\Delta r)^3-r^3 \right)$$

$$\displaystyle V=\frac{2\pi}{3}\left(2500.05^3-2500^3 \right)\text{ cm}^3\approx1.96353467866359\text{ m}^3$$

This is very close to the estimate.

#### DreamWeaver

##### Well-known member

[I'm a painter and decorator, see, so I might just find a use for this... ]

#### karush

##### Well-known member
well a hemisphere would not be easy to paint especially .05 cm uniformly!!

with a big brush I guess