Question

Use differentials to estimate the amount of paint needed to apply a coat of paint 0.07 cm thick to a hemispherical dome with diameter 48 m . (Round your answer to two decimal places.)

Answer 1

Answer:

10.12 $m^{3}$

Step-by-step explanation:

Given:

• 0.07 cm thick = 0.007m and it is dr (the change in radius)

As we know that, the volume of a hemispherical dome with diameter 48 m is:

• V = $\frac{2}{3}$ π$r^{3}$

Take the derivative of V we have:

$\frac{dV}{dr}$ = 2π$r^{2}$  

<=> dV = 2π$r^{2}$*dr

<=> dV = 2π*$48^{2}$ *0.0007 = 10.12 $m^{3}$

Hope it will find you well.

Answer 2

Answer:

the amount of paint needed = 2.53 m³

Step-by-step explanation:

First of all,

Volume of a sphere is given by;

V = 4πr³/3

We are dealing with a hemispherical dome in this problem.

So the formula of a hemispherical dome is half of volume of sphere

Thus,

Volume of hemispherical dome is;

V = (1/2) x 4πr³/3 = 2πr³/3

Now, I'll take the derivative of V of hemispherical dome with respect to r to get ;

dv/dr = 2πr²

Thus, dv = 2πr²•dr

Where;

dv is the change in volume and is the extra amount of paint needed to apply the coat.

dr is the change in radius

r is the original radius before the paint is applied

We are given that;

diameter = 48m

So, radius; r = 48/2 = 24m

also, dr = 0.07cm = 0.0007m

Thus,

dv = 2πr²•dr = 2π(24)²•0.0007 = 2.53 m³