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.)

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

i think the correct option is the second option.

i think the amount would be $140.62