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

To estimate the amount of paint needed for a hemispherical dome, we can use differentials. By calculating the differential volume and integrating it over the range of radii, we can find the total volume of paint needed. In this case, the estimated amount of paint needed is approximately 146.647 cubic meters.

Step-by-step explanation:

To estimate the amount of paint needed, we can use differentials. The volume of a hemisphere is given by the formula V = (2/3)πr³, where r is the radius. In this case, the radius is half of the diameter, so r = 24 m. We want to find the volume of a layer of paint that is 0.07 cm thick, so we need to calculate the difference in volumes for radii r and r + 0.07 cm. Using differentials, we have dV ≈ (2/3)π(3r²)(0.07 cm), where dV is the differential volume. Plugging in the values, we get dV ≈ 2πr²(0.07 cm).

Now, we convert cm to meters, so dV ≈ 2πr²(0.07 cm)(0.01 m/cm) ≈ 0.0002πr² m³. To find the total volume of paint needed, we integrate this differential volume over the range of radii from 0 to 24 m: V = ∫(0 to 24) 0.0002πr² dr. Integrating, we get V ≈ 0.0002π(24³/3 - 0³/3) ≈ 146.647 m³.

Therefore, the estimated amount of paint needed to apply a coat 0.07 cm thick to the hemispherical dome is approximately 146.647 cubic meters.

Answer 2

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