Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.100000 cm thick to a hemispherical dome with a diameter of 75.000 meters.
wouldn't the volume of paint be approximated by the surface area of the dome multiplied by the thickness of the pain?
To estimate the amount of paint needed to apply a coat of paint to a hemispherical dome, we can use linear approximation. Linear approximation is a method used to approximate the value of a function near a point using the tangent line to that point.
Let's break down the problem step by step:
1. Calculate the radius of the hemispherical dome:
Given that the diameter of the dome is 75.000 meters, we can find the radius by dividing the diameter by 2:
Radius = Diameter / 2 = 75.000 m / 2 = 37.500 m
2. Calculate the surface area of the hemispherical dome:
The surface area of a hemisphere is given by the formula:
Surface Area = 2πr^2
where r is the radius of the hemisphere.
Surface Area = 2π(37.500m)^2
3. Calculate the area of the coat of paint:
The coat of paint has a thickness of 0.100000 cm. Since the paint is evenly spread, the area of the coat of paint is the same as the surface area of the hemispherical dome, but increased by the thickness.
Coat Area = Surface Area + 2πr * Thickness
Coat Area = 2π(37.500m)^2 + 2π(37.500m)(0.100000cm)
4. Convert the thickness from centimeters to meters:
The thickness is given in centimeters, but the radius and surface area are calculated in meters. Therefore, we need to convert the thickness to meters:
Thickness (in meters) = Thickness (in centimeters) / 100
Thickness (in meters) = 0.100000 cm / 100 = 0.001000 m
5. Calculate the volume of the coat of paint:
The volume of the coat of paint can be obtained by multiplying the coat area by the thickness:
Volume = Coat Area * Thickness
Volume = (2π(37.500m)^2 + 2π(37.500m)(0.100000cm)) * (0.001000 m)
By substituting the appropriate values into the equation, you can now calculate the estimated amount of paint needed in cubic meters.