Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.020000 cm thick to a hemispherical dome with a diameter of 40.000 meters. How do I do this?
To estimate the amount of paint needed to apply a thin coat on the surface of a hemispherical dome using linear approximation, you can follow these steps:
1. Calculate the surface area of the hemisphere:
The formula for the surface area of a hemisphere is given by A = 2πr^2, where r is the radius of the hemisphere. In this case, the radius is half the diameter, so r = 40.000 meters / 2 = 20.000 meters. Convert this to centimeters: 20.000 meters * 100 = 2000.000 centimeters. Substitute this value into the formula to find the surface area: A = 2 * π * (2000.000)^2.
2. Approximate the change in surface area:
The change in surface area will be equal to the area of the thin coat of paint applied, which is the thickness multiplied by the surface area. In this case, the thickness is given as 0.020000 cm. Multiply this by the surface area previously calculated to find the change in surface area.
3. Apply linear approximation:
To use linear approximation, we assume that the change in surface area is small compared to the total surface area. Therefore, we can approximate the change using a linear equation. In this case, we can use the equation: ΔA ≈ dA/dr * Δr. Here, dA/dr represents the derivative of the surface area with respect to the radius, and Δr represents the change in radius.
4. Find the derivative of the surface area:
Take the derivative of the surface area A with respect to the radius r. In this case, dA/dr = 4πr.
5. Calculate the change in radius:
To find the change in radius, we need to consider the thickness of the paint layer, which is given as 0.020000 cm. Since the paint covers both the inside and outside of the hemisphere, we divide the thickness by 2 to get the change in radius. Therefore, Δr = 0.020000 cm / 2.
6. Substitute the values into the linear approximation equation:
ΔA ≈ dA/dr * Δr. Substitute the values for dA/dr (4πr) and Δr (0.020000 cm / 2). Simplify the equation.
7. Calculate the estimated change in surface area:
Evaluate the expression obtained from the previous step to find the approximate change in surface area.
Now you have estimated the change in surface area, which represents the amount of paint needed to apply a coat of 0.020000 cm thickness on the hemispherical dome.