Final answer:
To estimate the amount of rust-proofer needed for a hemisphere-shaped dome with a radius of 50ft and a coat thickness of 0.01 inches, we use differentials to calculate the additional surface area due to the coating. Convert the thickness to feet, use the differential surface area formula, and evaluate to find the increased surface area that the rust-proofer must cover.
Step-by-step explanation:
Estimating the Amount of Rust-Proofer Required for a Hemisphere The amount of rust-proofer needed to coat a hemisphere-shaped dome with a radius of 50ft and a coat thickness of 0.01 inches can be estimated using the surface area formula of a hemisphere and considering the thickness of the coating. The surface area A of a hemisphere is given by A = 2πr2, where r is the radius. To estimate the change in surface area caused by the thin layer of rust-proofer, we can use differentials.
Let dr be the increase in radius due to the coating (0.01 inches, which we convert to feet by dividing by 12 to obtain 0.01/12 feet). The differential surface area dA can be approximated by dA = 4πrdr. We substitute the radius, r = 50ft, and dr = 0.01/12ft into the differential formula to find dA. This estimate gives us the additional surface area that needs to be covered with rust-proofer.
Calculation Steps:
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- Convert the coating thickness from inches to feet: dr = 0.01 inches * (1 foot/12 inches) = 0.01/12 feet.
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- Calculate the differential surface area: dA = 4π(50)(0.01/12).
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- Evaluate the expression to find dA, which gives us the estimated additional surface area.
By calculating the differential surface area, we can determine the necessary amount of rust-proofer to cover the added thickness.