Final answer:
The estimated amount of paint needed to apply a 0.04 cm thick coat to a hemispherical dome with a diameter of 42 meters is approximately 0.084 liters.
Step-by-step explanation:
To estimate the amount of paint needed to cover a hemispherical dome using differentials, we calculate the surface area and then factor in the thickness of the paint layer. For a hemisphere with a diameter of 42 meters, the radius (r) is 21 meters. The surface area (A) of a hemisphere is 2πr². With r = 21 m, A = 2π(21 m)² = 2772π m². The differential of the surface area with respect to the radius (dA/dr) is 4πr. Multiplying dA/dr by the differential of the radius (dr), which is the additional thickness of the paint (0.04 cm or 4×10⁻´ m), gives us the increase in surface area due to the paint thickness. So, dA = 4πr • dr = 4π(21 m) • 4×10⁻´ m ≈ 0.21 m². The volume of paint (V) needed is the product of the surface area increase and the paint thickness: V = dA • dr = 0.21 m² • 4×10⁻´ m. Simplifying, we get a volume of paint required of approximately 0.000084 m³, or 84 cm³. Since 1 liter equals 1000 cm³, this converts to roughly 0.084 liters of paint.