Final answer:
To estimate the amount of paint needed to apply a coat of paint 0.07 cm thick to a hemispherical dome with a diameter of 58 m, use differentials. Find the surface area of the dome, then estimate the change in volume using differentials. Convert the differential volume to m³, round to two decimal places for the final answer.
Step-by-step explanation:
To estimate the amount of paint needed to apply a coat of paint 0.07 cm thick to a hemispherical dome with a diameter of 58 m, we can use differentials. First, we need to find the surface area of the dome. The surface area of a hemisphere can be found using the formula 2πr², where r is the radius. Since the diameter is given as 58 m, the radius is half of that, which is 29 m. Therefore, the surface area is 2π(29)² = 16862π m².
Next, we can use differentials to estimate the change in volume when the thickness increases by 0.07 cm. The volume of a hemisphere is given by (2/3)πr³, so the volume can be approximated as (2/3)π(29)³. Then, we can differentiate this expression with respect to the thickness (t) to find the differential volume, which is (2/3)π(29)³ dt.
Finally, we can plug in the value of dt (0.07 cm) to find the differential volume and convert it to the desired units (m³) by multiplying by 0.000001 (1 cm³ = 0.000001 m³). This will give us the estimate for the amount of paint needed to apply a coat 0.07 cm thick to the hemispherical dome. Rounding this value to two decimal places will give the final answer.