Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.03 cm thick to a hemispherical dome with a diameter of 45.000 meters.

To estimate the amount of paint needed to apply a coat of paint 0.03 cm thick to a hemispherical dome, we can use linear approximation. Here's how you can do it step-by-step:

Step 1: Find the radius of the hemispherical dome:
The radius can be calculated using the diameter,
r = d/2
r = 45.000 meters / 2
r = 22.500 meters

Step 2: Calculate the surface area of the hemisphere:
The surface area of a hemisphere can be calculated using the formula,
A = 2πr²
A = 2 * 3.14 * (22.500 meters)²
A ≈ 3182.66 square meters

Step 3: Convert the surface area to square centimeters:
Since we're working with cubic centimeters, we need to convert the surface area to square centimeters. We know that 1 meter = 100 centimeters. So,
1 square meter = 100 * 100 = 10,000 square centimeters
3182.66 square meters ≈ 31,826,600 square centimeters

Step 4: Calculate the volume of the paint needed:
To calculate the volume, we need to multiply the surface area with the thickness of the coat of paint.
Volume = Surface Area * Thickness
Volume = 31,826,600 square centimeters * 0.03 cm
Volume ≈ 954,798 cubic centimeters

Therefore, to apply a coat of paint 0.03 cm thick to a hemispherical dome with a diameter of 45.000 meters, you would need an estimated amount of 954,798 cubic centimeters of paint.

To use linear approximation to estimate the amount of paint needed to apply a coat to a hemispherical dome, we can break down the problem into smaller steps. Here's how you can do it:

Step 1: Find the radius of the hemispherical dome
The diameter of the dome is given as 45.000 meters. To find the radius, divide the diameter by 2.
radius = diameter / 2 = 45.000 m / 2 = 22.500 m

Step 2: Find the surface area of the dome
The surface area of a hemispherical dome can be calculated using the formula:
surface area = 2πr^2
where r is the radius.
surface area = 2π(22.500 m)^2 = 2π(506.250) m^2

Step 3: Calculate the change in surface area
Since we are applying a coat of paint that is 0.03 cm thick, the change in surface area can be estimated as the change in the surface of a hemisphere with a slightly larger radius.
change in surface area = 2π(r + Δr)^2 - 2πr^2
where Δr is the change in radius due to the thickness of the paint coat.

Step 4: Approximate the change in surface area using linear approximation
Linear approximation utilizes the concept of differentials, which allows us to approximate small changes. The change in surface area can be approximated using the linear approximation formula:
ΔA ≈ dA = 2πr * (dr)
where dr is the change in radius.

Step 5: Find the volume of the paint
The volume of the paint needed can be estimated by multiplying the change in surface area with the thickness of the paint coat.
volume of paint = ΔA * thickness of paint = dA * thickness of paint

Step 6: Convert the units to cubic centimeters
Since the thickness of the paint coat is given in centimeters, we need to convert the volume to cubic centimeters. Recall that 1 cubic meter equals 1,000,000 cubic centimeters (cm³).
volume of paint (in cm³) = volume of paint (in m³) * 1,000,000 cm³ / 1 m³

By following these steps, you can estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.03 cm thick to the hemispherical dome.