Use linear approximation or differentials to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.010000cm thick to a hemispherical dome with a diameter of 75.000meters.
Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.100000 cm thick to a hemispherical dome with a diameter of 55.000 meters.
Well, let's start by finding the volume of the hemispherical dome. The formula for the volume of a hemisphere is given by V = (2/3)πr³, where r is the radius.
We know that the diameter of the dome is 75.000 meters, so the radius (r) would be half of that, which is 37.500 meters.
Now, we want to paint a coat that is 0.010000 cm thick. Let's convert that to meters: 0.010000 cm = 0.000100 meters.
To estimate the amount of paint needed, we can calculate the change in volume when we increase the radius by 0.000100 meters using differentials.
The differential of the volume can be approximated by: dV = (4/3)πr²dr
Substituting the values, we have: dV = (4/3)π(37.500 meters)²(0.000100 meters)
Calculating this gives us: dV ≈ 0.1570796327 cubic meters.
Since we want the answer in cubic centimeters, we can convert cubic meters to cubic centimeters: 1 cubic meter = 1,000,000 cubic centimeters.
So, dV ≈ 0.1570796327 cubic meters x 1,000,000 cubic cm/m³ ≈ 157,079.6327 cubic centimeters.
Therefore, the amount of paint needed to apply a coat 0.010000 cm thick to the hemispherical dome is approximately 157,079.6327 cubic centimeters. But don't worry, I'm sure the paint store will have plenty in stock!
To estimate the amount of paint needed to apply a coat of paint 0.010000 cm thick to a hemispherical dome with a diameter of 75.000 meters, we can use linear approximation or differentials.
First, let's calculate the radius of the hemisphere by dividing the diameter by 2:
Radius = diameter / 2 = 75.000 meters / 2 = 37.500 meters
Next, let's find the surface area of the hemisphere. The surface area of a hemisphere is given by the formula:
Surface Area = 2πr^2
where π is a constant approximately equal to 3.14159 and r is the radius of the hemisphere. Plugging in the values, we get:
Surface Area = 2 * 3.14159 * (37.500 meters)^2
Now, let's convert the thickness of the paint from centimeters to meters, since the radius is given in meters:
Thickness = 0.010000 cm * (1 meter / 100 centimeters) = 0.000100 meters
To estimate the amount of paint needed, we will multiply the surface area by the thickness of the paint:
Volume of Paint = Surface Area * Thickness
Now, divide the volume of paint by 1000 to convert it to cubic centimeters:
Volume of Paint (in cubic centimeters) = (Surface Area * Thickness) * 1000
Finally, substitute the values into the formula:
Volume of Paint (in cubic centimeters) = (2 * 3.14159 * (37.500 meters)^2 * 0.000100 meters) * 1000
Calculating this expression will give you an estimate of the amount of paint needed to apply a coat of paint 0.010000 cm thick to the hemispherical dome with a diameter of 75.000 meters.
Here's the same answer in the correct units (cubic centimetres).
V=(4/3)πr³
dV/dr=4πr²
Approximate dV/dr with
ΔV/Δr, we get
ΔV=4πr²Δr (in cubic cm)
where
r=75.000/2*100 cm
=3750 cm, after conversion to cm
Δr=0.01cm
Actually, the formula for the volume of a HEMIsphere is (4/6)pi(r)^2
V=(4/3)πr³
dV/dr=4πr²
Approximate dV/dr with
ΔV/Δr, we get
ΔV=4πr²Δr
where
r=75.000/2, and
Δr=0.0001m after conversion to metres.