Use differentials to estimate the amount of paint needed to apply a coat of paint 0.03 cm thick to a hemispherical dome with diameter 54 m. (Round your answer to two decimal places.)
v = 2/3 pi r^3 (half a sphere)
dv = 2 pi r^2 dr
now just plug in the given r and dr
watch the units.
To estimate the amount of paint needed, we can use differentials to approximate the change in surface area of the dome when a layer of paint with thickness 0.03 cm is applied.
The surface area of a hemisphere is given by the formula:
A = 2πr^2
Where A is the surface area and r is the radius.
Given that the diameter of the dome is 54 m, we can find the radius by dividing the diameter by 2:
r = 54 m / 2 = 27 m
Now, we can find the initial surface area of the dome:
A = 2π(27 m)^2 = 14526π m^2
To estimate the change in surface area, we consider the differential of the surface area, dA:
dA = 4πr dr
Where dr is the change in radius, which in this case is equal to the thickness of the paint layer, 0.03 cm.
Substituting the values:
dA = 4π(27 m)(0.03 cm) = 3.24π m²
Therefore, the approximate change in surface area when a layer of paint 0.03 cm thick is applied is 3.24π m².
To estimate the amount of paint needed, we can divide the change in surface area by the thickness of the paint layer:
Amount of paint = (3.24π m²) / (0.03 cm) = 108π cm³ ≈ 340.06 cm³
Rounding to two decimal places, the estimated amount of paint needed to apply a coat of paint 0.03 cm thick to the hemispherical dome is approximately 340.06 cm³.
To estimate the amount of paint needed to apply a coat of paint 0.03 cm thick to a hemispherical dome, we can use differentials.
First, let's start by finding the surface area of the hemispherical dome. The formula for the surface area of a hemisphere is given by:
A = 2πr^2
Where A is the surface area and r is the radius of the hemisphere. Since the diameter of the dome is given as 54 m, the radius can be calculated as r = 54/2 = 27 m.
Now, let's calculate the differential dA, which represents an infinitesimal change in the surface area when the radius changes by a small increment dr. We can find it using differentials:
dA = (dA/dr) * dr
To find (dA/dr), we take the derivative of the surface area formula with respect to r:
(dA/dr) = d/dx (2πr^2)
= 4πr
Since we want to estimate the paint needed to apply a coat of paint 0.03 cm thick, we need to consider the infinitesimal change in volume dV, which is equal to the surface area multiplied by the thickness dx:
dV = A * dx
Now, let's substitute the values into the equations. The surface area A is 2πr^2, where r = 27 m. The thickness dx is given as 0.03 cm, which needs to be converted to meters (1 cm = 0.01 m):
dx = 0.03 cm * 0.01 m/cm
= 0.0003 m
So, we have:
A = 2πr^2
= 2π * (27)^2
= 14526π m^2
(dA/dr) = 4πr
= 4π * 27
= 108π
dV = A * dx
= 14526π m^2 * 0.0003 m
= 4.3578π m^3
Finally, to estimate the amount of paint needed, we can round the result to two decimal places:
dV ≈ 4.36π m^3
Therefore, the estimated amount of paint needed to apply a coat of paint 0.03 cm thick to a hemispherical dome with a diameter of 54 m is approximately 4.36π cubic meters.