You plan to paint a sphere of radius 200 cm with a coat of pain .2 cm thick. Use differentials to approximate the amount of paint that is needed. Use the equation for the volume of a sphere.
So far I have C=81, X=82, f(x)= x^1/4 and C-x = 1 (If correct) what is the next step?
dr=0.2cm
r=200cm
dV/dr = 4*pi*r^2
V=4*pi*r^2*dr
V=4*pi*200^2*0.2
To approximate the amount of paint needed to coat the sphere, we can use differentials. The equation for the volume of a sphere is given as V = (4/3) * π * r^3, where V represents the volume and r represents the radius.
Given that the radius of the sphere is 200 cm, we can substitute this value into the volume formula:
V = (4/3) * π * (200^3) = (4/3) * π * 8000000 ≈ 335,103,218.38 cm³
We want to find the change in volume when the radius increases by 0.2 cm. Let's call this change in radius as Δr = 0.2 cm.
Using differentials, we have the formula:
ΔV ≈ dV = V'(r) * Δr
To find V'(r) or the derivative of V with respect to r, we can differentiate the volume formula:
V'(r) = dV/dr = (4/3) * π * 3 * (r^2) = 4πr^2
Substituting the given radius, we have:
V'(200) = 4π(200)^2 = 4 * 3.14159 * 40000 ≈ 502,654.824 cm²
Finally, we can calculate the change in volume using the formula:
ΔV ≈ V'(200) * Δr = 502,654.824 cm² * 0.2 cm ≈ 100,530.9648 cm³
Therefore, the approximate amount of paint needed to coat the sphere is approximately 100,530.9648 cm³.