Imagine slicing through a sphere with a plane (sheet of paper). the smaller piece produced is called a radius of the sphere. Its volume is V=(pi)h^2(3r-h)/3, where r is the radius of the sphere and h is the thickness of the cap. find dr/dh for a sphere with a volume of 5pi/3m^3. evaluate the derivative when r=2m and h=1m.
we have
π/3 h^2(3r-h) = 5π/3
3rh^2-h^3 = 5
3h^2 dr/dh + 6rh - 3h^2 = 0
h dr/dh + 2r - h = 0
dr/dh = (h-2r)/h = 1 - 2r/h
...
To find dr/dh, we need to differentiate the volume equation V=(πh²(3r-h))/3 with respect to both r and h.
Let's start by differentiating with respect to r:
dV/dr = (πh²(3(1) - h))/3
Simplifying, we get:
dV/dr = (πh²(3 - h))/3
Next, we need to differentiate with respect to h:
dV/dh = (π(2h(3r - h) - h²))/3
Simplifying further, we get:
dV/dh = (π(6rh - 2h² - h²))/3
dV/dh = (π(h(6r - 3h - 2h)))/3
dV/dh = (π(h(6r - 5h)))/3
Now, we know that the volume V is given as 5π/3m³. Plugging this into the equation, we get:
5π/3 = (π(h(6r - 5h)))/3
To find dr/dh, we need to isolate it in the equation. So, let's rearrange the equation:
5π = h(6r - 5h)
6rh - 5h² = 5π
6rh = 5h² + 5π
r = (5h² + 5π)/(6h)
Now, we can differentiate r with respect to h:
dr/dh = [(2(5h)(6h) - (5h² + 5π)(6))/((6h)²]
dr/dh = [(60h² - 30h² - 30π)/(36h²)]
dr/dh = [(30h² - 30π)/(36h²)]
dr/dh = (5h² - 5π)/(6h²)
To evaluate the derivative when r = 2m and h = 1m, we substitute these values into the derivative equation:
dr/dh = (5(1)² - 5π)/(6(1)²)
dr/dh = (5 - 5π)/6
Therefore, when r = 2m and h = 1m, dr/dh = (5 - 5π)/6.