Suppose you have a sphere of radius r, and you also have a cylinder with radius r whose height is also r. The sum of the volumes of the sphere and the cylinder is
V=πr2⋅r+43πr3.
How can you rearrange the given equation to isolate r?
you almost have it already
v = πr^2*r + 4/3 πr^3
v = (1 + 4/3)πr^3
v = 7/3 πr^3
...
V = π r^2 + (4/3) π r^3
V = (3/3) π r^2 + (4/3)π r^3
V = (7 π / 3) (r^2 + r^3)
V / (7 π / 3) = (r^2 + r^3 )
r^3 + r^2 - 3 V / (7 π) = 0
solve cubic for r
Ignore what I said, had typo at start, left height out. Use what oobleck said.
To isolate the variable r in the given equation V = πr^2⋅r + 4/3πr^3, we can follow these steps:
Step 1: Write the equation: V = πr^3 + 4/3πr^3.
Step 2: Combine the terms with the same base, π: V = (1 + 4/3)r^3π.
Step 3: Simplify the expression inside the parentheses: V = (7/3)r^3π.
Step 4: Divide both sides of the equation by (7/3)π to isolate r^3: V / [(7/3)π] = r^3.
Step 5: Multiply both sides of the equation by (3/7) / π to solve for r^3: (3V/7π) * (3/7π) = r^3.
Step 6: Simplify the expression on the left side of the equation: r^3 = (9V / 49π^2).
Step 7: Take the cube root of both sides of the equation to find r: ∛(r^3) = ∛[(9V) / (49π^2)].
Therefore, by rearranging the given equation, we can isolate r as ∛(r^3) = ∛[(9V) / (49π^2)].