Find the rule for the volume of a cylinder in terms of r only if the height is equal to its circumference.
The surface area of a cylinder is 2pi square units. Find a rule for h in terms of r.
v = pi r^2 h = pi r^2 r ...
2pi r (r+h) = 2pi
r(r+h) = 1
...
To find the rule for the volume of a cylinder in terms of r only when the height is equal to its circumference, we need to use the given information.
Let's denote the radius of the cylinder as r and the height as h. We are told that the height is equal to the circumference, which is given by 2πr.
The volume of a cylinder is given by the formula V = πr^2h. Since h is equal to 2πr, we can substitute this value into the formula:
V = πr^2(2πr)
Simplifying this expression, we get:
V = 2π^2r^3
Therefore, the rule for the volume of a cylinder in terms of r only when the height is equal to its circumference is V = 2π^2r^3.
Now, let's move on to the second question about finding a rule for h in terms of r when the surface area of a cylinder is 2π square units.
The surface area of a cylinder is given by the formula A = 2πr^2 + 2πrh. We are given that the surface area is equal to 2π.
Setting up the equation, we have:
2π = 2πr^2 + 2πrh
Dividing both sides of the equation by 2π, we get:
1 = r^2 + rh
Rearranging the equation to isolate h, we have:
h = (1 - r^2)/r
So, the rule for h in terms of r when the surface area of a cylinder is 2π square units is h = (1 - r^2)/r.
To find the rule for the volume of a cylinder in terms of the radius (r) only when the height is equal to its circumference, we need to use the formula for the volume of a cylinder and substitute the given condition.
The formula for the volume of a cylinder is V = πr²h, where V represents the volume, r is the radius, and h is the height.
Given that the height (h) is equal to the circumference (C), we can rewrite the formula as V = πr²C.
Now, the formula for the circumference of a circle is C = 2πr, where C represents the circumference and r is the radius.
Substituting this into the previous equation, we have V = πr²(2πr).
Simplifying further, we get V = 2π³r³.
Therefore, the rule for the volume of a cylinder in terms of the radius only, when the height is equal to its circumference, is V = 2π³r³.
Now, let's move on to finding the rule for h in terms of r when the surface area of a cylinder is 2π square units.
The formula for the surface area of a cylinder is A = 2πrh + 2πr². Given that the surface area is 2π square units, we can rewrite the formula as 2π = 2πrh + 2πr².
Dividing both sides of the equation by 2π, we have 1 = rh + r².
To isolate h, we subtract r² from both sides of the equation, resulting in 1 - r² = rh.
Finally, dividing both sides of the equation by r, we obtain the rule for h in terms of r: h = (1 - r²) / r.