a cylindrical container with a circular base has a surface area of 64ft squared.express its volume as a function of its radius
To express the volume of the cylindrical container as a function of its radius, we need to use the formula for the surface area of a cylinder.
The formula for the surface area of a cylinder is:
S = 2πr^2 + 2πrh
Where:
S is the surface area
r is the radius of the base
h is the height of the cylinder
Given that the surface area is 64 square feet, we can set up the equation as follows:
64 = 2πr^2 + 2πrh
To express the volume as a function of the radius, we need to eliminate the height from the equation. We can do this by rearranging the formula for the surface area:
64 - 2πr^2 = 2πrh
Now, solve for h:
h = (64 - 2πr^2) / (2πr)
The volume of a cylinder is given by the formula:
V = πr^2h
Substitute the expression for h into the formula for the volume:
V = πr^2 * ((64 - 2πr^2) / (2πr))
Simplifying, we get:
V = (32πr^2 - 2π^2r^4) / πr
Therefore, the volume of the cylindrical container can be expressed as a function of its radius, r, by the equation:
V(r) = 32r - 2πr^3
To express the volume of a cylindrical container as a function of its radius, we first need to understand the relationship between the surface area and the dimensions of the cylinder.
The surface area of a cylindrical container can be divided into two parts: the area of the circular base and the lateral area (the curved surface). The formula for the surface area of a cylinder is:
Surface Area = 2πr² + 2πrh
Where:
- r is the radius of the base
- h is the height of the cylinder
- π is a mathematical constant, approximately equal to 3.14
In this case, we are given that the surface area is 64 square feet. Therefore:
64 = 2πr² + 2πrh
Now, let's express the volume of the cylinder as a function of its radius. The formula for the volume of a cylinder is:
Volume = πr²h
To express the volume as a function of the radius, we need to eliminate the height variable from the equation because it is not given. We can solve the surface area equation for h and substitute it into the volume equation:
64 = 2πr² + 2πrh
64 - 2πr² = 2πrh
h = (64 - 2πr²) / (2πr)
Substituting this value of h into the volume equation:
Volume = πr²h
Volume = πr² * (64 - 2πr²) / (2πr)
Simplifying further:
Volume = (πr² * (64 - 2πr²)) / (2πr)
Volume = (r² * (64 - 2πr²)) / (2r)
Volume = (32r² - πr⁴) / r
Volume = 32r - πr³
Therefore, the volume of the cylindrical container can be expressed as a function of its radius: V(r) = 32r - πr³.
pi*r(2r + L) = 64
L = (64/(pi*r))- 2 r
V = pi r^2 L
Substitute the equation for L in terms of r.