Problem 3
Question
The volume of a right circular cylinder is calculated by a function of two variables, \(V(x, y)=\pi x^{2} y,\) where \(x\) is theradius of the right circular-cylinder and \(y\) represents the height of the cylinder. Evaluate \(V(2,5)\) and explain what this means.
Step-by-Step Solution
Verified Answer
The volume of the cylinder is \(20\pi\).
1Step 1: Understanding the Formula
The formula for the volume of a right circular cylinder is given by \(V(x, y) = \pi x^2 y\), where \(x\) is the radius and \(y\) is the height of the cylinder. We need to substitute the values \(x = 2\) and \(y = 5\) into this function.
2Step 2: Substituting the Values
Substitute \(x = 2\) and \(y = 5\) into the formula: \(V(2, 5) = \pi (2)^2 (5)\).
3Step 3: Calculating the Radius Squared
Compute \((2)^2 = 4\). Now the formula becomes \(V(2, 5) = \pi \cdot 4 \cdot 5\).
4Step 4: Calculating the Volume
Multiply the numbers: \(4 \times 5 = 20\). Thus the formula now is \(V(2, 5) = \pi \times 20\).
5Step 5: Finalizing the Calculation
The volume of the cylinder with radius 2 and height 5 is \(20\pi\). This represents the total space enclosed by such a cylinder.
Key Concepts
Right Circular CylinderVolume CalculationFunction of Two Variables
Right Circular Cylinder
A right circular cylinder is a three-dimensional shape used in various fields, from geometry to engineering. It is characterized by having two circular bases that are parallel and identical.
A key feature is its flat surfaces and straight alignment, which means that the sides of the cylindrical surface are perpendicular to the base.
This perpendicular alignment makes it 'right' as opposed to inclined cylinders that have a slant. The center of each of these bases is connected by a straight line passing through the midpoints, which is known as the axis of the cylinder.
The height of the cylinder is the distance between the two parallel bases along this axis, and the radius is the distance from the center to the edge of the base.
A key feature is its flat surfaces and straight alignment, which means that the sides of the cylindrical surface are perpendicular to the base.
This perpendicular alignment makes it 'right' as opposed to inclined cylinders that have a slant. The center of each of these bases is connected by a straight line passing through the midpoints, which is known as the axis of the cylinder.
The height of the cylinder is the distance between the two parallel bases along this axis, and the radius is the distance from the center to the edge of the base.
Volume Calculation
The volume of any three-dimensional object refers to the amount of space it occupies. For a right circular cylinder, the volume can be calculated using the formula:
\[V(x, y) = \pi x^2 y\]
where \(x\) is the radius of the base and \(y\) is the height of the cylinder. This formula essentially expresses the cylindrical volume as a product of the area of its base and its height.
Since the base is a circle, its area is calculated as \(\pi x^2\). The formula then multiplies this area by the cylinder's height to give the total volume.
This makes sense intuitively: imagine stacking several identical circular disks the size of the base on top of each other, reaching up to the height; the volume is simply the space taken up by all these disks stacked together.
\[V(x, y) = \pi x^2 y\]
where \(x\) is the radius of the base and \(y\) is the height of the cylinder. This formula essentially expresses the cylindrical volume as a product of the area of its base and its height.
Since the base is a circle, its area is calculated as \(\pi x^2\). The formula then multiplies this area by the cylinder's height to give the total volume.
This makes sense intuitively: imagine stacking several identical circular disks the size of the base on top of each other, reaching up to the height; the volume is simply the space taken up by all these disks stacked together.
Function of Two Variables
In mathematics, functions of two variables expand upon the idea of a single-variable function. Here, the volume of a cylinder is modeled as a function, \(V(x, y)\), to see how the volume changes in relation to both the radius \(x\) and height \(y\).
This function helps visualize that as one or both of these variables change, the volume changes accordingly. It acts as a handy tool in sophisticated calculations to predict outcomes and allows us to plug in different values to see resultant changes.
For example, by evaluating \(V(2, 5)\), which uses the specific values \(x = 2\) and \(y = 5\), we find the particular volume of this cylinder to be \(20\pi\).
This result is derived by first computing the radius squared \((2)^2 = 4\), then multiplying by the height \(5\), resulting in a base volume of \(20\). Therefore, the function shows how these parameters interact to form the total cylindrical volume.
This function helps visualize that as one or both of these variables change, the volume changes accordingly. It acts as a handy tool in sophisticated calculations to predict outcomes and allows us to plug in different values to see resultant changes.
For example, by evaluating \(V(2, 5)\), which uses the specific values \(x = 2\) and \(y = 5\), we find the particular volume of this cylinder to be \(20\pi\).
This result is derived by first computing the radius squared \((2)^2 = 4\), then multiplying by the height \(5\), resulting in a base volume of \(20\). Therefore, the function shows how these parameters interact to form the total cylindrical volume.
Other exercises in this chapter
Problem 1
For the following exercises, evaluate each function at the indicated values. $$ W(x, y)=4 x^{2}+y^{2} . \text { Find } W(2,-1), \quad W(-3,6) $$
View solution Problem 2
For the following exercises, evaluate each function at the indicated values. $$ W(x, y)=4 x^{2}+y^{2} . \text { Find } W(2+h, 3+h) $$
View solution Problem 4
An oxygen tank is constructed of a right cylinder of height \(y\) and radius \(x\) with two hemispheres of radius \(x\) mounted on the top and bottom of the cyl
View solution Problem 5
For the following exercises, find the domain of the function. $$ V(x, y)=4 x^{2}+y^{2} $$
View solution