Problem 2
Question
A Wisconsin cheese factory makes its cheese in solid cylinders of radius 2 inches. A wedge of cheese is cut from the cylinder by chopping through the diameter of the base at an angle of 45 degrees with the base. Find the volume of the wedge of cheese.
Step-by-Step Solution
Verified Answer
The volume of the wedge of cheese is \( 2\pi \) cubic inches.
1Step 1: Calculate the area of the complete circular base
The area (A) of a circle is calculated with the formula \( A = \pi r^2 \), where r is the radius. Given a radius of 2 inches, the area of the circle is \( A = \pi (2)^2 = 4\pi \) square inches.
2Step 2: Find the area of the sector
The sector area of a circle is calculated as a proportion of the full circle area using the formula \( S = A \times \frac{\theta}{360} \), where \(\theta\) is the sector angle. For a 45-degree sector, the area (S) is therefore \( S = 4\pi \times \frac{45}{360} = \frac{\pi}{2} \) square inches.
3Step 3: Calculate the volume of the cylinder representing the wedge of cheese
After finding the sector area, it can be extended into the third dimension to find the volume of the cylinder. The formula for the volume of a cylinder is \( V = S \times h \), where h is the height. As the height of the cylinder equals the diameter of the base (twice the radius), the volume becomes \( V = \frac{\pi}{2} \times 4 = 2\pi \) cubic inches.
Key Concepts
Circular Sector AreaGeometry in CalculusVolume of a Cylinder
Circular Sector Area
Understanding the area of a circular sector is crucial for solving a variety of geometry problems. A circular sector is a pie-shaped part of a circle, bounded by two radii and the arc between them. Think of it as a slice of pizza, with the tip at the circle's center and the crust along the circle's edge.
To calculate the area of a circular sector, we need two things: the circle's radius and the angle at the center of the sector, known as the central angle. The area of the full circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius. To find the area of just the sector, we use a fraction of the full area, proportional to the central angle as compared to a full 360 degrees.
So the formula for the sector's area becomes \( S = A \times \frac{\theta}{360} \) where \( \theta \) is the central angle in degrees. Imagine cutting a cake into equal slices—each slice's area corresponds to the slice's central angle as a fraction of the entire cake's area. This concept is pivotal when calculating the volume of objects with circular bases, like our wedge of cheese.
To calculate the area of a circular sector, we need two things: the circle's radius and the angle at the center of the sector, known as the central angle. The area of the full circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius. To find the area of just the sector, we use a fraction of the full area, proportional to the central angle as compared to a full 360 degrees.
So the formula for the sector's area becomes \( S = A \times \frac{\theta}{360} \) where \( \theta \) is the central angle in degrees. Imagine cutting a cake into equal slices—each slice's area corresponds to the slice's central angle as a fraction of the entire cake's area. This concept is pivotal when calculating the volume of objects with circular bases, like our wedge of cheese.
Geometry in Calculus
Geometry and calculus often intersect, particularly when dealing with volumes and areas. In calculus, we sometimes use the method of integration to calculate areas and volumes of shapes that are not regular. When dealing with a circular sector, we can actually simplify the problem by using basic geometry, provided the shapes are regular and not deformed.
In our cheese wedge problem, we've leveraged this intersection by using the geometry of the circular sector to inform the calculus of volume. To add depth to the concept, let's consider if the wedge was not a straight cut but rather curved or irregular. In that case, integrating along the length of the cheese would give us a more accurate volume, but for regular shapes like cylinders, simple geometry will suffice.
This approach simplifies complex problems and allows us to use the principles of geometry within the realm of calculus to find practical solutions to real-world problems like slicing cheese in a factory.
In our cheese wedge problem, we've leveraged this intersection by using the geometry of the circular sector to inform the calculus of volume. To add depth to the concept, let's consider if the wedge was not a straight cut but rather curved or irregular. In that case, integrating along the length of the cheese would give us a more accurate volume, but for regular shapes like cylinders, simple geometry will suffice.
This approach simplifies complex problems and allows us to use the principles of geometry within the realm of calculus to find practical solutions to real-world problems like slicing cheese in a factory.
Volume of a Cylinder
The volume of a cylinder can be imagined as how much space an object occupies or how much cheese you're getting in your wedge! To find it, we utilize the area of one of the circular bases and multiply it by the height (or length) of the cylinder. The formula is \( V = \text{base area} \times \text{height} \), where the base area is the area of the circle (since the base of a cylinder is a circle) and the height is the distance between the two bases.
In simple terms, if you stack a bunch of circular pieces of paper precisely on top of one another until it reaches a certain height, the volume of the stack would be the volume of a cylinder. This is akin to stacking our circular sector cheese slices to the height of the cheese cylinder. In our cheese example, the volume of the entire wedge is found by multiplying the sector area by the height, which is equal to the cylinder's diameter since the wedge is cut right through the middle.
The resulting volume tells us exactly how much cheese is in the wedge. This core concept is essential when calculating the volume of all cylindrical objects, whether it's cheese, a roll of paper, or a container filled with liquid.
In simple terms, if you stack a bunch of circular pieces of paper precisely on top of one another until it reaches a certain height, the volume of the stack would be the volume of a cylinder. This is akin to stacking our circular sector cheese slices to the height of the cheese cylinder. In our cheese example, the volume of the entire wedge is found by multiplying the sector area by the height, which is equal to the cylinder's diameter since the wedge is cut right through the middle.
The resulting volume tells us exactly how much cheese is in the wedge. This core concept is essential when calculating the volume of all cylindrical objects, whether it's cheese, a roll of paper, or a container filled with liquid.
Other exercises in this chapter
Problem 1
A tent has a base that is an isosceles triangle. The mouth of the tent measures 8 feet and the length is 12 feet. The tent is constructed so that the cross sect
View solution Problem 2
Approximate each length with error less than 0.05. The length of one arc of the cosine curve, say from \(x=\frac{-\pi}{2}\) to \(x=\frac{\pi}{2}\).
View solution Problem 3
Find the volume generated when the region in the first quadrant bounded by \(y=x^{2}\) and \(y=3 x\) is rotated about (a) the \(x\) -axis, (b) the \(y\) -axis,
View solution Problem 4
Find the volume generated when the region bounded by the \(y\) -axis, \(y=x^{2}\), and \(y=4\) is rotated about the \(x\) -axis. Do this in three ways. (a) Chop
View solution