Problem 10
Question
A wedge is cut from a solid in the shape of a right-circular cone having a base radius of \(5 \mathrm{ft}\) and an altitude of \(20 \mathrm{ft}\) by two half planes through the axis of the cone. The angle between the two planes has a measurement of \(30^{\circ}\). Find the volume of the wedge cut out.
Step-by-Step Solution
Verified Answer
The volume of the wedge is \( \frac{125\pi}{9} \) cubic feet.
1Step 1 - Calculate the volume of the cone
Use the formula for the volume of a cone: \( V_{cone} = \frac{1}{3} \pi r^2 h \). Here, the radius \( r = 5 \) ft and the height \( h = 20 \) ft. Substitute these values to get: \( V_{cone} = \frac{1}{3} \pi (5^2)(20) \) \( V_{cone} = \frac{1}{3} \pi (25)(20) \) \( V_{cone} = \frac{1}{3} \pi (500) \) \( V_{cone} = \frac{500\pi}{3} \) cubic feet.
2Step 2 - Determine the fraction of the cone represented by the wedge
The wedge represents a portion of the cone based on the angle between the planes relative to the full circle (360 degrees). Since the angle is 30 degrees, the fraction of the cone is: \( \text{Fraction} = \frac{30^{\circ}}{360^{\circ}} = \frac{1}{12} \).
3Step 3 - Calculate the volume of the wedge
Multiply the volume of the cone by the fraction: \( V_{wedge} = V_{cone} \times \frac{1}{12} \). Using the previously computed volume of the cone: \( V_{wedge} = \left( \frac{500\pi}{3} \right) \times \frac{1}{12} \) \( V_{wedge} = \frac{500\pi}{36} \) \( V_{wedge} = \frac{125\pi}{9} \) cubic feet.
Key Concepts
Volume CalculationRight-Circular ConeGeometry
Volume Calculation
Understanding how to calculate volume is crucial in solving geometry problems, particularly for three-dimensional shapes like cones. Volume measures how much space an object takes up. When determining the volume of a cone, we use the formula: \( V = \frac{1}{3} \pi r^2 h \), where \( r \) is the radius of the base, and \( h \) is the height of the cone.
For instance, if you have a cone with a radius of 5 feet and a height of 20 feet, the volume can be computed as follows:
\( V_{cone} = \frac{1}{3} \pi (5^2)(20) \)
\( V_{cone} = \frac{1}{3} \pi (25)(20) \)
\( V_{cone} = \frac{1}{3} \pi (500) \)
\( V_{cone} = \frac{500\pi}{3} \) cubic feet.
This tells you the total volume of a solid cone before any section, like a wedge, is cut out.
For instance, if you have a cone with a radius of 5 feet and a height of 20 feet, the volume can be computed as follows:
\( V_{cone} = \frac{1}{3} \pi (5^2)(20) \)
\( V_{cone} = \frac{1}{3} \pi (25)(20) \)
\( V_{cone} = \frac{1}{3} \pi (500) \)
\( V_{cone} = \frac{500\pi}{3} \) cubic feet.
This tells you the total volume of a solid cone before any section, like a wedge, is cut out.
Right-Circular Cone
A right-circular cone is a three-dimensional geometric shape with a circular base and a pointed top, known as the apex, directly above the center of the base. The 'right' indicates that the line from the apex to the base's center is perpendicular to the base.
In this problem, we cut a wedge from the right-circular cone. The base radius of the cone is 5 feet, and the altitude (or height) from the base to the apex is 20 feet.
Knowing these dimensions helps accurately compute the entire volume of the cone. Here's the formula to keep in mind: \( V = \frac{1}{3} \pi r^2 h \). Thus, substituting 5 for the radius and 20 for the height, we determine the volume of the right-circular cone as previously calculated.
By understanding this concept, you can tackle more complex problems involving different parts of a cone.
In this problem, we cut a wedge from the right-circular cone. The base radius of the cone is 5 feet, and the altitude (or height) from the base to the apex is 20 feet.
Knowing these dimensions helps accurately compute the entire volume of the cone. Here's the formula to keep in mind: \( V = \frac{1}{3} \pi r^2 h \). Thus, substituting 5 for the radius and 20 for the height, we determine the volume of the right-circular cone as previously calculated.
By understanding this concept, you can tackle more complex problems involving different parts of a cone.
Geometry
Geometry is the branch of mathematics that studies shapes, sizes, and properties of space. It is fundamental to solving problems related to figures like cones, spheres, and cubes.
In this exercise, geometry helps us understand how to slice a cone to form a wedge. A wedge is a fraction of a cone, determined by the angle between two planes that intersect through the cone’s apex.
Here, the angle between the planes is \( 30^\circ \). To find the fraction of the original cone, it's essential to note that a full circle measures \( 360^\circ \). Hence, the wedge is \( \frac{30^\circ}{360^\circ} = \frac{1}{12} \) of the original cone.
Multiplying this fraction by the total cone volume gives the wedge's volume:
\( V_{wedge} = V_{cone} \times \frac{1}{12} \)
Using our calculated cone volume:
\( V_{wedge} = \left( \frac{500\pi}{3} \right) \times \frac{1}{12} \)
\( V_{wedge} = \frac{500\pi}{36} \)
\( V_{wedge} = \frac{125\pi}{9} \) cubic feet.
Geometry allows us to break down complex shapes into simpler parts and understand their properties.
In this exercise, geometry helps us understand how to slice a cone to form a wedge. A wedge is a fraction of a cone, determined by the angle between two planes that intersect through the cone’s apex.
Here, the angle between the planes is \( 30^\circ \). To find the fraction of the original cone, it's essential to note that a full circle measures \( 360^\circ \). Hence, the wedge is \( \frac{30^\circ}{360^\circ} = \frac{1}{12} \) of the original cone.
Multiplying this fraction by the total cone volume gives the wedge's volume:
\( V_{wedge} = V_{cone} \times \frac{1}{12} \)
Using our calculated cone volume:
\( V_{wedge} = \left( \frac{500\pi}{3} \right) \times \frac{1}{12} \)
\( V_{wedge} = \frac{500\pi}{36} \)
\( V_{wedge} = \frac{125\pi}{9} \) cubic feet.
Geometry allows us to break down complex shapes into simpler parts and understand their properties.
Other exercises in this chapter
Problem 10
The face of a gate of a dam is vertical and in the shape of an isosceles trapezoid \(3 \mathrm{ft}\) wide at the top, \(4 \mathrm{ft}\) wide at the bottom, and
View solution Problem 10
A bucket weighing \(20 \mathrm{lb}\) containing \(60 \mathrm{lb}\) of sand is attached to the lower end of a \(100 \mathrm{ft}\) long chain that weighs \(10 \ma
View solution Problem 10
Find by integration the volume of a right-circular cone of altitude \(h\) units and base radius \(a\) units.
View solution Problem 11
The length of a rod is \(L \mathrm{ft}\) and the center of mass of the rod is at the point \(\frac{3}{4} L \mathrm{ft}\) from the left end. If the measure of th
View solution