Problem 30
Question
In \(29-34\) find the volume and draw a typical slice. A pyramid \(P\) has height 6 and square base of side 2 . Its volume is \(\frac{1}{3}(6)(2)^{2}=8\). (a) Find the volume up to height 3 by horizontal slices. What is the length of a side at height \(y ?\) (b) Recompute by removing a smaller pyramid from \(P\).
Step-by-Step Solution
Verified Answer
Volume by slices up to height 3 is 1; subtracting smaller pyramid's volume gives 7 overall.
1Step 1: Understand the problem
We are dealing with a pyramid that has a height of 6 and a square base with side length 2. We seek to find the volume up to a height of 3 using horizontal slices and by removing a smaller pyramid from the whole.
2Step 2: Determine the slice dimensions
First, consider the proportion of a side at height \(y\). The pyramid's side changes linearly, meaning at full height (6), the side is 2. Therefore, the side at height \(y\) can be expressed as \( s(y) = \frac{2}{6}y = \frac{y}{3} \).
3Step 3: Set up the integral for volume using slicing method
The area of a horizontal slice at height \(y\) is \(s(y)^2\). At height \(y\), this becomes \(\left(\frac{y}{3}\right)^2 = \frac{y^2}{9}\). The slice's thickness is \( dy \), leading to the volume element \( dV = \frac{y^2}{9} dy \).
4Step 4: Integrate to find total volume up to height 3
Integrate the volume element from \(y = 0\) to \( y = 3 \):\[ V = \int_{0}^{3} \frac{y^2}{9} \, dy = \frac{1}{9} \int_{0}^{3} y^2 \, dy. \] Compute the integral: \[ V = \frac{1}{9} \left[ \frac{y^3}{3} \right]_{0}^{3} = \frac{1}{9} \times \left( \frac{27}{3} \right) = 1. \]
5Step 5: Find volume by removing a smaller pyramid
The smaller pyramid subtracted has the same base size as at height 3 (i.e., height of 3 and side \( \frac{3}{3} = 1 \)), so its volume is \( \frac{1}{3}(3)(1)^2 = 1 \).
6Step 6: Calculate the total volume by subtraction
The main pyramid's total volume is 8. Removing the smaller pyramid's volume of 1 from this gives 8 - 1 = 7.
Key Concepts
volume by integrationhorizontal slicing methodgeometry of pyramidsmathematical problem solving
volume by integration
Calculating the volume of a pyramid using integration is a powerful mathematical approach. This method relies on the idea of summing infinitesimally small quantities to find the total volume. For a pyramid, we consider horizontal slices stacked from base to tip. Each slice has a volume, which we integrate across the height of the pyramid to get the total volume.
The formula for the volume of each horizontal slice involves its area and thickness. For the pyramid described, the area of the slice at height \( y \) is \( \left(\frac{y}{3}\right)^2 \). With an infinitesimally small thickness \( dy \), the volume element is \( dV = \frac{y^2}{9} \, dy \). By integrating this volume element from \( y = 0 \) to the desired height, you can find the pyramid's volume up to any given height.
The formula for the volume of each horizontal slice involves its area and thickness. For the pyramid described, the area of the slice at height \( y \) is \( \left(\frac{y}{3}\right)^2 \). With an infinitesimally small thickness \( dy \), the volume element is \( dV = \frac{y^2}{9} \, dy \). By integrating this volume element from \( y = 0 \) to the desired height, you can find the pyramid's volume up to any given height.
horizontal slicing method
The horizontal slicing method is an intuitive way to visualize and calculate the volume of three-dimensional shapes like pyramids. Imagine cutting the pyramid into many thin, flat layers, like slicing a cake horizontally. Each layer has a certain area and, when stacked, these layers approximate the shape of the pyramid.
To apply this method for the pyramid described, think of a layer at height \( y \). Its size shrinks linearly as it approaches the apex. Here, the side length \( s(y) \) at height \( y \) is \( \frac{y}{3} \). Thus, the square area's slice becomes \( \left(\frac{y}{3}\right)^2 \). Calculating the volume involves summing all these slice volumes (using integration). It gives a clear picture of how the pyramid fills up, piece by piece, up to the target height.
To apply this method for the pyramid described, think of a layer at height \( y \). Its size shrinks linearly as it approaches the apex. Here, the side length \( s(y) \) at height \( y \) is \( \frac{y}{3} \). Thus, the square area's slice becomes \( \left(\frac{y}{3}\right)^2 \). Calculating the volume involves summing all these slice volumes (using integration). It gives a clear picture of how the pyramid fills up, piece by piece, up to the target height.
geometry of pyramids
Understanding the geometry of pyramids helps in solving problems involving their volume. These structures have a base that is any polygon, and their sides taper uniformly to a single vertex, the apex. In this exercise, the pyramid has a square base, making calculations straightforward.
A key property in such problems is how the dimensions change proportionally from the base to the apex. For instance, the length of the side in a slice at height \( y \) can be found by proportion: \( s(y) = \frac{2}{6}y = \frac{y}{3} \). This linear relationship plays a crucial role in determining slice areas and, subsequently, the volume.
A key property in such problems is how the dimensions change proportionally from the base to the apex. For instance, the length of the side in a slice at height \( y \) can be found by proportion: \( s(y) = \frac{2}{6}y = \frac{y}{3} \). This linear relationship plays a crucial role in determining slice areas and, subsequently, the volume.
mathematical problem solving
Mathematical problem solving is a methodical approach that involves breaking down a problem into digestible parts. In the context of calculating pyramid volume, this means identifying known quantities, establishing relationships between variables, and applying mathematical principles like integration.
The given task tests these skills by asking for the volume of a partial pyramid both by direct calculation and by deductive reasoning (i.e., subtracting volumes). It's crucial to approach such problems with clarity:
The given task tests these skills by asking for the volume of a partial pyramid both by direct calculation and by deductive reasoning (i.e., subtracting volumes). It's crucial to approach such problems with clarity:
- Understand the geometry involved.
- Determine variable relationships proportionally.
- Set up the integration accurately for precise volume calculation.
- Consider alternative approaches, such as volume subtraction, for verification.
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