/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 A nose cone for a space reentry ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 45

A nose cone for a space reentry vehicle is designed so that a cross section, taken \(x\) ft from the tip and perpendicular to the axis of symmetry, is a circle of radius \(\frac{1}{4} x^{2}\) ft. Find the volume of the nose cone given that its length is \(20 \mathrm{ft}\).

Short Answer

Expert verified
The volume of the nose cone is \(40000\pi\) cubic feet.

Step by step solution

01

Understand the Shape and Problem

The problem describes a solid of revolution where the cross-section is a circle whose radius is a function of its position along the length. This directly relates to a volume problem suitable for integration techniques, notably the disk method.
02

Identify the Function for Radius

The cross section taken at a distance of \(x\) feet from the tip of the nose cone has a radius given by the function \(r(x) = \frac{1}{4}x^2\). The volume of the solid can be calculated by integrating the area of these circular cross sections as \(x\) ranges from 0 to 20 feet, the length of the nose cone.
03

Determine the Area of Cross Section

The area of each circular cross-section is given by the formula \(A(x) = \pi (r(x))^2\). Substituting the radius function, we get \(A(x) = \pi \left(\frac{1}{4}x^2\right)^2 = \pi \frac{1}{16}x^4\).
04

Set Up the Integral for Volume

Using the disk method, the volume \(V\) of the nose cone is computed by integrating the area function from 0 to 20: \[ V = \int_{0}^{20} \pi \frac{1}{16}x^4 \, dx. \]
05

Simplify the Integral and Solve

Factor out the constants outside of the integral to simplify computing:\[ V = \frac{\pi}{16} \int_{0}^{20} x^4 \, dx. \]Now integrate \(x^4\) to get\[ V = \frac{\pi}{16} \left[ \frac{x^5}{5} \right]_{0}^{20}. \]
06

Evaluate the Integral Bounds

Plug in the bounds:\[ V = \frac{\pi}{16} \left( \frac{20^5}{5} - 0^5 \right) = \frac{\pi}{16} \times \frac{3200000}{5}. \]
07

Calculate the Final Volume

Compute the multiplication and division to find:\[ V = \frac{\pi}{16} \times 640000 = 40000\pi. \] Thus, the volume of the nose cone is \(40000\pi\) cubic feet.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Disk Method
The Disk Method is a technique used in calculus to calculate the volume of a solid of revolution. This means finding the 3D shape created by rotating a 2D shape around an axis. This is particularly useful when dealing with objects like the nose cone in our problem.
In this method:
  • The solid is visualized as a stack of disks or cylinders.
  • Each disk has a radius defined by the shape's boundary at that point.
  • We calculate the volume by summing up the volumes of these disks using integration.
This technique simplifies the process of finding volumes by translating a solid shape into a more manageable series of simpler shapes.
Integration
Integration is a cornerstone of calculus and is used to aggregate values over a range into one whole. In our volume calculation, integration compiles the areas of infinitely small disk slices to find the total volume of the object.
Key points about integration:
  • It involves finding the antiderivative or the area under a curve over a specified interval.
  • In the context of the disk method, the area function of each disk is integrated along the axis of rotation.
  • The result provides the total volume of the revolved body, summing up all disk volumes.
Mastering integration is crucial for solving problems involving solids of revolution, as it allows for the precise calculation of complex volumes.
Solid of Revolution
A Solid of Revolution is a three-dimensional solid formed by rotating a two-dimensional area around an axis. In our specific problem, a parabola described by the radius function is revolved around the axis to form a cone-like structure.
Things to remember:
  • The original shape is usually a function graph, area, or line.
  • The solid is defined by the path it takes around a specific line, usually the x-axis or y-axis.
  • The method used (e.g., disk or shell) will depend on how the solid is being rotated and what symmetries it has.
Understanding the notion of turning a 2D shape into a 3D solid through rotation is key to applying the disk method effectively.
Volume Calculation
Calculating the volume of a solid is an essential skill in many fields, including engineering and physics. In our nose cone problem, the volume calculation is done step-by-step using integration and the disk method.
Main steps involved:
  • Determine the function for the radius of each disk, which here was given as \( r(x) = \frac{1}{4}x^2 \).
  • Express the area of each disk, integrating along the length of the nose cone. This gives \( A(x) = \pi \frac{1}{16}x^4 \).
  • Setup and solve the integral to find the total volume, \( V = \int_{0}^{20} \pi \frac{1}{16}x^4 \, dx \).
  • Evaluate this integral to obtain the result as \( 40000\pi \) cubic feet.
Correct setup and evaluation of integrals are critical for accurate volume determination and reflect the precision demanded in real-world applications.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A 3 lb bucket containing 20 lb of water is hanging at the end of a \(20 \mathrm{ft}\) rope that weighs 4 oz/f. The other end of the rope is attached to a pulley. How much work is required to wind the length of rope onto the pulley, assuming that the rope is wound onto the pulley at a rate of \(2 \mathrm{ft} / \mathrm{s}\) and that as the bucket is being lifted, water leaks from the bucket at a rate of \(0.5 \mathrm{lb} / \mathrm{s} ?\)

Evaluate the integrals. $$\int_{0}^{\sqrt{3}} \frac{d t}{\sqrt{t^{2}+1}}$$

Find \(d y / d x\). $$y=\operatorname{csch}(1 / x)$$

The design of the Gateway Arch in St. Louis, Missouri, by architect Eero Saarinan was implemented using equations provided by Dr. Hannskarl Badel. The equation used for the centerline of the arch was $$y=693.8597-68.7672 \cosh (0.0100333 x) \mathrm{ft}$$ for \(x\) between -299.2239 and 299.2239 (a) Use a graphing utility to graph the centerline of the arch. (b) Find the length of the centerline to four decimal places. (c) For what values of \(x\) is the height of the arch \(100 \mathrm{ft} ?\) Round your answers to four decimal places. (d) Approximate, to the nearest degree, the acute angle that the tangent line to the centerline makes with the ground at the ends of the arch.

Find the mass and center of gravity of the lamina with density \(\delta .\) A lamina bounded by the \(x\) -axis, the line \(x=1,\) and the curve \(y=\sqrt{x} ; \delta=2\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.