Q. 25 Use four slices to construct an ... [FREE SOLUTION]

91影视

Chapter 6: Q. 25 (page 556)

Use four slices to construct an approximation of each of the
quantities described in Exercises 25鈥�30. In each case include
a labeled diagram and an explicit list of the values of $鈭� x$ and
each $x_{k}$ and ${x_{k}}^{}$ (or $鈭� y$ and each $y_{k}$ and ${y_{k}}^{}$ ).
The mass of a cylindrical rod with radius of 2 centimeters and length of 24 centimeters whose density at a point x centimeters from the left end is 蚁(x) = 10.5 + 0.01527 $x^{2}$ grams per cubic centimeter.

Short Answer

Expert verified

$k_{y} = 2.35 f t k_{x} = 1.8 f t$

Step by step solution

Step1. Given Information

radius=2cm

length=24cm

density at left end = $蚁 (x) = 10.5 + 0.01527 x^{2} \frac{g m}{c m^{3}}$

Step2. Calculate

$d m = 蚁 d v = 蚁 (蟺 x^{2}) d y d m = 蚁蟺 \frac{1}{y^{2}} d y d I_{y} = \frac{1}{2} d m x^{2}$

Step3. Continue

$d I_{y} = \frac{1}{2} (蚁蟺 \frac{1}{y^{2}} d y) (\frac{1}{y^{2}}) = \frac{1}{2} \frac{蚁蟺 d y}{y^{4}} 鈭� d I_{y} = \frac{1}{2} 蚁蟺 {鈭�}_{0.25}^{4} \frac{1}{y^{4}} d y = I_{y} = 134 + 蚁$

Step4. Continue

$c = \frac{1}{2} [蚁蟺 (4^{2}) (0.25)] (4^{2}) = 32 蟺蚁鈭� d m = m = 蚁蟺 {鈭�}_{0.25}^{4} \frac{1}{y^{2}} d y 鈬� m = 24.34 蚁$

Step5. Continue

$k_{x} = \sqrt{\frac{I_{x}}{m}}, k_{y} = \sqrt{\frac{I_{y}}{m}}$

Step6. Continue

$k_{y} = \sqrt{\frac{134 蚁}{94.34 蚁}} 鈬� k_{y} = 2.35 f t .$

Step7. Continue

$d {I^{'}}_{x} = \frac{1}{4} d m x^{2} + d m y^{2} = \frac{1}{4} (蚁蟺 \frac{d y}{y}) \frac{1}{y^{2}} + (蚁蟺 \frac{d y}{y}) y^{2} = 蚁蟺 (\frac{1}{4 y^{4}} + 1) d y 鈭� d {I^{'}}_{x} = {鈭�}_{0.25}^{4} 蚁蟺 (\frac{1}{4 y^{2}} + 1) d y$

Step8. Continue

${I^{'}}_{x} = (\frac{- 1}{12 y^{3}} + y) {|_{0.25}}^{4} 鈬� {I^{'}}_{X} = 28.23 y$

Step9. Continue

${I^{''}}_{x} = \frac{1}{4} [蚁蟺 (4^{2}) (0.25)] (4^{2}) + [蚁蟺 (4^{2}) (0.25)] (0 . 125^{2}) {I^{"}}_{x} = 50.46 蚁$

Step10. Continue

$I_{x} = {I^{'}}_{x} + {I^{"}}_{x} = 28.53 蚁 + 50.46 蚁 I_{X} = 78.99 蚁 k_{x} = \sqrt{\frac{78.99 蚁}{24.37 蚁}} 鈬� k_{x} = 1.8 f t .$

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!

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Step1. Given Information

Step2. Calculate

Step3. Continue

Step4. Continue

Step5. Continue

Step6. Continue

Step7. Continue

Step8. Continue

Step9. Continue

Step10. Continue

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Pure Maths

Calculus

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.