Calculus (MindTap Course List)
Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
Chapter 12.P, Problem 1P

Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same radius r. The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the box. Thus the balls are tightly packed in the box (see the figure). Find r. (If you have trouble with this problem, read about the problem-solving strategy entitled Use Analogy on page 98.)

Chapter 12.P, Problem 1P, Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same

Expert Solution & Answer
Check Mark
To determine

To find:

The radius r

Answer to Problem 1P

Solution:

The radius of each ball is

r =3-32m

Explanation of Solution

1) Concept:

We create an analogous problem in two dimensions and use the strategy in two dimension to solve the problem in three dimension.

2) Calculation:

Since three dimensional situations are often diffucult to visualize and work with, we create an analogous problem in two dimensions and use the strategy in two dimension to solve the problem in three dimension.

Let us try to find the analogous problem in two dimensions. The analogue of a cube is a square, and the analogue of a sphere is a circle.Thus, a simillar problem in two dimensions is as follows:

What is r, if five circles with the same radius r are contained in a square of side 1m so that the circles touch each other and four of the circles touch two sides of square, as shown in the figure,

Calculus (MindTap Course List), Chapter 12.P, Problem 1P , additional homework tip  1

By pythagorean Theorem, the diagonal of the square is 2

Also from the figure, the diagonal of the square is 4r+2x

Therefore, 4r+2x=2

But, x is the diagonal of the smallar square of side r.

Therefore,

 x=2 r

Therefore,

4r+2x=2 implies that

4r+22 r =2

That is, 4+22 r =2

Therefore,

 r =24+22

Let’s use this idea to solve the original three dimensional problem.

Calculus (MindTap Course List), Chapter 12.P, Problem 1P , additional homework tip  2

The diagonal of the cube is 12+12+12=3

And the diagonal of the cube is also 4r+2x.

Therefore, 4r+2x=3

 where x is the diagonal of the cube with edge r.

Therefore,

x=r2+r2+r2=3 r

Therefore,

4r+2x=3 implies that  4r+23 r=3

That is, 4+2 3 r =3

Therefore,

 r =34+23

By rationalization,

r =23-32

Therefore, the radius of each ball is

r =3-32m

Conclusion:

The radius of each ball is

r =3-32m

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Chapter 12 Solutions

Calculus (MindTap Course List)

Ch. 12.1 - Determine whether the points lie on a straight...Ch. 12.1 - Prob. 12ECh. 12.1 - Find an equation of the sphere with centre (3,2,5)...Ch. 12.1 - Find an equation of the sphere with centre (2,6,4)...Ch. 12.1 - Find an equation of the sphere that passes through...Ch. 12.1 - Prob. 16ECh. 12.1 - Prob. 17ECh. 12.1 - Prob. 18ECh. 12.1 - Prob. 19ECh. 12.1 - Prob. 20ECh. 12.1 - a Prove that the midpoint of the line segment from...Ch. 12.1 - Find an equation of a sphere if one of its...Ch. 12.1 - Prob. 23ECh. 12.1 - Find an equation of the largest sphere with centre...Ch. 12.1 - Prob. 25ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 27ECh. 12.1 - Prob. 28ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 30ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 33ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 36ECh. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Prob. 38ECh. 12.1 - Prob. 39ECh. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - Write inequalities to describe the region. The...Ch. 12.1 - The figure shows a line L1 in space and a second...Ch. 12.1 - Consider the points P such that the distance from...Ch. 12.1 - Prob. 45ECh. 12.1 - Find the volume of the solid that lies inside both...Ch. 12.1 - Find the distance between the spheres x2+y2+z2=4...Ch. 12.1 - Describe and sketch a solid with the following...Ch. 12.2 - Are the following quantities vectors or scalars?...Ch. 12.2 - What is the relationship between the point 4, 7...Ch. 12.2 - Name all the equal vectors in the parallelogram...Ch. 12.2 - Prob. 4ECh. 12.2 - Prob. 5ECh. 12.2 - Prob. 6ECh. 12.2 - In the figure, the tip of c and the tail of d are...Ch. 12.2 - If the vectors in the figure satisfy |u|=|v|=1 and...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Prob. 10ECh. 12.2 - Prob. 11ECh. 12.2 - Prob. 12ECh. 12.2 - Prob. 13ECh. 12.2 - Prob. 14ECh. 12.2 - Prob. 15ECh. 12.2 - Prob. 16ECh. 12.2 - Prob. 17ECh. 12.2 - Prob. 18ECh. 12.2 - Prob. 19ECh. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=5i+3j,b=i2jCh. 12.2 - Prob. 21ECh. 12.2 - Find a+b,4a+2b,|a|, and |ab|. a=8,1,4,b=5,2,1Ch. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - Prob. 25ECh. 12.2 - Prob. 26ECh. 12.2 - What is the angle between the given vector and the...Ch. 12.2 - Prob. 28ECh. 12.2 - If v lies in the first quadrant and makes an angle...Ch. 12.2 - Prob. 30ECh. 12.2 - Prob. 31ECh. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - Prob. 34ECh. 12.2 - Prob. 35ECh. 12.2 - Prob. 36ECh. 12.2 - A block-and-tackle pulley hoist is suspended in a...Ch. 12.2 - The tension T at each end of a chain has magnitude...Ch. 12.2 - A boatman wants to cross a canal that is 3 km wide...Ch. 12.2 - Three forces act on an object. Two of the forces...Ch. 12.2 - Prob. 41ECh. 12.2 - Prob. 42ECh. 12.2 - If A, B, and C are the vertices of a triangle,...Ch. 12.2 - Prob. 44ECh. 12.2 - a Draw the vectors a=3,2,b=2,1 and c=7,1. b Show,...Ch. 12.2 - Prob. 46ECh. 12.2 - Prob. 47ECh. 12.2 - Prob. 48ECh. 12.2 - Prob. 49ECh. 12.2 - Prove Property 5 of vectors algebraically for the...Ch. 12.2 - Use vectors to prove that the line joining the...Ch. 12.2 - Prob. 52ECh. 12.3 - Which of the following expressions are meaningful?...Ch. 12.3 - Prob. 2ECh. 12.3 - 210 Find ab. a=1.5,0.4, b=4,6Ch. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.3 - Prob. 7ECh. 12.3 - Prob. 8ECh. 12.3 - Prob. 9ECh. 12.3 - Prob. 10ECh. 12.3 - If u is a unit vector, find uv and uw.Ch. 12.3 - If u is a unit vector, find uv and uw.Ch. 12.3 - Prob. 13ECh. 12.3 - Prob. 14ECh. 12.3 - Prob. 15ECh. 12.3 - Find the angle between the vectors. First find an...Ch. 12.3 - Prob. 17ECh. 12.3 - Prob. 18ECh. 12.3 - Prob. 19ECh. 12.3 - Prob. 20ECh. 12.3 - 2122 Find, correct to the nearest degree, the...Ch. 12.3 - Prob. 22ECh. 12.3 - Prob. 23ECh. 12.3 - 2324 Determine whether the given vectors are...Ch. 12.3 - Prob. 25ECh. 12.3 - Prob. 26ECh. 12.3 - Prob. 27ECh. 12.3 - Prob. 28ECh. 12.3 - Prob. 29ECh. 12.3 - Prob. 30ECh. 12.3 - 3132 Find the acute angles between the curves at...Ch. 12.3 - 3132 Find the acute angles between the curves at...Ch. 12.3 - Prob. 33ECh. 12.3 - Prob. 34ECh. 12.3 - Prob. 35ECh. 12.3 - Prob. 36ECh. 12.3 - Prob. 37ECh. 12.3 - Prob. 38ECh. 12.3 - 3944 Find the scalar and vector projections of b...Ch. 12.3 - Prob. 40ECh. 12.3 - Prob. 41ECh. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Prob. 44ECh. 12.3 - Prob. 45ECh. 12.3 - Prob. 46ECh. 12.3 - If a=3,0,1, find a vector b such that compab=2.Ch. 12.3 - Prob. 48ECh. 12.3 - Prob. 49ECh. 12.3 - Prob. 50ECh. 12.3 - Prob. 51ECh. 12.3 - Prob. 52ECh. 12.3 - Use a scalar projection to show that the distance...Ch. 12.3 - If r=x,y,z,a=a1,a2,a3, and b=b1,b2,b3, show that...Ch. 12.3 - Prob. 55ECh. 12.3 - Prob. 56ECh. 12.3 - Prob. 57ECh. 12.3 - If c=|a|b+|b|a, where a, b, and c are all nonzero...Ch. 12.3 - Prob. 59ECh. 12.3 - Prob. 60ECh. 12.3 - Use Theorem 3 to prove the Cauchy-Schwarz...Ch. 12.3 - Prob. 62ECh. 12.3 - The Parallelogram Law states that...Ch. 12.3 - Prob. 64ECh. 12.3 - Prob. 65ECh. 12.4 - Prob. 1ECh. 12.4 - Prob. 2ECh. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - 17 Find the cross product ab and verify that it is...Ch. 12.4 - Prob. 5ECh. 12.4 - Prob. 6ECh. 12.4 - Prob. 7ECh. 12.4 - If a=i2k and b=j+k, find ab. Sketch a, b, and ab....Ch. 12.4 - Prob. 9ECh. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - 912 Find the vector, not with determinants, but by...Ch. 12.4 - Prob. 13ECh. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - Prob. 17ECh. 12.4 - If a=1,0,1,b=2,1,1, and c=0,1,3, show that...Ch. 12.4 - Find two unit vectors orthogonal to both 3,2,1 and...Ch. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - Prob. 25ECh. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Find the area of the parallelogram with vertices...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - 2932 a Find a nonzero vector orthogonal to the...Ch. 12.4 - Prob. 32ECh. 12.4 - Prob. 33ECh. 12.4 - Prob. 34ECh. 12.4 - Prob. 35ECh. 12.4 - Prob. 36ECh. 12.4 - Prob. 37ECh. 12.4 - Prob. 38ECh. 12.4 - Prob. 39ECh. 12.4 - Prob. 40ECh. 12.4 - A wrench 30 cm long lies along the positive y-axis...Ch. 12.4 - Prob. 42ECh. 12.4 - Prob. 43ECh. 12.4 - a Find all vectors v such that 1,2,1v=3,1,5 b...Ch. 12.4 - Prob. 45ECh. 12.4 - a Let P be a point not on the plane that passes...Ch. 12.4 - Prob. 47ECh. 12.4 - Prob. 48ECh. 12.4 - Prob. 49ECh. 12.4 - Prob. 50ECh. 12.4 - Prob. 51ECh. 12.4 - Prob. 52ECh. 12.4 - Suppose that a0. a If ab=ac, does it follow that...Ch. 12.4 - If v1, v2, and v3 are noncoplanar vectors, let...Ch. 12.5 - Determine whether each statement is true or false...Ch. 12.5 - Prob. 2ECh. 12.5 - Prob. 3ECh. 12.5 - Prob. 4ECh. 12.5 - Prob. 5ECh. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - Prob. 7ECh. 12.5 - Prob. 8ECh. 12.5 - Prob. 9ECh. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - 612 Find parametric equations and symmetric...Ch. 12.5 - Prob. 13ECh. 12.5 - Is the line through (2,4,0) and (1,1,1)...Ch. 12.5 - a Find symmetric equations for the line that...Ch. 12.5 - a Find parametric equations for the line through...Ch. 12.5 - Prob. 17ECh. 12.5 - Prob. 18ECh. 12.5 - 1922 Determine whether the lines L1 and L2 are...Ch. 12.5 - Prob. 20ECh. 12.5 - Prob. 21ECh. 12.5 - Prob. 22ECh. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - 2340 Find an equation of the plane. The plane that...Ch. 12.5 - Prob. 40ECh. 12.5 - Prob. 41ECh. 12.5 - Prob. 42ECh. 12.5 - Prob. 43ECh. 12.5 - Prob. 44ECh. 12.5 - 4547 Find the point at which the line intersects...Ch. 12.5 - Prob. 46ECh. 12.5 - Prob. 47ECh. 12.5 - Prob. 48ECh. 12.5 - Find direction numbers for the line of...Ch. 12.5 - Prob. 50ECh. 12.5 - Prob. 51ECh. 12.5 - Prob. 52ECh. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - 5156 Determine whether the planes are parallel,...Ch. 12.5 - Prob. 57ECh. 12.5 - Prob. 58ECh. 12.5 - Prob. 59ECh. 12.5 - 5960 Find symmetric equations for the line of...Ch. 12.5 - Prob. 61ECh. 12.5 - Prob. 62ECh. 12.5 - Prob. 63ECh. 12.5 - Prob. 64ECh. 12.5 - Prob. 65ECh. 12.5 - Prob. 66ECh. 12.5 - Prob. 67ECh. 12.5 - Prob. 68ECh. 12.5 - Prob. 69ECh. 12.5 - Prob. 70ECh. 12.5 - 7172 Find the distance from the point to the given...Ch. 12.5 - Prob. 72ECh. 12.5 - Prob. 73ECh. 12.5 - Prob. 74ECh. 12.5 - Prob. 75ECh. 12.5 - Prob. 76ECh. 12.5 - Show that the lines with symmetric equations x=y=z...Ch. 12.5 - Prob. 78ECh. 12.5 - Prob. 79ECh. 12.5 - Prob. 80ECh. 12.5 - Prob. 81ECh. 12.5 - Prob. 82ECh. 12.5 - If a, b, and c are not all 0, show that the...Ch. 12.6 - a What does the equation y=x2 represent as a curve...Ch. 12.6 - a Sketch the graph of y=ex as a curve in 2. b...Ch. 12.6 - 38 Describe and sketch the surface. x2+z2=1Ch. 12.6 - 38 Describe and sketch the surface. 4x2+y2=4Ch. 12.6 - 38 Describe and sketch the surface. z=1y2Ch. 12.6 - 38 Describe and sketch the surface. y=z2Ch. 12.6 - 38 Describe and sketch the surface. xy=1Ch. 12.6 - 38 Describe and sketch the surface. z=sinyCh. 12.6 - a Find and identify the traces of the quadric...Ch. 12.6 - a Find and identify the traces of the quadric...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 1120 Use traces to sketch and identify the...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - Prob. 24ECh. 12.6 - Prob. 25ECh. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - 2128 Match the equation with reasons for your...Ch. 12.6 - Prob. 28ECh. 12.6 - 2930 Sketch and identify a quadric surface that...Ch. 12.6 - Prob. 30ECh. 12.6 - Prob. 31ECh. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - Prob. 33ECh. 12.6 - Prob. 34ECh. 12.6 - Prob. 35ECh. 12.6 - 3138 Reduce the equation to one of the standard...Ch. 12.6 - Prob. 37ECh. 12.6 - Prob. 38ECh. 12.6 - Prob. 39ECh. 12.6 - Prob. 40ECh. 12.6 - Prob. 41ECh. 12.6 - Prob. 42ECh. 12.6 - Prob. 43ECh. 12.6 - Prob. 44ECh. 12.6 - Find an equation for the surface obtained by...Ch. 12.6 - Prob. 46ECh. 12.6 - Prob. 47ECh. 12.6 - Find an equation for the surface consisting of all...Ch. 12.6 - Traditionally, the earths surface has been modeled...Ch. 12.6 - A cooling tower for a nuclear reactor is to be...Ch. 12.6 - Show that if the point (a,b,c) lies on the...Ch. 12.6 - Show that the curve of intersection of the...Ch. 12.6 - Prob. 53ECh. 12.R - What is the difference between a vector and a...Ch. 12.R - Prob. 2CCCh. 12.R - Prob. 3CCCh. 12.R - Prob. 4CCCh. 12.R - Prob. 5CCCh. 12.R - Prob. 6CCCh. 12.R - Prob. 7CCCh. 12.R - Prob. 8CCCh. 12.R - Prob. 9CCCh. 12.R - Prob. 10CCCh. 12.R - Prob. 11CCCh. 12.R - Prob. 12CCCh. 12.R - Prob. 13CCCh. 12.R - Prob. 14CCCh. 12.R - Prob. 15CCCh. 12.R - Prob. 16CCCh. 12.R - Prob. 17CCCh. 12.R - What are the traces of a surface? How do you find...Ch. 12.R - Prob. 19CCCh. 12.R - Prob. 1TFQCh. 12.R - Prob. 2TFQCh. 12.R - Prob. 3TFQCh. 12.R - Prob. 4TFQCh. 12.R - Prob. 5TFQCh. 12.R - Prob. 6TFQCh. 12.R - Prob. 7TFQCh. 12.R - Prob. 8TFQCh. 12.R - Prob. 9TFQCh. 12.R - Prob. 10TFQCh. 12.R - Prob. 11TFQCh. 12.R - Prob. 12TFQCh. 12.R - Prob. 13TFQCh. 12.R - Prob. 14TFQCh. 12.R - Prob. 15TFQCh. 12.R - Prob. 16TFQCh. 12.R - Prob. 17TFQCh. 12.R - Prob. 18TFQCh. 12.R - Prob. 19TFQCh. 12.R - Prob. 20TFQCh. 12.R - Prob. 21TFQCh. 12.R - Prob. 22TFQCh. 12.R - Prob. 1ECh. 12.R - Prob. 2ECh. 12.R - Prob. 3ECh. 12.R - Calculate the given quantity if...Ch. 12.R - Prob. 5ECh. 12.R - Prob. 6ECh. 12.R - Prob. 7ECh. 12.R - Prob. 8ECh. 12.R - Prob. 9ECh. 12.R - Prob. 10ECh. 12.R - Prob. 11ECh. 12.R - Prob. 12ECh. 12.R - Prob. 13ECh. 12.R - Prob. 14ECh. 12.R - Prob. 15ECh. 12.R - Prob. 16ECh. 12.R - Prob. 17ECh. 12.R - Prob. 18ECh. 12.R - Prob. 19ECh. 12.R - Prob. 20ECh. 12.R - Find the point in which the line with parametric...Ch. 12.R - Prob. 22ECh. 12.R - Prob. 23ECh. 12.R - a Show that the planes x+yz=1 and 2x3y+4z=5 are...Ch. 12.R - Prob. 25ECh. 12.R - a Find an equation of the plane that passes...Ch. 12.R - Prob. 27ECh. 12.R - Prob. 28ECh. 12.R - Prob. 29ECh. 12.R - Prob. 30ECh. 12.R - Prob. 31ECh. 12.R - Prob. 32ECh. 12.R - Prob. 33ECh. 12.R - Prob. 34ECh. 12.R - Prob. 35ECh. 12.R - Prob. 36ECh. 12.R - Prob. 37ECh. 12.R - Prob. 38ECh. 12.P - Each edge of a cubical box has length 1 m. The box...Ch. 12.P - Let B be a solid box with length L, width W, and...Ch. 12.P - Let L be the line of intersection of the planes...Ch. 12.P - A plane is capable of flying at a speed of 180...Ch. 12.P - Prob. 5PCh. 12.P - Find an equation of the largest sphere that passes...Ch. 12.P - Prob. 7PCh. 12.P - A solid has the following properties. When...
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