# 5.7E: Excersies

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## Exercise $$\PageIndex{1}$$

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. For the following exercises, the cylindrical coordinates $$\displaystyle (r,θ,z)$$ of a point are given. Find the rectangular coordinates $$\displaystyle (x,y,z)$$ of the point.

1) $$\displaystyle (4,\frac{π}{6},3)$$

$$\displaystyle (2\sqrt{3},2,3)$$

2) $$\displaystyle (3,\frac{π}{3},5)$$

3) $$\displaystyle (4,\frac{7π}{6},3)$$

$$\displaystyle −2\sqrt{3},−2,3)$$

4) $$\displaystyle (2,π,−4)$$

For the following exercises, the rectangular coordinates $$\displaystyle (x,y,z)$$ of a point are given. Find the cylindrical coordinates $$\displaystyle (r,θ,z)$$of the point.

5) $$\displaystyle (1,\sqrt{3},2)$$

$$\displaystyle (2,\frac{π}{3},2)$$

6) $$\displaystyle (1,1,5)$$

7) $$\displaystyle (3,−3,7)$$

$$\displaystyle (3\sqrt{2},−\frac{π}{4},7)$$

8) $$\displaystyle (−2\sqrt{2},2\sqrt{2},4)$$

## Exercise $$\PageIndex{2}$$

For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.

9) [T] $$\displaystyle r=4$$

A cylinder of equation $$\displaystyle x^2+y^2=16,$$ with its center at the origin and rulings parallel to the z-axis, 10) [T] $$\displaystyle z=r^2cos^2θ$$

11) [T] $$\displaystyle r^2cos(2θ)+z^2+1=0$$

Hyperboloid of two sheets of equation $$\displaystyle −x^2+y^2−z^2=1,$$ with the y-axis as the axis of symmetry, 12) [T] $$\displaystyle r=3sinθ$$

13) [T] $$\displaystyle r=2cosθ$$

Cylinder of equation $$\displaystyle x^2−2x+y^2=0,$$ with a center at $$\displaystyle (1,0,0)$$ and radius $$\displaystyle 1$$, with rulings parallel to the z-axis, 14) [T] $$\displaystyle r^2+z^2=5$$

15) [T] $$\displaystyle r=2secθ$$

Plane of equation $$\displaystyle x=2,$$ 16) [T] $$\displaystyle r=3cscθ$$

## Exercise $$\PageIndex{3}$$

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates.

17) $$\displaystyle z=3$$

$$\displaystyle z=3$$

18) $$\displaystyle x=6$$

19) $$\displaystyle x^2+y^2+z^2=9$$

$$\displaystyle r^2+z^2=9$$

20) $$\displaystyle y=2x^2$$

21) $$\displaystyle x^2+y^2−16x=0$$

$$\displaystyle r=16cosθ,r=0$$

22) $$\displaystyle x^2+y^2−3\sqrt{x^2+y^2}+2=0$$

## Exercise $$\PageIndex{4}$$

For the following exercises, the spherical coordinates $$\displaystyle (ρ,θ,φ)$$ of a point are given. Find the rectangular coordinates $$\displaystyle (x,y,z)$$ of the point.

23) $$\displaystyle (3,0,π)$$

$$\displaystyle (0,0,−3)$$

24) $$\displaystyle (1,\frac{π}{6},\frac{π}{6})$$

25) $$\displaystyle (12,−\frac{π}{4},\frac{π}{4})$$

$$\displaystyle (6,−6,\sqrt{2})$$

26) $$\displaystyle (3,\frac{π}{4},\frac{π}{6})$$

## Exercise $$\PageIndex{5}$$

For the following exercises, the rectangular coordinates $$\displaystyle (x,y,z)$$ of a point are given. Find the spherical coordinates $$\displaystyle (ρ,θ,φ)$$ of the point. Express the measure of the angles in degrees rounded to the nearest integer.

27) $$\displaystyle (4,0,0)$$

$$\displaystyle (4,0,90°)$$

28) $$\displaystyle (−1,2,1)$$

29) $$\displaystyle (0,3,0)$$

$$\displaystyle (3,90°,90°)$$

30) $$\displaystyle (−2,2\sqrt{3},4)$$

## Exercise $$\PageIndex{6}$$

For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.

31) [T] $$\displaystyle ρ=3$$

Sphere of equation $$\displaystyle x^2+y^2+z^2=9$$ centered at the origin with radius $$\displaystyle 3$$, 32) [T] $$\displaystyle φ=\frac{π}{3}$$

33) [T] $$\displaystyle ρ=2cosφ$$

Sphere of equation $$\displaystyle x^2+y^2+(z−1)^2=1$$ centered at $$\displaystyle (0,0,1)$$ with radius $$\displaystyle 1$$, 34) [T] $$\displaystyle ρ=4cscφ$$

35) [T] $$\displaystyle φ=\frac{π}{2}$$

The xy-plane of equation $$\displaystyle z=0,$$ 36) [T] $$\displaystyle ρ=6cscφsecθ$$

## Exercise $$\PageIndex{7}$$

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface.

37) $$\displaystyle x^2+y^2−3z^2=0, z≠0$$

$$\displaystyle φ=\frac{π}{3}$$ or $$\displaystyle φ=\frac{2π}{3};$$ Elliptic cone

38) $$\displaystyle x^2+y^2+z^2−4z=0$$

39) $$\displaystyle z=6$$

$$\displaystyle ρcosφ=6;$$ Plane at $$\displaystyle z=6$$

40) $$\displaystyle x^2+y^2=9$$

## Exercise $$\PageIndex{8}$$

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle φ

in radians rounded to four decimal places.

41) [T] $$\displaystyle (1,\frac{π}{4},3)$$

$$\displaystyle (\sqrt{10},\frac{π}{4},0.3218)$$

42) [T] $$\displaystyle (5,π,12)$$

43) $$\displaystyle (3,\frac{π}{2},3)$$

$$\displaystyle (3\sqrt{2},\frac{π}{2},\frac{π}{4})$$

44) $$\displaystyle (3,−\frac{π}{6},3)$$

## Exercise $$\PageIndex{9}$$

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

45) $$\displaystyle (2,−\frac{π}{4},\frac{π}{2})$$

$$\displaystyle (2,−\frac{π}{4},0)$$

46) $$\displaystyle (4,\frac{π}{4},\frac{π}{6})$$

47) $$\displaystyle (8,\frac{π}{3},\frac{π}{2})$$

$$\displaystyle (8,\frac{π}{3},0)$$

48) $$\displaystyle (9,−\frac{π}{6},\frac{π}{3})$$

## Exercise $$\PageIndex{10}$$

For the following exercises, find the most suitable system of coordinates to describe the solids.

49) The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length $$\displaystyle a$$, where $$\displaystyle a>0$$

Cartesian system, $$\displaystyle {(x,y,z)|0≤x≤a,0≤y≤a,0≤z≤a}$$

50) A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of $$\displaystyle a$$ and $$\displaystyle b$$, respectively, where $$\displaystyle b>a>0$$

51) A solid inside sphere $$\displaystyle x^2+y^2+z^2=9$$ and outside cylinder $$\displaystyle (x−\frac{3}{2})^2+y^2=\frac{9}{4}$$

Cylindrical system, $$\displaystyle {(r,θ,z)∣r^2+z^2≤9,r≥3cosθ,0≤θ≤2π}$$

52) A cylindrical shell of height $$\displaystyle 10$$ determined by the region between two cylinders with the same center, parallel rulings, and radii of $$\displaystyle 2$$ and $$\displaystyle 5$$, respectively

## Exercise $$\PageIndex{11}$$

53) [T] Use a CAS to graph in cylindrical coordinates the region between elliptic paraboloid $$\displaystyle z=x^2+y^2$$ and cone $$\displaystyle x^2+y^2−z^2=0.$$

The region is described by the set of points $$\displaystyle {(r,θ,z)∣∣0≤r≤1,0≤θ≤2π,r^2≤z≤r}.$$ 54) [T] Use a CAS to graph in spherical coordinates the “ice cream-cone region” situated above the xy-plane between sphere $$\displaystyle x^2+y^2+z^2=4$$ and elliptical cone $$\displaystyle x^2+y^2−z^2=0.$$

## Exercise $$\PageIndex{12}$$

55) Washington, DC, is located at $$\displaystyle 39°$$ N and $$\displaystyle 77°$$ W (see the following figure). Assume the radius of Earth is $$\displaystyle 4000$$ mi. Express the location of Washington, DC, in spherical coordinates. $$\displaystyle (4000,−77°,51°)$$

56) San Francisco is located at $$\displaystyle 37.78°N$$ and $$\displaystyle 122.42°W.$$ Assume the radius of Earth is $$\displaystyle 4000$$mi. Express the location of San Francisco in spherical coordinates.

57) Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are $$\displaystyle (4000,−43.17°,102.91°).$$

$$\displaystyle 43.17°W, 22.91°S$$

58) Find the latitude and longitude of Berlin if its spherical coordinates are $$\displaystyle (4000,13.38°,37.48°).$$

## Exercise $$\PageIndex{13}$$

59) [T] Consider the torus of equation $$\displaystyle (x^2+y^2+z^2+R^2−r^2)^2=4R^2(x^2+y^2),$$ where $$\displaystyle R≥r>0.$$

a. Write the equation of the torus in spherical coordinates.

b. If $$\displaystyle R=r,$$ the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is $$\displaystyle ρ=2Rsinφ.$$

c. Use a CAS to graph the horn torus with $$\displaystyle R=r=2$$ in spherical coordinates.

$$\displaystyle a. ρ=0, ρ+R2−r2−2Rsinφ=0;$$

c. 60) [T] The “bumpy sphere” with an equation in spherical coordinates is $$\displaystyle ρ=a+bcos(mθ)sin(nφ)$$, with $$\displaystyle θ∈[0,2π]$$ and $$\displaystyle φ∈[0,π]$$, where $$\displaystyle a$$ and $$\displaystyle b$$ are positive numbers and $$\displaystyle m$$ and $$\displaystyle n$$ are positive integers, may be used in applied mathematics to model tumor growth.

a. Show that the “bumpy sphere” is contained inside a sphere of equation $$\displaystyle ρ=a+b.$$ Find the values of $$\displaystyle θ$$ and $$\displaystyle φ$$ at which the two surfaces intersect.

b. Use a CAS to graph the surface for $$\displaystyle a=14, b=2, m=4,$$ and $$\displaystyle n=6$$ along with sphere $$\displaystyle ρ=a+b.$$

c. Find the equation of the intersection curve of the surface at b. with the cone $$\displaystyle φ=\frac{π}{12}$$. Graph the intersection curve in the plane of intersection.

5.7E: Excersies is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.