Answer:Section 6.1
Solutions and Hints
by Brent M. Dingle
for the book:
Precalculus, Mathematics for Calculus 4
th Edition
by James Stewart, Lothar Redlin and Saleem Watson.
If you remember nothing else from this section remember:
arc length = radius * angle
s = r * q
where the angle, q, is measured in radians.
There are other formulas, but that one is pretty important.
48. A circular arc of length 3 feet subtends a central angle of 25.
Find the radius of the circle.
Start with s = r*q , s = 3 feet, q = 25 = 25*(p/180) = (5/36)p radians
3 = r * (5/36)p Å 3*(36/5) = r*p Å 21.6 = p*r Å r = 21.6/p feet
52. Memphis, Tennessee and New Orleans Louisiana lie approximately on
the same meridian. Memphis has latitude 35 N and New Orleans 30 N.
Find the distance between the cities, given the radius of the earth is
3960 miles.
Again start with s = r*q,
with r = 3960 miles and q = 35 – 30 = 5 = 5*(p/180) = (1/36)p.
and we need to find s = arc length = distance between the cities.
s = 3960*(1/36)p = 110p miles.
60. A sector of a circle of radius 24 miles has an area of 288 square miles.
Find the central angle of the sector.
For this you need a new formula.
The area of a sector of circle = A = ½*r
2
*q ,
where q is the central angle of the sector measured in radians
and r of course is the radius of the circle.
For this problem r = 24 miles, A = 288 sq. miles and we need to find q.
288 = (½)*24
2
*q Å 288 = 288*q Å 1 radian = q (or about 57.3)
62. Three circles with radii 1, 2 and 3 feet are externally tangent to one
another. Find the area of the sector of the circle of radius 1 that is
cut off by the line segments joining the center of that circle to the
centers of the other two circles.
So you start out with:
Notice you know the length of ALL the sides of the triangle, because you know
the radius of each circle. From this you might discern that: 5
2
= 3
2
+ 4
2
. Thus you
have the (length of the hypotenuse)
2
= (length of side A)
2
+ (length of side B)
2
.
And from that you may conclude the triangle is a right triangle, or rather the angle
we are interested in is 90. Thus we use the area formula given in the text:
The area of a sector of circle = A = ½*r
2
*q ,
For this problem r = 1 foot, q = 90 = p/2, and we need to find A
A = ½ * 1
2
* p/2 = p/4 square feet.
Step-by-step explanation:
