Math For Liberal Arts I
 

Homework:  §4.9,  page 123,  #1, 2, 3, 4, 5, 6, 7

#1, page 123.   For each of the following pairs of numbers, find both the arithmetic mean and the geometric mean, and compare them.

        (a)    3, 27

SOLUTION:  The arithmetic mean is the average:  A = (3 + 27) / 2 = 30 / 2 = 15.  The geometric mean is the square root of the product:  G = Ö(3 × 27) = Ö81 = 9.  In this case, the geometric mean (9) is less than the arithmetic mean (15).

        (b)    16, 4

SOLUTION:  As above, we compute that the arithmetic mean is  A = (16 + 4) / 2 = 20 / 2 = 10,  and the geometric mean is  G = Ö(16 × 4) = Ö64 = 8.  Again, the geometric mean (8) is less than the arithmetic mean (10).

        (c)    8, 18

SOLUTION:  Again, we compute that the arithmetic mean is  A = (8 + 18) / 2 = 26 / 2 = 13,  and the geometric mean is  G = Ö(8 × 18) = Ö144 = 12.  Again, the geometric mean (12) is less than the arithmetic mean (13).

        (d)    8, 10

SOLUTION:  The arithmetic mean is  A = (8 + 10) / 2 = 18 / 2 = 9,  and the geometric mean is  G = Ö(8 × 10) = Ö80,  which is approximately  8.94.  Therefore, in this case also, the geometric mean (8.94) is less than the arithmetic mean (9).

        (e)    2, 5

SOLUTION:  The arithmetic mean is  A = (2 + 5) / 2 = 7 / 2 = 3.5,  and the geometric mean is  G = Ö(2 × 5) = Ö10,  which is approximately  3.16.  In this case also, the geometric mean (3.16) is less than the arithmetic mean (3.5).

        (f)     4, 8

SOLUTION:  The arithmetic mean is  A = (4 + 8) / 2 = 12 / 2 = 6,  and the geometric mean is  G = Ö(4 × 8) = Ö32,  which is approximately  5.66.  Again in this case, the geometric mean (5.66) is less than the arithmetic mean (6).

#2, page 123.   Show that the geometric mean of the numbers  10,  16,  and  50  is less than their arithmetic mean.

SOLUTION:  The arithmetic mean is the average:  A = (10 + 16 + 50) / 3 = 76 / 3,  which is approximately  25.33.  In this case, the geometric mean is the cube root of the product:  G = 3Ö(10 × 16 × 50) = 3Ö8000 = 20.  As in the examples above, the geometric mean (20) is less than the arithmetic mean (25.33).

#3,  page 123.   For each of the following triplets, find the arithmetic and geometric means.

        (a)    3, 4, 7

SOLUTION:  The arithmetic mean is the average:  A = (3 + 4 + 7) / 3 = 14 / 3,  which is approximately  4.67.  The geometric mean is the cube root of the product:  G = 3Ö(3 × 4 × 7) = 3Ö84,  which is approximately  4.38.

        (b)    2, 12, 9

SOLUTION:  The arithmetic mean is  A = (2 + 12 + 9) / 3 = 23 / 3,  which is approximately  7.67.  The geometric mean is  G = 3Ö(2 × 12 × 9) = 3Ö216 = 6.

        (c)    2, 4, 8

SOLUTION:  The arithmetic mean is  A = (2 + 4 + 8) / 3 = 14 / 3,  which is approximately  4.67.  The geometric mean is  G = 3Ö(2 × 4 × 8) = 3Ö64 = 4.

#4, page 123.   Show that  Öxy £ (x + y) / 2  by noting the inequality  (Öx - Öy)2 ³ 0,  and expanding the left-hand side.

SOLUTION:  First note that the inequality  (Öx - Öy)2 ³ 0  does indeed hold for all choices of  x  and  y,  because the square of a number is never negative.  Multiplying out the left hand side, we get

            (Öx - Öy)2 = (Öx - Öy)(Öx - Öy) = (Öx)(Öx) - (Öx)(Öy) - (Öy)(Öx) + (Öy)(Öy) = x - 2Öxy + y

This leads to the inequality  x - 2Öxy + y ³ 0.  Now add  2Öxy  to both sides, which gives the inequality  x + y ³ 2Öxy.  Finally, divide both sides by  2,  which yields the inequality  (x + y) / 2 ³ Öxy.  That is, the arithmetic mean is greater than or equal to the geometric mean.

#5, page 123.   Suppose you have a rectangle of length  9  and width  4,  and you want to construct a square with the same area.  What will the lengths of the sides of the square have to be?  What has this to do with the subject of this section?

SOLUTION:  The area of a  9  by  4  rectangle is  9 × 4 = 36.  If the square has side of length  s,  then its area is  s × s = s2.  Now if we want the square to have the same area as the rectangle, then we want  s2 = 36.  Taking the square root of both sides yields the side of the square  s = Ö36 = 6.  Notice that a side of the square is the geometric mean of the sides of the rectangle:  s = Ö(9 × 4).

#6, page 123.   Give a geometric interpretation of the geometric mean of three numbers.  (See problem #5 above.)

SOLUTION:  The geometric mean of three numbers is the cube root of their product.  If you had a "rectangular" box of length  x,  width  y,  and height  z,  then the geometric mean of these three dimensions would be  s = 3Ö(x × y × z),  which has the property that a cubic box of length, width, and height  s  would have volume  s3 = x × y × z.  That is, the geometric mean of three numbers gives you a side of a cube whose volume is the same as the volume of the rectangular box with the three given numbers as dimensions.

#7, page 123.   Why do we say that  7  times  7  is  7  squared?

SOLUTION:  Because the area of a  7  by  7  square is  7 × 7 = 72.