Beauty by Numbers

As I walk the shores of Lake Huron, I am put in mind of Oliver Wendell Holmes's The Chambered Nautilus, which please join me in taking as a natural (ha!) link from last week's post to this week's exploration of Fibonacci.  Having read these two pieces, consider the following, adapted from Arup Guha's links to his TOK past:

Most math problems have multiple solutions that are significantly different. In this assignment, choose a single math problem from the 4 below and solve it two different ways, the more different, the better. After showing each solution to your problem, discuss which of the two solutions you feel is more elegant/beautiful.

An example problem is shown here:

Find the area of a triangle with side lengths 5, 12, and 13:

1)    Solve for the angle across the side 13 using the law of cosines:

cos C = (5² + 12² – 13²)/(2*5*12)

cos C = 0

C = 90°, thus, 5 and 12 form a base and height for the triangle and the

area is (.5)(5)(12) = 30.

2)    Use Heron’s formula: A = √(s)(s-a)(s-b)(s-c), where s is the semiperimeter of the triangle.

S = (5+12+13)/2 = 15

Area =  √(15)(15-5)(15-12)(15-13) = √(15)(10)(3)(2) = √900 = 30

Please choose one of the following problems for your posting:

1) If log(xy³) = 1 and log(x²y) = 1, what is log (xy)?

2) Mr. Earl E. Bird leaves his house for work at exactly 8:00A.M. every morning. When he averages 40 miles per hour, he arrives at his workplace three minutes late. When he averages 60 miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?

3) What is the area of the triangle bounded by the lines y = x, y = -x and y = 6?

4) Given that x²+y² = 14x + 6y + 6, what is the largest possible value that 3x+4y can have?