An Answer Key of Sorts

Here's help for getting in in the 1890s.

ADVANCED ALGEBRA

Do your own proof, but doctoral candidate Sam Lichtenstein offers the key bit of information.

Three numbers in Harmonic Progression are defined as:

a ,

        a
b = ———  ,
      1+d

         a
c =  ———
      1+2d

FRENCH

Doctoral candidate Alison Stiner translates:

N’auriez vous pas honte d’avouer que vous n’avez point regretté ni repenti le fait de l’avoir frappé, mais au contraire que vous en êtes content?

PHYSICS

Tom Rossing, a visiting professor at the Center for Computer Research in Music and Acoustics and the co-author of Physics of Musical Instruments, answers part of the question:

You could hold the vibrating tuning fork right at the top of a pipe that is standing vertically in water. Adjust the pipe up and down until the tuning fork gives its loudest sound. The height of the pipe outside the water is one-half the wavelength.

The number of vibrations per second, or frequency (f), is related to the wavelength (λ) and the velocity of sound (v) by f = v/λ .

Rick Pam, physics lecturer, takes over the rest:

If v = 1,100 feet per second, and f = 100 cycles per second (a vibration is the same as a cycle; in 1960, the Hertz [Hz, for short] was adopted as the modern name for cycles per second), then the wavelength of sound = 1100/100 = 11 feet. As above, the length of an open organ pipe will be half the wavelength, or 5.5 feet.

Alternatively, a student in 1892 could have waited about 50 years for the invention of the strobe tuner, an electric gadget that measures the frequency of the vibrating fork directly, or about 115 years for a smart phone app that does the same thing.

TRIGONOMETRY

Lichtenstein’s help with this trig problem:

            sinΘ                        cosΘ
tanΘ = ———    and   cotΘ = ———
            cosΘ                        sinΘ

To solve the problem, use the Pythagorean identity: sin²Θ + cos²Θ = 1

tan²Θ + cos²Θ =

sin²Θ       cos²Θ
————  +  ————   =
cos²Θ       sin²Θ

1-cos²Θ       cos²Θ
—————  +  —————
  cos²Θ        1-sin²Θ

Comments (1)

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This clears out some cobwebs. Thanks for sending. The second denominator in the solution should be 1-cos theta squared.

Posted by James T. Ruddell, P.E. on Feb 5, 2012 10:45 AM