3.2: Modeling with Trignometric Equations

Question

The temperature in Townhood varies between 14 and 71 degrees Fahrenheit during a typical year. Assume that the temperature on July 1 is 71. Find an equation for the temperature (denoted T) in Townhood in terms of months since Dec 1 (denoted x).

T(x) = 

Answer 1

Given:

T: [14, 71] ºF

Dec 1 ⇒ x = 0

July 1 ⇒ x = 7

T( x = 7 ) = 71 ºF

Find:

T(x) = ?

Solution:

T_avg = ( 71 + 14 ) / 2 = 85 / 2

T_alt = ( 71 - 14 ) / 2 = 57 / 2

T(x) = T_avg + T_alt sin ( Aπ * x )

T(x) = 85 / 2 + 57 / 2 * sin ( Aπ * x )

71 = 85 / 2 + 57 / 2 * sin ( 7Aπ )

57 / 2 = 57 / 2 * sin ( 7Aπ )

sin ( 7Aπ ) = 1

7Aπ = π / 2

A = 1 / 14

∴ T(x) = 57 / 2 * sin ( πx / 14 ) + 85 / 2

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