Mat126

Real World Applications

Mary D. Penick

MAT 126: Survey of Mathematical Methods

Tonya Meisner, Instructor

22 December 2013

Real World Applications

35. A person hired a firm to build a CB radio tower. The firm charges $100 for labor for the first 10 feet. After that, the cost of the labor for each succeeding 10 feet is $25 more than the preceding 10 feet. That is, the next 10 feet will cost $125, the next10 feet will cost $150, etc. How much will it cost to build a 90-foot tower?

This problem involves arithmetic sequence since labor cost for each successive 10 ft remains constant at $25.

In addition, the arithmetic sequence of the labor costs is 100, 125, 150, …For finding the cost for building a 90 ft tower, we sum the first 9 terms of the above sequence. To sum all the terms, the formula for an arithmetic series can be used:

Calculations

We know that S_n = (1/2) * n (a_1 + a_n ) The total cost of the tower is the average of the first and last terms times the number of terms (in this case, 9 terms)

a_n = a with a subscript of n (this is the nth term in the series)

a_1 = a with a subscript of 1 (this is the 1st term in the series)

n = number of terms

d = difference between consecutive terms (the common difference)

The first term in the series is known

a_1 = 100

The last term in the series can be computed using the following formula:

a_n = a_1 + (n - 1) * (d)

Applying the notation to your problem

a_1 = 100

d (the common difference) = 25

The 9th term (representing the last 10 feet of construction):

n = 9

a_n = a_1 + (n - 1) * (d)

a_9 = 100 + (9 - 1) * (25)

a_9 = 100 + (8) * (25)

a_9 = 100 + 200

a_9 = 300

The total cost of the tower is:

S_n = (1/2) * n (a_1 + a_n )

S_9 = (1/2) * 9 (100 + 300 )

S_9 = (1/2) * 9 * (400)

S_9 = 1800

Therefore, the total cost of the tower is : $1,800...