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90-foot tower - Problem Solving

by Latonya Smith
(Texas)










































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 next 10 feet will cost $150, etc. How much will it cost to build a 90-foot tower?

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Mar 05, 2012
90-foot tower
by: Staff


Question:

by Latonya Smith
(Texas)


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 next 10 feet will cost $150, etc. How much will it cost to build a 90-foot tower?



Answer:

each 10 feet represents a term in an arithmetic sequence

The 90 foot tower has 9 terms


To sum all the terms, the formula for an arithmetic series can be used:

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


>>> the final answer is: $1,800




Thanks for writing.

Staff
www.solving-math-problems.com



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