Problema Solution
From the magazine the “Mathematics Teacher” comes the following problem:
A building is 12 stories high and is covered entirely by windows on all four sides. Each floor has 38 windows on it. Once a year, all the windows are washed. The cost for washing the windows is $2.00 for each first-floor window, $2.50 for each second-floor window, $3.00 for each third floor window, and so on.
How much will it cost to wash the windows of this building?
What if the building is 30 stories tall?
What if the building is n stories tall?
Answer provided by our tutors
the cost for the first floor is: 2*38
the cost for the second floor is: (2 + 0.5)*38
the cost for the third floor is: (2 + 2*0.5)*30
....
the cost for the n th floor is: (2 + (n - 1)*0.5)*38
the total cost for n store building is the sum:
2*38 + (2 + 0.5)*38 + (2 + 2*0.5)*30 + .... + (2 + (n - 1)*0.5)*38 =
= 38(2 + (2 + 0.5) + (2 + 2*0.5) + ... + (2 + (n - 1)*0.5))
the sequence 2, (2 + 0.5), (2 + 2*0.5), ... , (2 + (n - 1)*0.5) is arithmetic progression with:
first term: a = 2
common difference: d = 0.5
The sum of the first n arithmetic sequence terms is:
S = (n/2)(2a + (n - 1)d)
for a = 2 and d = 0.5
S = (n/2)(2*2 + (n - 1)*0.5)
S = (n/2)(3.5 + 0.5n)
The cost to wash all the windows of the n stories high building is: 38S.
For a building that is 12 stories high n = 12
S = (12/2)(3.5 + 0.5*12)
S = 57
the cost to was all the windows is 38S = 38*57 = $2,166.
If the building is 30 stores high then n = 30 and the sum is:
S = (30/2)(3.5 + 0.5*30)
S = 277.5
the cost to was all the windows is 38S = 38*277.5 = $10,545.