The projected cost of a new high school is $20,066,340 of which $5,216,340 will be covered by annual grants. Using data from the last census the planning committee estimated the cost per town...

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Math

The projected cost of a new high school is $20,066,340 of which $5,216,340 will be covered by annual grants. Using data from the last census the planning committee estimated the cost per town resident. However new census data reveals that the population has increased by 2200 people reducing the cost of each resident by $75.

What is the current population of the town?

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If $5,216,340 will be covered by annual grants, this leaves  $20,066,340 - $5,216,340 = $14,850,000 left to be covered by the residents of the town.

Let x = original population
Let `y_n` = new cost per person
Let` y_o` = old cost per person

`y_n = 14,850,000/(x+2,200)`
`y_o = 14,850,000/x`

`y_o-y_n=75`

`14,850,000/x - 14,850,000/(x+2,200) = 75`

`14,850,000(x+2200) -14,850,000x = 75x(x+2,200)`

`14,850,000x+32,670,000,000 -14,850,000x = 75x^2 +165,000x`

`75x^2+165,000x-32,670,000,000=0`

`x^2+2,200x-435,600,000=0`

Use the quadratic formula:

a=1; b=2,200; c=435,600,000

`x=(-b+-sqrt(b^2-4ac))/(2a)`

`x=(-2,200+-sqrt(2,200^2-4(1)(-435,600,000)))/(2(1))`

`x=(-2,200+-41,800)/2`

Since the population could not have been negative we know that:

`x=(-2,200+41,800)/2 = 19,800`

Therefore, the current population of the town is 19,800+2,200 = 22,000.

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