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

BOSEOK
Aug 29, 2017

#1**0 **

Subtracting the value of the grants from the projected cost gives the cost that must be paid by the residents:

$20,066,340 - $5,216,340 = $14,850,000

If there were 'x' residents, the cost per resident would be: 14 850 000 / x

However, since there are 2200 more residents, the cost per resident would be: 14 850 000 / (x + 2200)

The difference between the two is $75; therefore, we get this equation:

14 850 000 / x - 14 850 000 / (x + 2200) = 75

To get rid of all denominators, multiply each term by (x)(x + 2200)

(14 850 000)(x + 2200) - (14 850 000)(x) = (75)(x)(x + 2200)

Multiplying out:

14 850 000x + 32 670 000 000 - 14 850 000x = 75x^{2} + 165 000x

Simplifying:

32 670 000 000 = 75x^{2} + 165 000x

Dividing all terms by 75:

435 600 000 = x^{2} + 2200x

Subtracting 435 600 000 from both sides:

0 = x^{2} + 2200x - 435 600 000

Factoring: (Actually, I would either graph this or use the quadratic formula.)

0 = (x - 19 800)(x + 22 000)

So, x is either 19,800 or - 22,000

Since -22,000 is impossible, x = 19,800

So, the current population is 19,800 + 2,200 = 22,000

geno3141
Aug 29, 2017