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The projected cost of a new high school is $20,066,340, of which $5,216,340 will be...

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loishy | Student, Grade 10 | Salutatorian

Posted June 27, 2013 at 6:34 PM via web

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The projected cost of a new high school is $20,066,340, of which $5,216,340 will be covered by available grants. Useing 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 per town resident by $75. What is the current population of the town.

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted June 27, 2013 at 7:08 PM (Answer #1)

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The cost of the new high school is $20,066,340, of which $5,216,340 is covered by a grant leaving $14,850,000 for the city to pay.

Let C be the initial cost per resident and let R be the initial number of residents.

Then `C=(14,850,000)/R` . With the revised census data, the cost is reduced by $75 and the number of residents is increased by 2200 so we have:

`C-75=(14,850,000)/(R+2200)` . Substituting `(14850000)/R` for C we get an equation in R:

`(14850000)/R-75=(14850000)/(R+2200)`

`=> 14850000(1/R-1/(R+2200))=75`

Multiply both sides by R(R+2200) (The least common denominator) to get:

`=>14850000(R)(R+2200)(1/R-1/(R+2200))=75(R)(R+2200)` `=>3267000000=75R^2+165000R`

This is quadratic in R -- rewrite in standard form:

`75R^2+165000R-3267000000=0`

Divide out the greates common factor 75:

`R^2+2200R-435600000=0`

Use the quadratic formula:

`R=(-2200+-sqrt(2200^2-4(435600000)))/2`

`R=(-2200+-41800)/2`

`R=-1100+-20900`

`R=19800"or"-11000`

The second answer does not make sense in the context of the problem.

Thus the number of residents before the census was 19,800.

With the revised figures, we add 2200 to the population to get the current population.

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The current number of residents is 22000 people.

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Check: The cost under the original numbers per resident was:

`C=(14,850,000)/(19,800)=$750`

With the revised numbers we get:

`C=(14,850,000)/(22000)=$675` which is $75 less than the original figure as required.

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