Can you please help me ?

Question

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

Answer 1

Total cost covered by residents = $20,066,340 - $5,216,340  = $14,850,000

Assume the current population to be x.

Cost per resident = $14,850,000/x

So population, according to previous census =  x- 2200.

So previous estimated cost = $14,850,000/(x-2200)

According to problem

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

 

Multiplying both sides by x(x-2200), we get

 

14,850,000x - 75x(x-2200) = 14850000(x-2200)

14,850,000x -75x^2 + 75*2200x = 14,850,000x  -2200*14850000

dividing both sides by -75,

 

x^2  - 2200x = 435600000 

x^2 -2200x - 435600000 = 0

 

Using the formula for roots of a quadratic equation, roots are ½[2200 ± √(2200^2 + 4 * 435600000 )] = ½ [2200 ± 41800] 

Ignoring the negative root, x = 22000

 

So the current population is 22,000

Answer 2

Assign variables:

N = original number of residents

C = original cost per resident

 

Since there are 2 unknowns, we need two equations.

 

Use this section to get the first equation: 

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

Cost to be paid by residents = 20,066,340 - 5,216,340 = 14,850,000  = 1.485 X 10^7

(N) (C) = 1.485 X 10^7

therefore C = 1.485 X 10^7 / N   ("equation for C")

 

Using the new numbers, we get this:

(N + 2200) (C - 75 ) = 1.485 X 10^7

 

Substitute the equation for C into the above, rearrange some terms, and we have:

N^2 + 2200N - 4.356X10^8

 

Using the quadratic equation, solve this to get N = 19,800 (original residents)

 

So, the new number of residents is 19,800 + 2200 = 22,000