The cost in millions of dollars of building a threestory hig

The cost, in millions of dollars, of building a three-story high school in New York State was estimated to be C(x) = 2.1 + 0.14x 0.0001x2 where x is the number of thousands of square feet. Suppose that you are contemplating building a for-profit three-story high school and estimate that your total revenue will be $0.21 million dollars per thousand square feet. What is the profit function (in millions of dollars)? P(x) = What size school should you build in order to break even? (Round your answer to 3 decimal places.) thousand ft2

Solution

The cost, in millions of dollars, C(x) = 2.1 + 0.14x 0.0001x²

total revenue will be $0.21 million dollars per thousand square feet,x is the number of thousands of square feet.

revenue R(x)=0.21x

profit P(x) =revenue R(x) -cost C(x)

profit P(x) =0.21x - (2.1 + 0.14x 0.0001x²)

profit P(x) = - 2.1 + 0.07x + 0.0001x²

for break even

profit P(x) = - 2.1 + 0.07x + 0.0001x² =0

- 21000 + 700x +x² =0

for ax^2 +bx +c =0 ==>x=[-b+(b²-4ac)]/(2a),x=[-b-(b²-4ac)]/(2a)

a=1, b=700,c =-21000

==>x=[-700+(700²-4*1*(-21000))]/(2*1),x=[-700-(700²-4*1*(-21000))]/(2*1)

=>x =28.814,x =-728.81

x cannot be negative

so school should be build 28.814 thousands of square feet  in order to break even