An independent home builder's annual profit, in thousands of dollars, can be modeled by the function p(x)=5.152x^3 -143x^2+1102x-1673, where x is the number of houses built in a year. his company can build at most 13 houses in a years. a.) how many should the builder construct in order to have a profit of at least $400,000? b.) how many houses should the builder construct in order to maximize profit?

A) 3 houses B) 5 houses Frankly, for this question, using advanced mathematics isn't needed since x can only be an integer and the function needs to be evaluated at only 14 points. So let's evaluate the function for the values of x ranging from 0 to 13 and see what we get: f(0) = -1673 f(1) = -708.848 f(2) = 0.216 f(3) = 485.104 f(4) = 776.728 f(5) = 906 f(6) = 903.832 f(7) = 801.136 f(8) = 628.824 f(9) = 417.808 f(10) = 199 f(11) = 3.312 f(12) = -138.344 f(13) = -195.056 Looking at the above values, the answer to "How many should the builder construct in order to have a profit of at least $400,000?" is rather obvious. That would be 3 houses with a profit of about $485,000. Yes, building 9 houses would get a profit closer to $400,000; But doing three times the effort for less profit isn't a reasonable business choice. And to maximize profit, the obvious choice becomes 5 houses since that's the largest result.