IV. An independent home builder\'s annual profit, in thousands of dollars, can b
ID: 3019180 • Letter: I
Question
IV. An independent home builder's annual profit, in thousands of dollars, can be modeled by the function P(x) = 5.1 52x3-143x2 + 1102x-1673, where xis the number of houses built in a ver. His company can build at most 13 houses in a year (SLO #1,2,6) 10. Find the y-intercept and explain what it means in this context 11. Find the x-intercept(s) and explain what they mean in this context. 12. Determine the domain of the function in the context of this problem. 13. How many houses should the builder construct in order to have a profit of at least $400,000? 14. How many houses should the builder construct in order to maximize profit?Explanation / Answer
IV)
10)
for y intrercept ,x=0
=>P(0)=-1673
when no houses are built , home builder will incurr the loss of 1673 thousand dollars
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11)
foe x intercept,P(x) =0
=>5.152x3-143x2+1102x-1673=0
by calculator
=>x2,11,14.7375
home builder's profit will be zero when he builds 2 houses in a year ,home builder's profit will be zero when he builds 11 houses in a year ,
x14.7375 has no significance , since it is given in the problem that he buils atmost 13houses in year.
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12)
domain is 0 x 13
domain is x[0,13] and x has to be an integer
domain is {0,1,2,3,4,5,6,7,8,9,10,11,12,13}
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14)
profit 400000 dollars
=>P(x)400
=>5.152x3-143x2+1102x-1673400
=>2.79x9.08
builder can build atleast 3 houses and atmost 9 houses per year to make profit of atleast 400000$
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14)
P(x)=5.152x3-143x2+1102x-1673
P'(x)=15.456x2-286x+1102
for critical points P'(x)=0
15.456x2-286x+1102=0
=>x=5.47, 13
P(0)=-1673
P(5)=906
P(6)=904
P(13)=-195
builder should build 5 houses per year to maximize profit
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