Question 588892
a. The length of the diagonal of the 30-ft by 10ft rectangle, in feet, is
{{{sqrt(30^2+10^2)=sqrt(900+100)=sqrt(1000)=10sqrt(10)}}}
The cost if the contractor runs the pipe entirely under the driveway would be ${{{4*(10sqrt(10))}}}=${{{40sqrt(10)}}} (approx $126.49)
b. If the contractor runs the pipe 30ft alongside the driveway and the 10ft straight across, the cost would be
${{{3*30+4*10}}}=${{{90+40}}}=${{{130}}}
d. Going alongside the driveway for some distance, x, and then going under the drive diagonally to the terminal would include a distance, in feet, under the driveway of
{{{sqrt((30-3)^2+10^2)=sqrt(900-60x+x^2+100)=sqrt(x^2-60x+1000)}}}.
The cost, in $, would be {{{C(x)=3x+4sqrt(x^2-60x+1000)}}}.
{{{graph(300,300,-5,30,-10,140,3x+4sqrt(x^2-60x+1000))}}}
c. The contractor claims that he can doo the job for $120 by going longside the driveway for some distance, x, and then going under the drive diagonally.
That means
{{{120=3x+4sqrt(x^2-60x+1000)}}} --> {{{120-3x=4sqrt(x^2-60x+1000)}}} --> {{{(120-3x)^2=16(x^2-60x+1000)}}} --> {{{1440-720x+9x^2=16x^2-960x+16000}}} --> {{{7x^2-240x+1600=0}}}
Applying the quadratic formula,
{{{x = (240 +- sqrt(240^2-4*7*1600 ))/(2*7)=(240 +- sqrt(57600-44800 ))/14=(240 +- sqrt(12800))/14 }}} (approximately 9.06 and 25.22 ft)
e. Use the minimum feature of a graphing calcutor to find the appropriate value for x that will minimize the cost. I do not have a graphing calculator handy but calculus says it's about 18.66 ft.
f. What is the minimum cost (to the nearest cent)of which the job can be done.
$116.46 according to my calculations