Pipe is to be laid from point A in a rectangular plot of size 10 x 20 feet to a point P on side BC and from there on to C as shown in red in the figure below. The cost of laying pipe within the plot is$c per foot (it must be underground), while the cost of laying pipe along the side of the plot is $d per foot.10A20(a) Determine the total cost, f(x), as a function of c, d and x, where x is the distance from P to B. (Note that f(x) is discontinuous at x = 0.)f(x) =dollars(b) What is the cost of laying the pipe on the side of the plot (x = 0) when c = 30 and d = 25? (Round your answer to the nearest cent.)(c) What is the least cost of laying the pipe when c = 30 and d = 25? (Round your answer to the nearest cent.)

In the above figure, AB=10, BP=x, BC=20AP=AB2+BP2=100+x2PC=BC-BP=20-x Total cost of laying pipe, fx=cAP+dPC⇒fx=c100+x2+d20-xWe need to find the value of f0 when c=30, d=25.f0=30100+02+2520-0 =3010+2520 =300+500 =$800The cost of laying the pipe on the side of the plot is $800.fx=30100+x2+2520-xf'x=ddx30100+x2+2520-x =302x2100+x2+250-1 =30x100+x2-25at critical points, f'x=0⇒30x100+x2-25=0⇒6x=5100+x2⇒36x2=25100+x2=25x2+2500⇒11x2=2500⇒x2=250011⇒x=5011f''x=ddx30x100+x2-25 =3000100+x232f''5011=3000100+5011232=0.507>0 ⇒x=5011 is a point of relative minima.f5011=30100+50112+2520-5011=5011+500=665.831≈$665.83The least cost of laying the pipe is $665.83