3 kmSIDEWALK↑ o1 kmPARKAThe fearless pizza delivery guy travels by foot and skateboard from the pizza shop to my house. My house is at point B on the north side of a grassy parkwhich is 1 km wide. The pizza shop is at point A on the south side of the park opposite the point O which is 3 km east of my house. The delivery guy knowsthat I am hungry and that I tip well, so he wants to minimize his travel time. He walks across the park in a straight line from point A to a point Con the northside of the park at 4 km hour, and then he skates along the sidewalk from point C to point B at 12 km per hour. See the diagram. What choice of the point Cminimizes the travel time? (The delivery guy is fearless, but not reckless. He always wears a helmet.)Your solution should clearly display a function to be minimized, an interval on which the function is to be minimized, a calculation of the minimum value, andan explanation of why the value you found is a minimum.

Name: 3 kmSIDEWALK↑ o1 kmPARKAThe fearless pizza delivery guy travels by fo
Uploaded: 2024-03-20T07:07:26.000Z
Channel: Bartleby
Description: 3 kmSIDEWALK↑ o1 kmPARKAThe fearless pizza delivery guy travels by foot and skateboard from the pizza shop to my house. My house is at point B on the north side of a grassy parkwhich is 1 km wide. The pizza shop is at point A on the south side of th

Let the the distance between O and C is = xThat means BC=3-x and AC=x2+12=x2+1On the grass the speed is 4 km/hour. So the time required for AC=x2+14On the road the speed is 12 km/hour.So the required time on road for BC=3-x12So the total time=T=x2+14+3-x12 Our goal is to minimize T=x2+14+3-x12 on the interval [0,3]For minimum T, dTdx=0T=x2+14+3-x12dTdx=18x2+1(2x)-112dTdx=x4x2+1-1120=x4x2+1-112x4x2+1=11212x=4x2+13x=x2+1(3x)2=x2+19x2=x2+18x2=1x2=18x=18=122=24 , this is a critical point. dTdx=x4x2+1-112d2Tdx2=14x2+1-x4xx2+14x2+12-0=4x2+1-4x2x2+116(x2+1)=4x2+4-4x216(x2+1)x2+1=416(x2+1)x2+1=14(x2+1)x2+1At x= 24d2Tdx2=14(x2+1)x2+1=1418+118+1>0 (second derivative test)So at x= 24 there is a minimum. Answer: C should be 24 km away from O.