5. IE. A box with a square base and open top must have a volume of 40% cm'. Find the dimensions of the hot that minimizes the amount of material used. A box with an open top is to be constructed from a square piece of cardboard, 3 m wide. by cutting out a square from each of the four corners and bending up the sides, as indicated in the gure. Find the largest Vlum that such a bolt can have. 7. The lifeguard at a public beach has 400 m of rope available to lay out a rectangular restricted swimming area using the straight shoreline as one side of the rectangle. (a) If she wants to maximize the swimming area, what will the dimensions of the rectangle be? (b) To ensure the safety of swimmers, she decides that nobody should be more than 50 m from shore. What should the dimensions of the swimming area be with this added restriction?2 16. A Norman window has the shape of a rectangle capped by a semicircular region as shown. If the perimeter of the window is 8 m, find the width of the window that will admit the greatest amount of light. 17. A boat leaves a dock at noon and heads west at a speed of 25 km/h. Another boat heads north at 20 km/h and reaches the same dock at 1:00 p.m. When were the boats closest to each other? 18. Find the largest possible volume of a right circular cylinder that is inscribed in a sphere of radius r.ANSWERS 1. 75, -75 2. 20. 10 3. both 25 cm 5. 20 cm by 20 cm by 10 cm 6. 2 m' 7. (a) 100 m by 200 m (b) 50 m by 300 m 8. 500v/2 m by 750V/2 m 10. (- 10. /500 11. (-2.2) 12. = 5.4 cm TT 13. (a) all wire for the circle 160 (b) 4+ T 22.4 cm for the square 14. 4 cm 15. Lay cable underground to a point 1165 m cast of P. 16 16. 2.24 m TT + 4 h after noon (about 12:23 p.m.) 4V3 18. 9 TP