Both sections

Really nice choices of problems. Both sections are very similar in content and variety, so I just gathered all those problems together in a huge crazy story-problem exam. If you satisfactorily finish these problems, you are welcome to download the adjacent warning and post it in your room door, or in your car, so people understand how bad-ass you are.

1. A farmer wants to fence an area of 24 million square feet and then divide it in half with a fence parallel to one of the sides of the rectangular area. What should the length of the two sides be to minimize fencing?

2. A ladder 12 ft long is resting against a vertical wall. If the ladder is sliding away from the wall at 0.5 ft/sec, how quickly is the angle between the ground and ladder changing when the ladder is 4 ft from the base of the wall?

3. A square box with an open top has a volume of 10 cubic meters. Material for the base costs $5 per square meter. Material for the sides costs $2 per square meter. What is the cost of materials for the cheapest such container?

4. A piece of wire 10 m long is cut into 2 pieces. One piece is built into a square and the other is built into an equilateral triangle.
   • How much wire should be used for the square in order to maximize the total area?
   • How much wire should be used for the square to minimize the total area?

5. A flashlight lying on the ground shines on a building 15 yards away as a 5-yard tall man walks towards the same building at 2 yd/s. How fast is the length of the shadow decreasing when he is exactly half way between the flashlight and the building?

6. A man is 6 feet tall and walking away from a streetlight that is 15 feet tall at a rate of 5 ft/sec. At what rate is his shadow increasing when he is 20 feet from the streetlight?

7. Professor Blanco-Silva wants to put a poster over the blackboard in room 412 in LeConte during Exam day, since there is writing on the blackboard but no eraser can be found. The poster is a picture of a student’s face to motivate us to do better on the exam. The area of the poster is to be fixed at 180 square inches with 2-inch margins on all sides. What dimensions will give the largest printed area?

   

8. A cylindrical tank is filled with water. The tank stands upright and has a radius of 10 meters. How fast does the height of the water drop when it is drained at 15 cubic meters per second?

9. At 3 pm, ship A is 120 km west of ship B. Ship A is sailing east at 25 km/hour and ship B is sailing north at 35 km/h. How fast is the distance between the ships changing at 7 pm?

10. Gravel is being dumped from a conveyor belt at a rate of 20 cubic feet per minute and forms the shape of a cone whose base, diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

11. The height of a rectangular box is 10 inches. Its length increases at a rate of 2 in/sec and its width decreases at the rate of 4 in/sec. When the length is 8 inches and the width is 6 inches, the volume of the box is changing at the rate of?

12. A man launches his boat from point A on a bank of a straight river, 3 km wide, and wants to reach point B, 8 km downstream on the opposite bank, as quickly as possible (see the figure below). He could row his boat directly across the river to point C and then run to B, or he could row directly to B, or he could row to some point D between C and B and then run to B. If he can row 6 km/h and run 8 km/h, where should he land to reach B as soon as possible? (We assume that the speed of the water is negligible compared to the speed at which the man rows.)

13. Find two numbers whose difference is 100 and whose product is a minimum