Thom sec3pt5
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This document contains multiple word problems involving optimization and related rates from a calculus textbook. The first problem asks to find the angle between two sides of a triangle that maximizes the triangle's area. The second problem asks to find the dimensions of a right circular cylinder of maximum volume that can fit inside a sphere. The third problem involves ships sailing at different speeds and calculating the distance between them over time.
Thom sec3pt5
- 1. Thomas' Calculus Tenth Edition Section 3.5- Modeling and Optimization 13. Two sides of a triangle have lengths a and b, and the angle between them is . What value of will maximize the triangle's area? (Hint: sin ) Note that a and b are constants. will be a maximum when a b cos when cos . The only feasible value of where this occurs is ° Therefore the maximum area will occur when the triangle is a right triangle. 19. Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 10 cm. What is the maximum volume? Let and be as shown in the diagram below: 10 x y The radius of the cylinder is and its height is The volume of a cylinder is Get as function of just one variable.
- 2. Therefore will be a maximum when The maximum volume will occur when the radius is 8.165 cm and the height is 2 11.547 cm . The maximum volume is 7 . 2418 cm 39. At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots (nautical miles per hour) and continued to do so all day. Ship B was sailing east at 8 knots and continued to do so all day. a) Start counting time with at noon and express the distance between the ships as a function of time. Use In t hours, ship A will go miles south and ship B will go miles east. So after hours we have the following: 12 12 t 8 t 12 - 12 t s The distance between the 2 ships is
- 3. b) How rapidly was the distance between the 2 ships changing at noon? One hour later? At noon, and . knots At 1:00, and . knots c) The visibility that day was 5 nautical miles. Did the ships ever sight each other? To answer this question we need to find the minimum value of This will occur when when the numerator . hours At this time nautical miles. This is the closest distance between the 2 ships, and consequently the 2 ships did not sight each other. d) Graph and together as functions of for Compare the graphs and reconcile what you see with your answers in parts b and c.
- 4. s(t) s'(t) When s' = 0, s is a minimum Horizontal Asymptote y = 14.42 The main thing to observe is that the minimum will occur when e) The graph of looks as if it has a horizontal asymptote in the 1st quadrant. This suggests that has a limiting value as What is this limiting value? What is it's relation to the ships' individual speeds? To find lim ignore all terms on the top and bottom expect for the terms of highest degree. lim lim This means that the graph of has a horizontal asymptote Note that . This is the square root of the sum of the squares of the 2 ship speeds.
- 5. 43. It costs you c dollars to manufacture and distribute backpacks. If the backpacks sell at x dollars each, the number sold is where and are positive constants. What selling price will bring the maximum profit? For each packpack sold they make a profit of The total profit is