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Chapter 4 | Applications of Derivatives
pizzas for a revenue of R ( x ) = ax and costs C ( x ) = b + cx + dx 2 , where x represents the number of pizzas. 332. Find the profit function for the number of pizzas. How many pizzas gives the largest profit per pizza? 333. Assume that R ( x ) =10 x and C ( x ) =2 x + x 2 . How many pizzas sold maximizes the profit? 334. Assume that R ( x ) =15 x , and C ( x ) =60+3 x + 1 2 x 2 . How many pizzas sold maximizes the profit? For the following exercises, consider a wire 4ft long cut into two pieces. One piece forms a circle with radius r and the other forms a square of side x . 335. Choose x to maximize the sum of their areas. 336. Choose x to minimize the sum of their areas. For the following exercises, consider two nonnegative numbers x and y such that x + y =10. Maximize and minimize the quantities. 337. xy 338. x 2 y 2
325. In the previous problem, assume the patient was nervous during the third measurement, so we only weight that value half as much as the others. What is the value that minimizes ( x −70) 2 +( x −80) 2 + 1 2 ( x −120) 2 ? 326. You can run at a speed of 6 mph and swim at a speed of 3 mph and are located on the shore, 4 miles east of an island that is 1 mile north of the shoreline. How far should you run west to minimize the time needed to reach the island?
For the following problems, consider a lifeguard at a circular pool with diameter 40m. He must reach someone who is drowning on the exact opposite side of the pool, at position C . The lifeguard swims with a speed v and runs around the pool at speed w =3 v .
339. y − 1 x 340. x 2 − y
327. Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, θ . 328. Find at what angle θ the lifeguard should swim to reach the drowning person in the least amount of time. 329. A truck uses gas as g ( v ) = av + b v , where v represents the speed of the truck and g represents the gallons of fuel per mile. At what speed is fuel consumption minimized? For the following exercises, consider a limousine that gets m ( v ) = (120−2 v ) 5 mi/gal at speed v , the chauffeur costs $15/h, and gas is $3.5/gal. 330. Find the cost per mile at speed v . 331. Find the cheapest driving speed. For the following exercises, consider a pizzeria that sell
For the following exercises, draw the given optimization problem and solve. 341. Find the volume of the largest right circular cylinder that fits in a sphere of radius 1. 342. Find the volume of the largest right cone that fits in a sphere of radius 1. 343. Find the area of the largest rectangle that fits into the triangle with sides x =0, y =0 and x 4 + y 6 =1. 344. Find the largest volume of a cylinder that fits into a cone that has base radius R and height h . 345. Find the dimensions of the closed cylinder volume V =16 π that has the least amount of surface area. 346. Find the dimensions of a right cone with surface area S =4 π that has the largest volume.
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