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 40 m. 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 = 3v.
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 - 2v)/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 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) = 10x and C(x) = 2x + x^2. How many pizzas sold maximizes the profit?
334. Assume that R(x) = 15x, and C(x) = 60 + 3x + 1/2x^2. How many pizzas sold maximizes the profit?
For the following exercises, consider a wire 4 ft 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
339. y - 1/x
340. x^2 - y
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 = 16π that has the least amount of surface area.
346. Find the dimensions of a right cone with surface area S = 4π that has the largest volume.