Download Calculus I - Maximizing Garden Size, Right Triangle Area, and More - Prof. Geoff Clement and more Assignments Calculus in PDF only on Docsity! Calculus I Optimization Problems Name ____________________ 1. A man wishes to have a rectangular-shaped garden in his backyard. He has 50 feet of fencing material with which to enclose his garden. Find the dimensions for the largest garden he can have if he uses all of the fencing material. 2. What is the largest possible area for a right triangle whose hypotenuse is 5 cm long, and what are its dimensions? 3. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 20” by 24” sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible? What is the resulting maximum volume? 4. You have been asked to design a one-liter (1,000 cm3 ) oil can shaped like a right circular cylinder. What dimensions will use the least material? 5. Acrosonic’s total profit (in dollars) from manufacturing and selling x units of their model F loudspeaker system is given by P x( ) = -0.02x2 + 300x – 200,000 (0 ≤ x ≤ 20,000) . How many units of the loudspeaker system must Acrosonic produce to maximize its profits? 6. The daily average cost function (in dollars per unit) of Elektra Electronics is given by C x( ) = 0.0001x2 – 0.08x + 40 + 5,000 x (x > 0) , where x stands for the number of programmable calculators that Elektra produces. Show that a production level of 500 units per day results in a minimum average cost for the company. 7. The altitude (in feet) of a rocket t seconds into flight is given by s = f t( ) = -t3+ 96t2 + 195t + 5 t ≥ 0( ) . Find the maximum altitude attained by the rocket, and find the maximum velocity attained by the rocket. 8. A manufacturer of tennis rackets finds that the total cost C(x) (in dollars) of manufacturing x rackets/day is given by C x( ) = 400 + 4x + 0.0001x2 . Each racket can be sold at a price of p dollars, where p is related to x by the demand equation p = 10 – 0.0004x. If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer. 9. A division of Chapman Corporation manufactures a pager. The weekly fixed cost for the division is $20,000, and the variable cost for producing x pagers/week is given by V x( ) = 0.000001x3 – 0.01x2 + 50x dollars. The company realizes a revenue of R x( ) = -0.02x2 + 150x (0 ≤ x ≤ 7,500) dollars from the sale of x pagers/week. Find the level of production that will yield a maximum profit for the manufacturer. 10. A grain silo has the shape of right circular cylinder surmounted by a hemisphere. If the silo is to have a capacity of 504π ft3 , find the radius and height of the silo that requires the least amount of material to construct. Hint. The volume of the silo is πr2h + 2 3 πr3 , and the surface area (including the floor) is π 3r2 + 2rh( ) .
