do the circle ones please

Note: For the L'Hopital's Rule problems, be sure to check whether L'Hopital's Rule applies before using it!!!! If you don't the graders will take off points!

Note: For the optimization problems, be sure that you find the interval your function is valid on. And be sure that you show that your answer is actually an absolute max/min. These are the two steps that students most often skip!

Exercises 4.3 Terms and Concepts What is the ma 1. T/F: An "optimization problem" is essentially an "extreme values" problem in a "story problem" setting. dard? cross section (w = h) that does not exceed the 10guare . T/F: This section teaches one to find the extreme values of a function that has more than 12. The strength S of e than one variable. to its cross section a wooden beam is direct width w and the square of its height h; that is , S = kwho for some constant k. Problems 3. Find the maximum product of two numbers (not necessar- ily integers) that have a sum of 100. Given a circular log with diameter of 12 inches, what sized 4. Find the minimum sum of two positive numbers whose beam can be cut from the log with max product is 500. maximum strength? 13. A power line is to be run to an offshore facility in the man- . Find the maximum sum of two positive numbers whose ner described in Example 4.3.3. The offshore fachuman- product is 500. miles at sea and 5 miles along d 5 miles along the shoreline from the power plant. It costs $50,000 per 6. Find the m in to yoomaximum sum of two numbers, each of which is the line underwater. 0, 300] whose product is 500. How much of the power line should be run underground to minimize the overall costs? 7. Find the maximal area of a right triangle with hypotenuse of length 1. 14. A power line is to be run to an offshore facility in the man- miles at con d in Example 4.3.3. The offshore facility is 5 t sea and 2 miles along the shoreline from thenews 8.) A rancher has 1000 feet of fencing in which to construct It costs $50,000 per mile to lay a power line under- t, equally size andat dimensions ground and $80,000 to run the line underwater. should these pens have to maximize the enclosed area? How much of the power line should be run underground to minimize the overall costs? 15. A woman throws a stick into a lake for her dog to fetch; the stick is 20 feet down the shore line and 15 feet into the water from there. The dog may jump directly into the wa- ter and swim, or run along the shore line to get closer to the stick before swimming. The dog runs about 22ft/s and 9. A standard soda can is roughly cylindrical and holds 355cm swims about 1.5ft/s. of liquid. What dimensions should the cylinder be to min How far along the shore should the dog run to minimize imize the material needed to produce the can? Based on get to the stick ? (Hint: the figure from the time it takes to get to the st your dimensions, determine whether or not the standard Example 4.3.3 can be useful.) can is produced to minimize the material costs. 16. A woman throws a stick into a lake for her dog to fetch; 10. Find the dimensions of a cylindrical can with a volume of the stick is 15 feet down the shore line and 30 feet into the 206in that minimizes the surface area. water from there. The dog may jump directly into the wa- ter and swim, or run along the shore line to get closer to The "#10 can"is a standard sized can used by the restau- the stick before swimming. The dog runs about 22ft/s and rant industry that holds about 206in with a diameter of 6 swims about 1.5ft/s. 2/16in and height of 7in. Does it seem these dimensions How far along the shore should the dog run to minimize the were chosen with minimization in mind? time it takes to get to the stick? (Google "calculus dog" to learn more about a dog's ability to minimize times.) 11. The United States Postal Service charges more for boxes whose combined length and girth exceeds 108" (the 17. What are the dimensions of the rectangle with largest area "length" of a package is the length of its longest side; the that can be drawn inside the unit circle? girth is the perimeter of the cross section, i.e., 2w + 2h)