91Ó°ÊÓ

A problem situation is given. (a) Decide what is being asked for, and label the unknown quantities. (b) Translate the verbal statements in the problem and the relationships between the known and unknown quantities into mathematical language, using a table as in Examples \(1-3\) (pages \(101 103\) ). The table is provided in Exercises 1 and \(2 .\) You need not find an equation to be solved. A triangle has area 96 square inches, and its height is twothirds of its base. What are the base and height of the triangle?

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$x^{3}+4 x^{2}+10 x+15=0 ; \quad(-3,-2)$$

Set up the problem by labeling the unknowns, translating the given information into mathematical language, and finding an equation that will produce the solution to the problem. You need not solve this equation. A corner lot has dimensions 25 by 40 yards. The city plans to take a strip of uniform width along the two sides bordering the streets to widen these roads. How wide should the strip be if the remainder of the lot is to have an area of 844 square yards?

In the remaining exercises, solve the applied problems. A radiator contains 10 quarts of fluid, \(30 \%\) of which is antifreeze. How much fluid should be drained and replaced with pure antifreeze so that the new mixture is \(40 \%\) antifreeze?

A farmer has 1800 feet of fencing. He plans to enclose a rectangular region bordering a river (with no fencing needed along the river side). What dimensions should he use to have an enclosure of largest possible area?

A rectangular field will be fenced on all four sides. Fencing for the north and south sides costs \(\$ 5\) per foot and fencing for the other two sides costs \(\$ 10\) per foot. What is the maximum area that can be enclosed for \(\$ 5000 ?\)

In the remaining exercises, solve the applied problems. A train leaves New York for Boston, 200 miles away, at 3: 00 P.M. and averages 75 mph. Another train leaves Boston for New York on an adjacent set of tracks at 5: 00 P.M. and averages 45 mph. At what time will the trains meet?

In the remaining exercises, solve the applied problems. A rectangle is twice as long as it is wide. If it has an area of 24.5 square inches, what are its dimensions? [Hint: See Example \(2 .]\)

In the remaining exercises, solve the applied problems. Two cars leave a gas station at the same time, one traveling north and the other south. The northbound car travels at 50 mph. After 3 hours, the cars are 345 miles apart. How fast is the southbound car traveling?

A cylindrical waste container with no top, a diameter of at least 2 feet, and a volume of 25 cubic feet is to be constructed. What should its radius be if (a) 65 square feet of material are to be used to construct it? (b) the smallest possible amount of material is to be used to construct it? In this case, how much material is needed?