130 CHAPTER 2 Functions and Their Graphs connection box for the cable is located 5 miles down the road. See the figure. Box Stream House 2 mi x 5 mi (a) If the installation cost is $500 per mile along the road and $700 per mile off the road, build a model that expresses the total cost C of installation as a function of the distance x (in miles) from the connection box to the point where the cable installation turns off the road. Find the domain of C C x . ( ) = (b) Compute the cost if x 1 mile. = (c) Compute the cost if x 3 miles. = (d) Graph the function C C x . ( ) = Use TRACE to see how the cost C varies as x changes from 0 to 5. (e) What value of x results in the least cost? 23. Time Required to Go from an Island to a Town An island is 2 miles from the nearest point P on a straight shoreline. A town is 12 miles down the shore from P. See the figure. 2 mi 12 mi 12 2 x x d2 P d1 Island Town (a) If a person can row a boat at an average speed of 3 miles per hour and the same person can walk 5 miles per hour, build a model that expresses the time T that it takes to go from the island to town as a function of the distance x from P to where the person lands the boat. (b) What is the domain of T? (c) How long will it take to travel from the island to town if the person lands the boat 4 miles from P? (d) How long will it take if the person lands the boat 8 miles from P? 24. Constructing an Open Box An open box with a square base is required to have a volume of 10 cubic feet. (a) Express the amount A of material used to make such a box as a function of the length x of a side of the square base. (b) How much material is required for such a box with a base 1 foot by 1 foot? [Hint: At t 0, = the cars leave the intersection.] N S E W d 19. Uniform Motion Two cars are approaching an intersection. One is 2 miles south of the intersection and is moving at a constant speed of 30 miles per hour. At the same time, the other car is 3 miles east of the intersection and is moving at a constant speed of 40 miles per hour. (a) Build a model that expresses the distance d between the cars as a function of time t. [Hint: At t 0, = the cars are 2 miles south and 3 miles east of the intersection, respectively.] (b) Use a graphing utility to graph d d t . ( ) = For what value of t is d smallest? 20. Inscribing a Cylinder in a Sphere Inscribe a right circular cylinder of height h and radius r in a sphere of fixed radius R. See the figure. Express the volume V of the cylinder as a function of h. [Hint: V r h. 2 π = Note also the right triangle.] Sphere r R h 21. Inscribing a Cylinder in a Cone Inscribe a right circular cylinder of height h and radius r in a cone of fixed radius R and fixed height H. See the figure. Express the volume V of the cylinder as a function of r. [Hint: V r h. 2 π = Note also the similar triangles.] Cone r R H h 22. Installing Cable TV MetroMedia Cable is asked to provide service to a customer whose house is located 2 miles from the road along which the cable is buried. The nearest
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