Mini-Lecture 2.6 Mathematical Models; Building Functions (2.6) Learning Objectives: 1. Build and Analyze Functions Examples: 1. A rectangle is inscribed in a circle of radius 3 centered at the origin. Let P ( x, y ) be a point in quadrant I that is a vertex of the rectangle and is on the circle. Express the area A of the rectangle as a function of x and the perimeter P of the rectangle as a function of x. Find the maximum area. For what value of ݔis the area the largest? For what value of ݔis the perimeter the largest? 2. Let P ( x, y ) be a point on the graph of y x 3 . Express the distance d from P to the point (2, 0) as function of x. What is d if x 1 ? For what values of ݔis the distance ݀ the smallest? 3. A right triangle has one vertex on the graph of y 16 x 2 , x 0 , at ( x, y ) , another at the origin, and the third on the positive x-axis at ( x, 0) . Express the area A of the triangle as a function of x. 4. An open box with a square base is to be made from a square piece of cardboard 16 inches on a side by cutting out a square from each corner and turning up the sides. Express the volume V of the box as a function of the length x of the side of the square cut from each corner. Find the volume if a 2-inch square is cut out. 5. A media company is going to install cable from a house to their connection box B. The house is located at one end of a driveway 7 miles back from a road cable (see diagram). The other end of the 7 mi cable B x driveway and the nearest connection 25 mi box are on the same road, 25 miles apart. The cost of installing the cable is $656 per mile off the road and $375 per mile along the road. Let x be the distance from where the driveway meets the road to where the cable comes to the road. Develop a function C(x) that expresses the total installation cost as a function of x.Now use your calculator to graph C. Use the graph to determine the value of x that will produce the minimum cost. Round to the nearest thousandth of a mile. (Tip: use a window [0, 20, 1] x [ 12,000 , 20,000 , 1000 ].) State the minimum cost for that installation, rounded to the nearest cent.
Teaching Notes:
Have students review basic geometric formulas for perimeter, area, and volume. Encourage students to draw graphs or figures before attempting to work the problems in this section. Emphasize proper function notation when giving answers.
Answer: 1) A( x ) 4 x 9 x , P ( x ) 4 x 4 9 x ; 2
2
2) d ( x )
x 6 x 2 4 x 4 , d 10 ; 1 3 3) A( x ) 8 x x 2 4) V ( x) 4 x 8 x , V (2) 288 in 2
5)
2
C x 656 x 2 49 375(25 x) $13,142.74
