SECTION 2.6 Mathematical Models: Building Functions 129 3. Let P x y , ( ) = be a point on the graph of y x. = (a) Express the distance d from P to the point 1, 0 ( ) as a function of x. (b) Use a graphing utility to graph d d x . ( ) = (c) For what values of x is d smallest? (d) What is the smallest distance? 4. Let P x y , ( ) = be a point on the graph of y x 1 . = (a) Express the distance d from P to the origin as a function of x. (b) Use a graphing utility to graph d d x . ( ) = (c) For what values of x is d smallest? (d) What is the smallest distance? 5. A right triangle has one vertex on the graph of y x x , 0, 3 = > at x y , , ( ) another at the origin, and the third on the positive y-axis at y 0, , ( ) as shown in the figure. Express the area A of the triangle as a function of x. 6. A right triangle has one vertex on the graph of y x x 9 , 0, 2 = − > 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. 7. A rectangle has one corner in quadrant I on the graph of y x 16 ,2 = − another at the origin, a third on the positive y-axis, and the fourth on the positive x-axis. See the figure. (a) Express the area A of the rectangle as a function of x. (b) What is the domain of A? (c) Graph A A x . ( ) = For what value of x is A largest? (d) What is the largest area? 8. A rectangle is inscribed in a semicircle of radius 2. See the figure. Let P x y , ( ) = be the point in quadrant I that is a vertex of the rectangle and is on the circle. (a) Express the area A of the rectangle as a function of x. (b) Express the perimeter p of the rectangle as a function of x. (c) Graph A A x . ( ) = For what value of x is A largest? (d) Graph p p x . ( ) = For what value of x is p largest? (e) What is the largest area? What is the largest perimeter? 9. A rectangle is inscribed in a circle of radius 2. See the figure. Let P x y , ( ) = be the point in quadrant I that is a vertex of the rectangle and is on the circle. (a) Express the area A of the rectangle as a function of x. (b) Express the perimeter p of the rectangle as a function of x. (c) Graph A A x . ( ) = For what value of x is A largest? (d) Graph p p x . ( ) = For what value of x is p largest? 10. A circle of radius r is inscribed in a square. See the figure. (a) Express the area A of the square as a function of the radius r of the circle. (b) Express the perimeter p of the square as a function of r. 11. Geometry A wire 10 meters long is to be cut into two pieces. One piece will be shaped as a square, and the other piece will be shaped as a circle. See the figure. (a) Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the square. (b) What is the domain of A? (c) Graph A A x . ( ) = For what value of x is A smallest? 12. Geometry A wire 10 meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped as a circle. (a) Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the equilateral triangle. (b) What is the domain of A? (c) Graph A A x . ( ) = For what value of x is A smallest? 13. Geometry A wire of length x is bent into the shape of a circle. (a) Express the circumference C of the circle as a function of x. (b) Express the area A of the circle as a function of x. 14. Geometry A wire of length x is bent into the shape of a square. (a) Express the perimeter p of the square as a function of x. (b) Express the area A of the square as a function of x. 15. Geometry A semicircle of radius r is inscribed in a rectangle so that the diameter of the semicircle is the length of the rectangle. See the figure. (a) Express the area A of the rectangle as a function of the radius r of the semicircle. (b) Express the perimeter p of the rectangle as a function of r. 16. Geometry An equilateral triangle is inscribed in a circle of radius r. See the figure. Express the circumference C of the circle as a function of the length x of a side of the triangle. r x Hint:First show that 3 . 2 2 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 17. Geometry An equilateral triangle is inscribed in a circle of radius r. See the figure in Problem 16. Express the area A within the circle, but outside the triangle, as a function of the length x of a side of the triangle. 18. Uniform Motion Two cars leave an intersection at the same time. One is headed south at a constant speed of 30 miles per hour, and the other is headed west at a constant speed of 40 miles per hour (see the figure). Build a model that expresses the distance d between the cars as a function of the time t. [See Hint on next page]. y (0, y) (0, 0) x y 5 x 3 (x, y) x y 4 (x, y) y 5 16 2 x 2 (0, 0) 8 16 x y 2 P 5 (x, y) 4 2 x 2 y 5 22 x y 2 P 5 (x, y) x 2 1 y 2 5 4 22 2 22 r 4x x 10 2 4x 10 m r x x x r
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