When solving application problems, you should have some procedure that you follow (draw a diagram, define unknown values, setup equations, solve)


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1 When solving application problems, you should have some procedure that you follow (draw a diagram, define unknown values, setup equations, solve) Once again when solving applied problems I will include questions in my notes to help setup equations. Keep in mind that the questions below in red will not appear in the homework, or on quizzes and exams; these are simply questions that you should be asking yourself when you see problems like these in order to help get equations that you can solve. Also keep in mind the various method for solving quadratic equations:  solve by factoring o write the equation as a polynomial set equal to zero, factor, use Zero Factor Theorem  solve as a special quadratic equation (perfect squares) o isolate the perfect square and take the square root of both sides of the equation  solve by completing the square o divide by the leading coefficient, isolate the constant, add half the square of the coefficient of x to both sides of the equation, factor, solve as a special quadratic equation  solve using the quadratic formula o identify a, b, c, plugin to the formula, simplify completely
2 Example 1: A rock is thrown directly upward with an initial velocity of 96 feet per second from a cliff 200 feet above a beach. The number of feet above the ground (f), after t seconds, is given by the equation f = 16t t a. When will the rock be 328 feet above the ground? Which variable will 328 replace in the equation f = 16t t + 200, feet above the ground f or time t? b. When will the rock hit the ground? (round to the nearest hundredth of a second). When the rock is on the ground, what is the value of f?
3 Example 2: A 12 x 15 inch picture is mounted on a wall and surrounded by a border of uniform width. The total area of the picture with the border around it is 304 in 2. What is the width of just the border? What is the width of the border? (if you don t know, assign a variable to represent this value) Draw a diagram of the picture with the border around it (a rectangle inside of another rectangle), and list the dimensions of the inner rectangle and the dimensions of the outer rectangle. Write an equation for the total area of the picture with the border around it, and use that equation to solve for your variable. total width total length = total area
4 Example 3: The blueprints for a new bathroom show a raised, rectangular bathtub with an area of 9 ft 2. The blueprints show that the raised tub will be surrounded by tile on all four sides; half a foot of tile on each side of the rectangle tub as well as at the bottom of the rectangle, and 1 foot of tile at the top of the rectangle. The total length of the raised tub including the tile will be twice the total width. Find the dimensions of the bathtub only. Draw a diagram of the bathtub and the surrounding tile (a rectangle inside of another rectangle), and list the dimensions of the inner rectangle and the dimensions of the outer rectangle. Write an equation for the area of the bathtub, and use that equation to solve for your variable. length of tub width of tub = area of tub
5 Example 4: A gardener plans to enclose a rectangular region using fencing on three sides and part of a shed on the fourth side. The side parallel to the shed needs to be twice the length of an adjacent side. If the area of the region is 6050 ft 2, how many feet of fencing should be purchased? Draw a diagram of the region, and list the length of each side. Write an equation for the area of the rectangular region. length of the region width of the region = area
6 Example 5: A family plans to have the hardwood floors in their square dining room refinished, and new baseboards installed. The cost of refinishing the hardwood floors is $4.50 per square foot and the cost of purchasing and installing the new baseboard is $12.00 per linear foot. If the family paid $1,224, what are the dimensions their dining room (width and length)? Draw a diagram of the square dining room and labels the sides. (if you don t know the side length, assign a variable to represent this value) The floor represent the area of the room, so write an equation for the cost to refinish the hardwood floors based on the area of the room. The baseboard represent the perimeter of the room, so write an equation for the cost to install new baseboards based on the perimeter of the room. Write an equation for the total cost of the job, and use that equation to solve for your variable. cost of floors + cost of baseboard = total cost
7 Example 6: A rectangular piece of cardboard is 2 inches longer than it is wide. From each corner, a 2 2 inch square is cut out and the flaps are then folded up to form an open box. If the volume of the box is 96 in 3, find the length and width of the original piece of cardboard. What is the width of the rectangular piece of cardboard? (if you don t know, assign a variable to represent this value) What is the length of the rectangular piece of cardboard? (remember that the length is 2 inches more than the width) Draw a diagram of the rectangular piece of cardboard and list its dimensions. Then draw the 2 2 inch squares in each corner and the flaps, and list the dimensions of the box. Keep in mind that the width and length of the box will be LESS than the width and length of the cardboard. Write an equation for the total volume of the box, and use that equation to solve for your variable. box width box length box height = box volume
8 Example 7: Two trains leave a station at 11:00am. One train travels north at a rate of 45 mph and another travels east at a rate of 60 mph. Assuming they do not stop, at what time will the trains be 200 miles apart? Draw a diagram of the distances traveled by each train, and the distance between them. This should produce a right triangle, which means we can use the Pythagorean Theorem to write an equation in terms of the length (distance) of each side. (distance traveled north ) 2 + (distance traveled east) 2 = (distance between them) 2
9 Answers to Exercises: 1a. t = 2, 4 seconds ; 1b. t = 7.64 seconds ; 2. 2 inches ; 3. 2 ft 4.5ft ; feet ; ft 12 ft ; ; 7. 1: 40pm ;
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