FC: Hill Word Problem | By Corey Andrade
1: Question 17: Calvin Butterball is driving along the highway. He starts up a long, straight hill. 114 meters from the bottom of the hill Calvin's car runs out of gas. He doesn't put on the brakes. So the car keeps rolling for a while, coasts to a stop, then starts rolling backwards. He finds that his distances from the bottom of the hill 4 and 6 seconds after he runs out of gas are 198 and 234 meters respectively. - There are three ordered pairs of time and distance in the information given above. What are they? - Write an equation expressing Calvin's distance from the bottom of the hill in terms of the number of seconds which have elapsed since he ran out of gas. Assume that a quadratic function is a reasonable mathematical model. - Use the model to predict Calvin's distance from the bottom of the hill 10 and 30 seconds after he ran out of gas - Last Chance Texaco Station is located 400 meters from the bottom of the hill. Based on your model, will Calvin's car reach the station before it stops and starts rolling backwards? Justify your answer.
2: Three ordered pairs of time | The three ordered pairs of time are: 0, 114 4,198 and 6, 234. The problem is only concerned with when the car runs out of gas, which is at 114 meters from the bottom of the hill. Because 114 is where the car first runs out of gas, that will be the starting point. At the starting point, the car has not moved therefore it's time will be 0 seconds at 114 meters. The problem mentions two more critical points of distance and time, which will be the other two ordered pairs. After 4 seconds, the car has traveled 198 meters and after 6 seconds, it traveled 234 meters.
3: The equation | In y=, Y1 should have the equation and Y2 should equal zero | To get the equation that expresses Calvin's distance from the bottom of the hill, put your ordered pairs in a graphing calculator. On your calculator click Stat, Edit..., put the X values (0,4,6) into L1, put the Y values (114,198,234) into L2, Stat, Scroll to Calc, Quadreg, 2nd 1 (L1), comma, 2nd 2 (L2), comma, Vars, scroll to Y-Vars, function..., Y1, enter You should get a=-.5, b=23, and c=114 therefore your equation is: y=-.5x^2+23x+114
4: Making a prediction | By zooming out enough, you can notice the graph of this equation is a parabola. By looking at the table of this graph on your graphing calculator (2nd, graph) you can see the distance traveled under Y1 and the amount of seconds elapsed under X. To find Calvin's distance from the bottom of the hill after 10 seconds, just scroll down to 10 in the X column. To find his distance after 30 seconds, scroll down to 30 under the X column. You should have that after 10 seconds, Calvin traveled 294 meters and after 30 seconds, he traveled 354 meters. By using this table, you can predict how far Calvin traveled in any amount of time.
5: Last Chance Texaco station | Last Chance Texaco Station is located 400 meters from the bottom of the hill. Based on the quadratic model, Calvin will not travel the distance of 400 meters. By going back to the table (2nd, graph) and scrolling down the X column, at 23 seconds, Calvin's car has traveled 378.5 meters. At 24 seconds, the car has now traveled 378 meters and then as time goes on, the distance keeps decreasing, showing that at this point, his car is rolling backwards. Since the station is 400 meters away and the car's maximum distance is 378.5 meters, Calvin never reaches the station before his car stops and starts rolling backwards.