Understanding Calculus BY a calculus student Michelle Oglesby
The Problem <ul><li>A car traveling up a mountain slope stops when a stray child runs into the street. The mountain road is one way, has no side walks and to the right is a straight down cliff. The child only out to retrieve the ball from its demolition soon departs the street and the car starts again. A bicyclist on the other side of the mountain path approaches the top of a blind pass and begins departing down the mountain. They collide and the person of the bicycle is doomed to a life in a wheel chair. The report from the police is listed below including the schematics of the hill, the average rate of the speed of the bicycle and car. </li></ul>http://www.youtube.com/watch?v=wjiB8DeKwv4
<ul><li>The equation to the hill, found by points, is </li></ul><ul><li>y= -.1x^2 -.15x + 4.5 </li></ul><ul><li>The distance between where the car stopped and the bike was positioned before descending the hill was at 7 miles apart. (The car was at x=-3.5 on a smaller scaled graph; the bike was at the top of the slope) </li></ul>The speed of the car 70 miles an hour until stopping point when the child is seen. When the car begins again the initial speed is at a rate of 6 miles a second. The car on reaching the top of the hill was at a speed of 25 miles an hour but on the way down hill the biker’s speed increased at .5 miles a second. The bike had traveled 5 seconds before being hit. The bicyclist is approximately 6 feet off the ground and can only see at a 45 degree angel
Problem One : <ul><li>You’re the defense attorney. Prove the driver is innocent. If you fail to complete this simple task with hard evidence than the consequences is a life in prison for the poor young driver in an unfortunate accident. </li></ul>First to find out if the point of impact would actually happen at the right spot based on the suppose position of the car and the bicycle.
You need to find the integral to find the equation to give distance. The car is6x. The anti for bike is .5x .5x + 6x = 7 miles ( the distance separating) X = 1.077 so the car and the bike would have collide in approximately 1 sec. The position of the car at this point was at 6 * 1.077 = 6.4625 The position of the bike at this point was .5 * 1.077 = .5385
The car in one second would have only been going 6 miles a second since in 1 seconds that’s the only amount of speed that could have been picked up but the position would have been 6 miles away from the original stopping point and the car to travel that far at the steady increase would have been going 36 miles an hour and therefore the data was recorded wrong and can be proven wrong and the driver is innocent .
Problem Two: <ul><li>Although with your proof you will be able to prove the case innocent, it turns out there's more money in locking away the poor and unfortunate. Prove the person is guilty. </li></ul>Find the maximum of the hill. At the maximum prove the car would have been able to see the biker at the maximum point and have enough time to stop using the rate of the increase and stopping of the car.
To find the maximum of the hill find the 0, and undefined points of the hill. -.1x^2 -.15x + 4.5 the derivative is -.2x - .15 There are no undefined points -.2x - .15 = 0 x=-.75 -.1(-.15) ^2 - .15(-.75) + 4.5 = 4.55625 Find the slope of the point the car is at. Take the derivative of the original equation -.2x-.15 plug in the point the car is at = 3.5 The slope of the line is m = .55 Y= mx + b 3.8= .55(-3.5) + b b= 5.725 y= .55x + 5.725
Draw the triangle created by y= 4.55625 and y=.55x + 5.725 and the horizontal line x = -3.5 Find the angel created at the intersection of the vertical line and the slope of the car tan x = .75625/1.375 Angle = 28.81 The angle of the car would be able to see the bicyclist and should have been able to stop (3.5, 4.55625) (3.5, 3.8) (-2.125, 4.55625)
<ul><li>The angle of the car would be able to see the bicyclist and should have been able to stop </li></ul><ul><li>Congratulations Your Rich and know Calculus </li></ul>