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Test Your Ellipse Skills!

Here is your chance to test your ellipse knowledge and skills after our lesson on the conic. This is an application problem and is used in an attempt to bring real world situations to this area of mathematics. Luckily, just like the parabola, the ellipse is a widely used conic in our lives and around our world. The orbit of earth about the Sun is an ellipse, as well as the shape of your local track field. In this case, we will examine the elliptical shape of a Indy Racing League race track, something many people, especially in the United States, are familiar with.

The Indy Racetrack

Indy car racing has been an American activity for almost one hundred years, and still thrives as a prominent sporting event. One of the most famed races, the Indy 500, is one of the most recognized and respected racing events to occur each year. Recently, the Indianapolis Motor Speedway, also known as the "Brickyard," received a makeover and renovation in order to prepare for the challenges of the new millennium and return of Formula One racing to the United States. In this question we will attempt to help out the architects and engineers in their quest for the ultimate racetrack. As of now, we know that they want to make the track 1100 feet long and 500 feet wide.

The Question!

The designers have decided that they need to add a few improvements to the track in order to aid in the supremacy of its technology. In attempting to achieve this quest, the designers think it would be excellent to include a brand new pit lane, to harness the new fire extinguishing stations and computer controlled fuel delivery pumps. They also want to incorporate dual scoreboards at the foci of the track so all spectators can know which driver holds what position. This is where they need our help. Our duty is to find the length of the pit lane, which they want 150 feet in front of the center of the track, so that the correct quantity of materials can be ordered. We also need to locate the foci so that construction of the scoreboard towers can begin as soon as possible. I have conveniently noted all of the information we need in order to complete the task in the diagram on the right. Lets get started!

The Answer!

In order to solve this problem we must be aware of that we are using an ellipse equation with an x-axis major axis. This means our equation looks like this: "(x-h)^2/a^2) + (y-k)^2/b^2) = 1." To find the foci, where we must put up the high-tech scoreboard towers, we have to first find a and b. Since a is the length of the center to either vertex, a equals 1100 ft /2 = 550 ft. B is the distance from the center to either minor vertex, thus b equals 500 ft / 2 = 250ft. Now we just have to use our equation "b^2 = a^2 - c^2" and solve for c: c^2 = 550^2 - 250^2 = 240000. Taking the square root of that number leaves us with c = 489.89 feet, thus each tower must be 489.89ft left or right of the center of the ellipse. To solve for the length of the pit lane, 150ft in front of the center of the track, we must use our original equation and plug in 150 for our y value. This gives us "(x)^2/550^2) + (150)^2/250^2) = 1." Now we must use algebra to solve for x, leaving us with the answer x = 440 feet. This is only one side right or left of the center, thus we must double our answer of 440 to get a total length across. Our pit lane must be 880 feet long to meet their requirements.