Instructions: For the following real-world problem, solve using any method. Use what you've learned to determine which method would be best. Put your answer in the context of the problem and determine the appropriate final answer.
Jessica is practicing her dives off the diving board. When she dives into the water, the path her body makes and where she resurfaces can be modeled by the equation [tex]$0=x^2-6 x$[/tex] where [tex]$x$[/tex] is the number of horizontal feet she travels. Solve the equation to find the number of feet she will travel before coming back to the surface of the water.
The two solutions to the equation are: [tex]$x=$[/tex] [$\\square$] or [$\\square$]
Which of your solutions gives the horizontal distance Jessica has traveled when she returns to the surface? [tex]$x=$[/tex] [$\\square$] feet
Answer (2)
Jessica travels 6 feet horizontally before coming back to the surface of the water. The solutions to the equation are x = 0 and x = 6 , where the latter represents the distance traveled. Thus, the final answer is x = 6 feet.
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Factor the quadratic equation: x 2 − 6 x = x ( x − 6 ) = 0 .
Solve for x by setting each factor to zero: x = 0 or x − 6 = 0 .
Determine the solutions: x = 0 or x = 6 .
Choose the solution that represents the horizontal distance Jessica traveled: 6 feet.
Explanation
Understanding the Problem We are given the equation 0 = x 2 − 6 x which models Jessica's dive, where x represents the horizontal distance in feet. We need to solve for x to find the points where Jessica enters and exits the water.
Factoring the Equation To solve the quadratic equation x 2 − 6 x = 0 , we can factor out an x from both terms: x ( x − 6 ) = 0
Solving for x Now, we set each factor equal to zero and solve for x :
x = 0 or x − 6 = 0
Solving the second equation for x , we get:
x = 6
Determining the Distance The two solutions are x = 0 and x = 6 . x = 0 represents the point where Jessica enters the water (at the diving board). x = 6 represents the point where she resurfaces. Since we are looking for the horizontal distance Jessica travels before resurfacing, we choose the positive value x = 6 .
Final Answer Therefore, Jessica travels 6 feet horizontally before coming back to the surface of the water.
Examples
Understanding quadratic equations can help model various real-world scenarios, such as the trajectory of a ball, the path of a projectile, or even the shape of a suspension bridge. In this case, we used a quadratic equation to model the path of a diver. By solving the equation, we found the distance the diver traveled horizontally. This concept can be applied in sports, engineering, and physics to analyze motion and optimize performance.
