Dolphin jump
[tex]y=-16 x^2+32 x-10[/tex]
This parabola has $\square$ [tex]x[/tex]-intercepts, representing the times when the dolphin's height above water is $\square$ feet.
Answer (1)
The problem asks for the number of x-intercepts of the parabola y = − 16 x 2 + 32 x − 10 .
Calculate the discriminant D = b 2 − 4 a c with a = − 16 , b = 32 , and c = − 10 .
Find D = ( 32 ) 2 − 4 ( − 16 ) ( − 10 ) = 384 .
Since 0"> D > 0 , there are two x-intercepts, meaning the dolphin's height is 0 feet at two points in time.
The parabola has 2 x -intercepts, representing the times when the dolphin's height above water is 0 feet.
Explanation
Understanding the Problem We are given the equation of the dolphin's jump as y = − 16 x 2 + 32 x − 10 , where y represents the height above water and x represents the time. We want to find the number of x -intercepts of this parabola, which represent the times when the dolphin's height above water is 0 feet.
Using the Discriminant To find the number of x -intercepts, we need to find the number of real solutions to the equation − 16 x 2 + 32 x − 10 = 0 . We can use the discriminant to determine the number of real solutions. The discriminant is given by D = b 2 − 4 a c , where a = − 16 , b = 32 , and c = − 10 .
Calculating the Discriminant Now, let's calculate the discriminant: D = b 2 − 4 a c = ( 32 ) 2 − 4 ( − 16 ) ( − 10 ) = 1024 − 640 = 384
Determining the Number of x-intercepts Since 0"> D = 384 > 0 , there are two distinct real solutions for x . This means the parabola has two x -intercepts. These x -intercepts represent the times when the dolphin's height above water is 0 feet.
Final Answer Therefore, the parabola has 2 x -intercepts, representing the times when the dolphin's height above water is 0 feet.
Examples
Understanding the trajectory of a dolphin's jump can be modeled using quadratic equations, similar to how engineers design the path of projectiles or athletes optimize their movements. For example, a coach might use this equation to analyze the dolphin's jump, determining the time it reaches a certain height or the distance it travels. By adjusting parameters in the equation, they can predict how changes in the dolphin's initial velocity or angle of launch would affect its jump. This helps in training and improving performance.
