Dolphin jump
[tex]y=-16 x^2+32 x-10[/tex]
This parabola has x-intercepts, representing the times when the dolphin's height above water is feet.
Answer (2)
The equation of the dolphin's jump is y = − 16 x 2 + 32 x − 10 .
Calculate the discriminant: D = b 2 − 4 a c = ( 32 ) 2 − 4 ( − 16 ) ( − 10 ) = 384 .
Since 0"> D > 0 , there are 2 x-intercepts.
The parabola has 2 x-intercepts, representing the times when the dolphin's height above water is 0 feet. 2
Explanation
Understanding the Problem We are given the equation of a dolphin's jump as y = − 16 x 2 + 32 x − 10 . We need to find the number of x-intercepts of this parabola, which represent the times when the dolphin's height above the water is 0 feet.
Setting up the Discriminant To find the x-intercepts, we need to solve the quadratic equation − 16 x 2 + 32 x − 10 = 0 . We can determine the number of real solutions by calculating the discriminant, D = b 2 − 4 a c , where a = − 16 , b = 32 , and c = − 10 .
Calculating the Discriminant Let's calculate the discriminant: D = ( 32 ) 2 − 4 ( − 16 ) ( − 10 ) = 1024 − 640 = 384
Determining the Number of x-intercepts Since the discriminant 0"> D = 384 > 0 , there are two distinct real roots (x-intercepts). This means the dolphin's height is 0 feet at two different times.
Conclusion 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 analyzing the path of a ball thrown in the air or designing the curve of a skateboard ramp. By determining the x-intercepts, we find when the dolphin enters and exits the water, which is crucial for studying its aquatic behavior. This mathematical approach is also fundamental in sports science for optimizing athletic performance and in engineering for designing efficient projectile motion.
The equation of the dolphin's jump is y = − 16 x 2 + 32 x − 10 . The discriminant of the equation shows that there are 2 x-intercepts, which represents the times when the dolphin's height above the water is 0 feet. Therefore, the dolphin exits and re-enters the water at two different moments.
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