conds. How long is the baseball in the air? What is the maximum height eball? $\newline$ $-2 x(8 x-45)$ $\newline$ A dolphin jumps out of the water. The quadratic function $y=-16 x^{2}+32 x$ models phin's height in feet above the water after $x$ seconds. How long is the dolphin the water? How high does the dolphin jump? $\newline$ $\frac{-32}{-32}$ $\newline$ The height of a rock thrown off a cliff can be modeled by $h=-16 t^{2}-8 t+120$ , wher height in feet and $t$ is time in seconds. How long

Full solution

Q. conds. How long is the baseball in the air? What is the maximum height eball?

\newline

-2 x(8 x-45)

\newline

A dolphin jumps out of the water. The quadratic function

y=-16 x^{2}+32 x

models phin's height in feet above the water after

x

seconds. How long is the dolphin the water? How high does the dolphin jump?

\newline

\frac{-32}{-32}

\newline

The height of a rock thrown off a cliff can be modeled by

h=-16 t^{2}-8 t+120

, wher height in feet and

t

is time in seconds. How long

Quadratic Formula Definition: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$ .
Dolphin's Jump Equation: For the dolphin's jump, the equation is $y = -16x^2 + 32x$ . Here, $a = -16$ , $b = 32$ , and $c = 0$ .
Applying Quadratic Formula: Plugging the values into the quadratic formula, we get $x = \frac{-32 \pm \sqrt{(32^2 - 4(-16)(0))}}{2(-16)}$ .
Simplifying Square Root: Simplifying inside the square root gives us $x = \frac{-32 \pm \sqrt{1024}}{-32}$ .
Solving for x: Since $\sqrt{1024} = 32$ , the equation becomes $x = (-32 \pm 32) / (-32)$ .
Time in the Air: This gives us two solutions for x: $x = 0$ or $x = 2$ . The dolphin is in the air from $x = 0$ to $x = 2$ seconds.
Finding the Vertex: To find the maximum height, we need to find the vertex of the parabola. The $x$ -coordinate of the vertex is given by $-\frac{b}{2a}$ .
Calculating x-coordinate: Plugging in the values, we get $-b/(2a) = -32/(2(-16)) = 1$ .
Calculating y-coordinate: Now we calculate the y-coordinate of the vertex by plugging $x = 1$ into the equation $y = -16x^2 + 32x$ .
Maximum Height: This gives us $y = -16(1)^2 + 32(1) = -16 + 32 = 16$ feet.
Maximum Height: This gives us $y = -16(1)^2 + 32(1) = -16 + 32 = 16$ feet.So, the dolphin is in the air for $2$ seconds and reaches a maximum height of $16$ feet.