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An orange is shot up into the air with a catapult. The function $h$ given by $h(t)=15+60t-16t^2$ models the orange's height, in feet, $t$ seconds after it was launched. Select all the true statements about the situation.

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Write a linear equation from a slope and y-intercept

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Q. An orange is shot up into the air with a catapult. The function

h

given by

h(t)=15+60t-16t^2

models the orange's height, in feet,

t

seconds after it was launched. Select all the true statements about the situation.

Identify Function Components: Identify the function and its components. $\newline$ The function given is $h(t) = 15 + 60t - 16t^2$ . This is a quadratic equation where: $\newline$ - $15$ is the initial height (in feet) from which the orange is launched. $\newline$ - $60t$ represents the initial velocity impact on height per second. $\newline$ - $-16t^2$ represents the acceleration due to gravity impacting the height.
Determine Initial Height: Determine the initial height of the orange at $t = 0$ seconds. $\newline$ Substitute $t = 0$ into the function: $\newline$ $h(0) = 15 + 60(0) - 16(0)^2 = 15$ feet. $\newline$ This means the orange starts $15$ feet above the ground.
Calculate Vertex for Maximum Height: Calculate the vertex of the parabola to find the maximum height and the time it occurs. $\newline$ The vertex formula for a parabola given by $ax^2 + bx + c$ is $t = -\frac{b}{2a}$ . $\newline$ Here, $a = -16$ and $b = 60$ . $\newline$ $t = -\frac{60}{2*(-16)} = -\frac{60}{-32} = 1.875$ seconds. $\newline$ Substitute $t = 1.875$ back into the height function to find the maximum height: $\newline$ $h(1.875) = 15 + 60(1.875) - 16(1.875)^2$ $\newline$ $= 15 + 112.5 - 56.25 = 71.25$ feet.

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