Questions: A rocket is launched in the air. Its height in feet is given by h=-16 t^2+24 t where t represents the time in seconds after launch. Interpret the coordinates of the vertex in context.

A rocket is launched in the air. Its height in feet is given by h=-16 t^2+24 t where t represents the time in seconds after launch. Interpret the coordinates of the vertex in context.

Transcript text: A rocket is launched in the air. Its height in feet is given by $h=-16 t^{2}+24 t$ where $t$ represents the time in seconds after launch. Interpret the coordinates of the vertex in context.

Solution

Solution Steps

Step 1: Identify the Vertex of the Parabola

The height of the rocket is given by the quadratic equation $ h = -16t^2 + 24t $. This is a standard form of a quadratic equation $ ax^2 + bx + c $, where $ a = -16 $, $ b = 24 $, and $ c = 0 $.

The vertex of a parabola given by $ ax^2 + bx + c $ can be found using the formula for the $ t $-coordinate of the vertex:

\[ t = -\frac{b}{2a} \]

Step 2: Calculate the $ t $-coordinate of the Vertex

Substitute $ a = -16 $ and $ b = 24 $ into the formula:

\[ t = -\frac{24}{2 \times -16} = \frac{24}{32} = 0.75 \]

The $ t $-coordinate of the vertex is 0.75 seconds.

Step 3: Calculate the $ h $-coordinate of the Vertex

Substitute $ t = 0.75 $ back into the original equation to find the $ h $-coordinate:

\[ h = -16(0.75)^2 + 24(0.75) \]

Calculate $ (0.75)^2 $:

\[ (0.75)^2 = 0.5625 \]

Substitute back: