Question

during a pumpkin launching contest, two pumpkins are launched at the same time. the path of pumpkin a is modeled by the equation \\(y = -0.01x^2 + 0.32x + 4\\). the path of pumpkin b is modeled by the equation \\(y = -0.01x^2 + 0.18x + 5\\). if \\(x\\) represents the distance the pumpkin traveled in feet and \\(y\\) represents height of the pumpkin in feet, what does the intersection point of the equations represent?

the height at which the pumpkins reach their greatest horizontal distance from the launch site
the horizontal distance of the pumpkins from the launch site when they reach the ground
the starting height of the pumpkins when they are at the launch site
the horizontal distance of the pumpkins from the launch site when their heights are the same

Explanation:

Identify the variables and their physical meanings

\(x\) represents the horizontal distance the pumpkin traveled in feet.
\(y\) represents the height of the pumpkin in feet.

Define the meaning of the intersection point

An intersection point \((x, y)\) of the two path equations satisfies both equations simultaneously.
This means at this specific horizontal distance \(x\), both pumpkin A and pumpkin B have the exact same height \(y\).

Evaluate the given options

• Option 1 describes the height at maximum horizontal distance, which is incorrect.
• Option 2 describes the ground landing distance (\(y = 0\)), which is incorrect.
• Option 3 describes the starting height (\(x = 0\)), which is incorrect.
• Option 4 correctly states that the intersection represents the horizontal distance of the pumpkins from the launch site when their heights are the same.

Answer:

• the height at which the pumpkins reach their greatest horizontal distance from the launch site
• the horizontal distance of the pumpkins from the launch site when they reach the ground
• the starting height of the pumpkins when they are at the launch site
• the horizontal distance of the pumpkins from the launch site when their heights are the same (Correct answer)