Question

melon incoming!
the target is 40 feet away. to travel that far, your melon will need to have a maximum height of 12 feet. use the graphing tool to explore the parabolic path of your melon, complete your assignment, and splat that target!

Explanation:

To determine the equation of the parabola for the melon's path, we know it's a projectile motion (parabola) opening downward, with vertex at \((20, 12)\) (since the maximum height is at the midpoint of the horizontal distance, \(40/2 = 20\)) and passing through \((0, 0)\) (launch point) and \((40, 0)\) (target point).

Step 1: Recall the vertex form of a parabola

The vertex form of a parabola is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. Here, \(h = 20\), \(k = 12\), so the equation becomes \(y = a(x - 20)^2 + 12\).

Step 2: Find the value of \(a\)

We know the parabola passes through \((0, 0)\). Substitute \(x = 0\) and \(y = 0\) into the equation:

$$
LATEXBLOCK0
$$

Step 3: Write the final equation

Substitute \(a = -\frac{3}{100}\) back into the vertex form:

$$ y = -\frac{3}{100}(x - 20)^2 + 12 $$

If we expand this, we get:

$$
LATEXBLOCK1
$$

Answer:

The equation of the parabola representing the melon's path is \(y = -\frac{3}{100}(x - 20)^2 + 12\) (or \(y = -\frac{3}{100}x^2+\frac{6}{5}x\)). To graph this, plot the vertex \((20, 12)\) and the roots \((0, 0)\) and \((40, 0)\), then draw the parabola opening downward.