32. Caleb and a friend are throwing snowbals at a red bucket on a hill. The flight of a snowball can be modeled by the equation $$ y=-0.02 x^{2}+x+4 $$, where $$ x $$ is the horizontal distance and $$ y $$ is the height of the snowball in feet. The hill's slope can be modeled by the equation $$ y=0.3 x-7.2 $$. What is the approximate location where the red bucket should be placed? Justify your answer.

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To determine where Caleb and his friend should place the red bucket, we need to find the point where the snowball lands on the hill. This occurs when the height of the snowball's flight equals the height of the hill at the same horizontal distance.

**Given Equations:**
1. **Snowball's Flight:** $$ y = -0.02x^2 + x + 4 $$
2. **Hill's Slope:** $$ y = 0.3x - 7.2 $$

**Steps to Find the Intersection Point:**

1. **Set the Equations Equal:**
   $$
   -0.02x^2 + x + 4 = 0.3x - 7.2
   $$

2. **Rearrange the Equation:**
   $$
   -0.02x^2 + x - 0.3x + 4 + 7.2 = 0
   $$
   $$
   -0.02x^2 + 0.7x + 11.2 = 0
   $$

3. **Eliminate Decimals by Multiplying by 100:**
   $$
   -2x^2 + 70x + 1120 = 0
   $$

4. **Simplify the Quadratic Equation:**
   $$
   x^2 - 35x - 560 = 0
   $$

5. **Solve Using the Quadratic Formula:**
   $$
   x = \frac{35 \pm \sqrt{35^2 + 4 \cdot 560}}{2} = \frac{35 \pm \sqrt{1225 + 2240}}{2} = \frac{35 \pm \sqrt{3465}}{2}
   $$
   $$
   \sqrt{3465} \approx 58.865
   $$
   $$
   x = \frac{35 + 58.865}{2} \approx \frac{93.865}{2} \approx 46.93
   $$
   $$
   x = \frac{35 - 58.865}{2} \approx \frac{-23.865}{2} \approx -11.93 \quad (\text{Not physically meaningful})
   $$

6. **Determine the Valid Solution:**
   The negative value doesn’t make sense in this context, so we take the positive solution:
   $$
   x \approx 46.93 \text{ feet}
   $$

**Conclusion:**
The red bucket should be placed approximately **47 feet** horizontally from the point where Caleb throws the snowball. This placement ensures that the snowball, following its parabolic trajectory, will land precisely into the bucket on the hill.

**Final Answer:**

After solving, the bucket should lie about 47 feet horizontally from the throw point. Thus, it should be placed approximately 47 ft from where Caleb throws the snowball.
The red bucket should be placed approximately 47 feet horizontally from where Caleb throws the snowball.

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