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.
Real Tutor Solution
Answer
Solution
Reviewed and approved by the UpStudy tutoring team
Explain 
Simplify this solution Bonus Knowledge
To find the approximate location of the red bucket, we need to determine the intersection point of the flights of the snowball and the slope of the hill. This means we need to set the equations equal to each other: \(-0.02 x^{2} + x + 4 = 0.3 x - 7.2\). Rearranging gives us \(-0.02 x^{2} + 0.7x + 11.2 = 0\). Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) (where \(a = -0.02\), \(b = 0.7\), and \(c = 11.2\)), we can find the value of \(x\). Solving this will yield two possible \(x\) values, but we need the corresponding \(y\) values to determine the actual position of the bucket. Once calculated, the intersection point will indicate where the snowball intersects with the slope of the hill, showing where to place the bucket for maximum snowball-catching efficiency! Another approach is to substitute realistic values for \(x\) to find approximate solutions quickly if you don't want to go through all the calculations. Starting with small integer values can help! Check for \(x = 10\) and \(x = 15\) or nearby and compare the corresponding \(y\) values. You can find where the two equations converge visually or via trial and error!
