Quadratic Baseball Question

Question

The height of a baseball is modeled by the function $h(x)=-0.005x^2+0.3x+1.5$, would an outfielder which is modeled by the function $m(x)=-0.06x+5.6$ where $50 \le x \le 90$, catch the ball?

Answer 1

Answer to your question:

Look at this graph of the equations. The blue represents the height of the baseball, and the red represents something about the outfielder. Notice one of the solutions is well outside the accepted range for $x$.

Some things that don't sit right with me:

Firstly, the height of a baseball is governed (without considering air resistance) by the equation $y = -\frac{g}{2}t^2 + v_0t + y_0$, where $g$ is the gravitational acceleration. So, this occurs on some planet (or with some units) where gravitational acceleration is $-0.005$. Whoever set the problem should have attached some units, or used an accurate gravitational acceleration.

Also, the process of setting the expressions equal to each other is very wrong. One function returns the elevation of the baseball, the other is the position of the outfielder. In order for setting them equal to make sense, we either need the function for the outfielder to return the outfielder's elevation with respect to time (which is very odd, to say the least). Or, we can have the function for the baseball return its distance in the horizontal plane. This makes more sense, but is against the problem definition.

Answer 2

It is more of a square function question than a baseball question. Although I had not expected this kind of question before clicked it, I want to discuss this. The function of baseball height is open downward, in other words, it has a maximum height. But this one is not necessarily covered to see if an outfielder is able to catch the ball. Let's suppose that f(x)=h(x)-g(x). Since f(x)=h(x)-g(x) is continuous between 50 and 100, it is reasonable to presume that there is a x intercept if f(50)*f(100)<0. Hence, you can conclude that there is an x value, which makes f(x)=0, from 50 to 100 because f(50) is above 0 and f(100) is below 0. So the answer is yes.