Throwing Horseshoes

Task

The height (in feet) of a thrown horseshoe $t$ seconds into flight can be described by the expression $$ 1\frac{3}{16} + 18t - 16t^2.$$ The expressions (a)–(d) below are equivalent. Which of them most clearly reveals the maximum height of the horseshoe's path? Explain your reasoning.
 

1. $\displaystyle 1\tfrac{3}{16} + 18t - 16t^2$
2. $\displaystyle -16\left(t-\frac{19}{16}\right)\left(t + \frac{1}{16}\right)$
3. $\displaystyle \frac{1}{16}(19-16t)(16t+1)$
4. $\displaystyle -16\left(t-\frac{9}{16}\right)^2 + \frac{100}{16}$.

IM Commentary

Variations on this problem might ask which expression is useful for finding the starting height of the horseshoe, or the time when it lands.

The task illustrates A-SSE.1a because it requires students to identify expressions as sums or products and interpret each summand or factor.

Solution

(d)

This expresses the height as the sum of a negative number, $-16$, times a squared expression, $\left(t-\frac{9}{16}\right)^2$, plus a positive number, $\frac{100}{16}$. Since the squared expression is always either positive or zero, the value of the entire expression for the height is always less than or equal to $\frac{100}{16}$, and is only equal to zero if the squared expression is zero, when $t = \frac{9}{16}$. So the maximum value is $\frac{100}{16}$.