To determine how much higher Player 1 jumped than Player 2, we can use the given hang time formula and the given times for the players.
The hang time formula is:
[tex]\[ t = 2\left(\frac{2h}{32}\right)^{\frac{1}{2}} \][/tex]
First, we need to rearrange this formula to solve for the height [tex]\( h \)[/tex] in terms of the hang time [tex]\( t \)[/tex].
Rearranging the formula:
[tex]\[ t = 2\left(\frac{2h}{32}\right)^{\frac{1}{2}} \][/tex]
[tex]\[ \frac{t}{2} = \left(\frac{2h}{32}\right)^{\frac{1}{2}} \][/tex]
[tex]\[ \left(\frac{t}{2}\right)^2 = \frac{2h}{32} \][/tex]
[tex]\[ \frac{t^2}{4} = \frac{2h}{32} \][/tex]
[tex]\[ \frac{t^2}{4} = \frac{h}{16} \][/tex]
[tex]\[ h = t^2 \times 4 \][/tex]
Now, apply this formula to calculate the height for each player's hang time.
For Player 1 [tex]\( t_1 = 0.9 \)[/tex] seconds:
[tex]\[ h_1 = (0.9)^2 \times 4 \][/tex]
[tex]\[ h_1 = 0.81 \times 4 \][/tex]
[tex]\[ h_1 = 3.24 \text{ feet} \][/tex]
For Player 2 [tex]\( t_2 = 0.8 \)[/tex] seconds:
[tex]\[ h_2 = (0.8)^2 \times 4 \][/tex]
[tex]\[ h_2 = 0.64 \times 4 \][/tex]
[tex]\[ h_2 = 2.56 \text{ feet} \][/tex]
Next, find the difference in the heights:
[tex]\[ \text{Difference in height} = h_1 - h_2 \][/tex]
[tex]\[ \text{Difference in height} = 3.24 \text{ feet} - 2.56 \text{ feet} \][/tex]
[tex]\[ \text{Difference in height} = 0.68 \text{ feet} \][/tex]
Since the problem asks for the answer in inches, we need to convert this difference from feet to inches:
[tex]\[ 1 \text{ foot} = 12 \text{ inches} \][/tex]
[tex]\[ \text{Difference in height in inches} = 0.68 \times 12 \][/tex]
[tex]\[ \text{Difference in height in inches} = 8.16 \text{ inches} \][/tex]
Rounding 8.16 to the nearest inch, we get:
[tex]\[ \text{Difference in height} \approx 8 \text{ inches} \][/tex]
Therefore, Player 1 jumped 8 inches higher than Player 2.
