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Sagot :
To determine the height difference between Player 1 and Player 2, we will solve for their jumping heights using the given equation for hang time:
[tex]\[ t = 2 \left(\frac{2h}{32}\right)^{\frac{1}{2}} \][/tex]
where [tex]\( t \)[/tex] is the hang time in seconds, and [tex]\( h \)[/tex] is the height in feet.
### Step 1: Rearrange the equation to solve for [tex]\( h \)[/tex]
First, we manipulate the equation to isolate [tex]\( h \)[/tex] on one side:
[tex]\[ t = 2 \left(\frac{2h}{32}\right)^{0.5} \][/tex]
Dividing both sides by 2:
[tex]\[ \frac{t}{2} = \left(\frac{2h}{32}\right)^{0.5} \][/tex]
Squaring both sides to remove the square root:
[tex]\[ \left(\frac{t}{2}\right)^2 = \frac{2h}{32} \][/tex]
Multiply both sides by 32 to solve for [tex]\( 2h \)[/tex]:
[tex]\[ 32 \left(\frac{t}{2}\right)^2 = 2h \][/tex]
Dividing both sides by 2 gives [tex]\( h \)[/tex]:
[tex]\[ h = \frac{32}{2} \left(\frac{t}{2}\right)^2 \][/tex]
This simplifies to:
[tex]\[ h = 16 \left(\frac{t}{2}\right)^2 \][/tex]
### Step 2: Calculate the heights for Player 1 and Player 2
For Player 1 with [tex]\( t_1 = 0.9 \)[/tex] seconds:
[tex]\[ h_1 = 16 \left(\frac{0.9}{2}\right)^2 \][/tex]
[tex]\[ h_1 = 16 \left(0.45\right)^2 \][/tex]
[tex]\[ h_1 = 16 \times 0.2025 \][/tex]
[tex]\[ h_1 = 3.24 \text{ feet} \][/tex]
For Player 2 with [tex]\( t_2 = 0.8 \)[/tex] seconds:
[tex]\[ h_2 = 16 \left(\frac{0.8}{2}\right)^2 \][/tex]
[tex]\[ h_2 = 16 \left(0.4\right)^2 \][/tex]
[tex]\[ h_2 = 16 \times 0.16 \][/tex]
[tex]\[ h_2 = 2.56 \text{ feet} \][/tex]
### Step 3: Convert feet to inches
Since 1 foot is equal to 12 inches, we convert the heights from feet to inches:
[tex]\[ h_1 \text{ in inches} = 3.24 \times 12 \][/tex]
[tex]\[ h_1 \text{ in inches} = 38.88 \text{ inches} \][/tex]
[tex]\[ h_2 \text{ in inches} = 2.56 \times 12 \][/tex]
[tex]\[ h_2 \text{ in inches} = 30.72 \text{ inches} \][/tex]
### Step 4: Calculate the height difference to the nearest inch
Find the difference in heights:
[tex]\[ \text{Height Difference} = 38.88 \text{ inches} - 30.72 \text{ inches} \][/tex]
[tex]\[ \text{Height Difference} = 8.16 \text{ inches} \][/tex]
Rounding 8.16 to the nearest inch, we get:
[tex]\[ \text{Height Difference} \approx 8 \text{ inches} \][/tex]
### Step 5: State the final result
To the nearest inch, Player 1 jumped 8 inches higher than Player 2.
[tex]\[ t = 2 \left(\frac{2h}{32}\right)^{\frac{1}{2}} \][/tex]
where [tex]\( t \)[/tex] is the hang time in seconds, and [tex]\( h \)[/tex] is the height in feet.
### Step 1: Rearrange the equation to solve for [tex]\( h \)[/tex]
First, we manipulate the equation to isolate [tex]\( h \)[/tex] on one side:
[tex]\[ t = 2 \left(\frac{2h}{32}\right)^{0.5} \][/tex]
Dividing both sides by 2:
[tex]\[ \frac{t}{2} = \left(\frac{2h}{32}\right)^{0.5} \][/tex]
Squaring both sides to remove the square root:
[tex]\[ \left(\frac{t}{2}\right)^2 = \frac{2h}{32} \][/tex]
Multiply both sides by 32 to solve for [tex]\( 2h \)[/tex]:
[tex]\[ 32 \left(\frac{t}{2}\right)^2 = 2h \][/tex]
Dividing both sides by 2 gives [tex]\( h \)[/tex]:
[tex]\[ h = \frac{32}{2} \left(\frac{t}{2}\right)^2 \][/tex]
This simplifies to:
[tex]\[ h = 16 \left(\frac{t}{2}\right)^2 \][/tex]
### Step 2: Calculate the heights for Player 1 and Player 2
For Player 1 with [tex]\( t_1 = 0.9 \)[/tex] seconds:
[tex]\[ h_1 = 16 \left(\frac{0.9}{2}\right)^2 \][/tex]
[tex]\[ h_1 = 16 \left(0.45\right)^2 \][/tex]
[tex]\[ h_1 = 16 \times 0.2025 \][/tex]
[tex]\[ h_1 = 3.24 \text{ feet} \][/tex]
For Player 2 with [tex]\( t_2 = 0.8 \)[/tex] seconds:
[tex]\[ h_2 = 16 \left(\frac{0.8}{2}\right)^2 \][/tex]
[tex]\[ h_2 = 16 \left(0.4\right)^2 \][/tex]
[tex]\[ h_2 = 16 \times 0.16 \][/tex]
[tex]\[ h_2 = 2.56 \text{ feet} \][/tex]
### Step 3: Convert feet to inches
Since 1 foot is equal to 12 inches, we convert the heights from feet to inches:
[tex]\[ h_1 \text{ in inches} = 3.24 \times 12 \][/tex]
[tex]\[ h_1 \text{ in inches} = 38.88 \text{ inches} \][/tex]
[tex]\[ h_2 \text{ in inches} = 2.56 \times 12 \][/tex]
[tex]\[ h_2 \text{ in inches} = 30.72 \text{ inches} \][/tex]
### Step 4: Calculate the height difference to the nearest inch
Find the difference in heights:
[tex]\[ \text{Height Difference} = 38.88 \text{ inches} - 30.72 \text{ inches} \][/tex]
[tex]\[ \text{Height Difference} = 8.16 \text{ inches} \][/tex]
Rounding 8.16 to the nearest inch, we get:
[tex]\[ \text{Height Difference} \approx 8 \text{ inches} \][/tex]
### Step 5: State the final result
To the nearest inch, Player 1 jumped 8 inches higher than Player 2.
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