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Two equations can be used to track the position of a baseball [tex]$t$[/tex] seconds after it is hit.

Suppose [tex]$h = -16t^2 + 45t + 4.8$[/tex] gives the height, in feet, of a baseball [tex]$t$[/tex] seconds after it is hit, and [tex]$s = 103.8t$[/tex] gives the horizontal distance, in feet, of the ball from home plate [tex]$t$[/tex] seconds after it is hit.

Use these equations to determine whether this particular baseball will clear a 10-foot fence positioned 330 feet from home plate.

A. Yes, the ball will clear the fence.
B. No, the ball will not clear the fence.



Answer :

GinnyAnswer
To determine whether the baseball will clear the 10-foot fence positioned 330 feet from home plate, we need to follow these steps:

1. Determine the time when the ball reaches 330 feet horizontally:
- The horizontal distance [tex]\( s \)[/tex] is given by [tex]\( s = 103.8 t \)[/tex].
- We need to find the time [tex]\( t \)[/tex] when [tex]\( s = 330 \)[/tex] feet.
[tex]\[ 103.8 t = 330 \][/tex]
- Solving for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{330}{103.8} \approx 3.179 \][/tex]

2. Calculate the height of the ball at this time:
- The height of the ball [tex]\( h \)[/tex] is given by the equation [tex]\( h = -16 t^2 + 45 t + 4.8 \)[/tex].
- Substitute [tex]\( t \approx 3.179 \)[/tex] into the height equation:
[tex]\[ h(3.179) = -16 (3.179)^2 + 45 (3.179) + 4.8 \][/tex]
- This evaluates to approximately:
[tex]\[ h(3.179) \approx -13.852 \][/tex]

3. Determine if the ball clears the 10-foot fence:
- We need to compare the height of the ball at [tex]\( t \approx 3.179 \)[/tex] seconds to the height of the fence.
- The height of the ball at this time is approximately [tex]\( -13.852 \)[/tex] feet.
- Since [tex]\( -13.852 \)[/tex] feet is significantly less than 10 feet, the ball does not reach a height greater than 10 feet at the point when it reaches the fence.

Therefore, the baseball will not clear the 10-foot fence. The correct answer is:
[tex]\[ \boxed{\text{No, the ball will not clear the fence.}} \][/tex]