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Answer :
To solve this problem, we need to determine two things:
1. When the object's height will reach 186 feet.
2. When the object will reach the ground.
Let's go through this step by step:
### Part 1: Finding when the height is 186 feet
The equation for the height [tex]\( h \)[/tex] of the object is:
[tex]\[ h = -16t^2 + 163t + 13 \][/tex]
We want to find when this height is 186 feet:
[tex]\[ 186 = -16t^2 + 163t + 13 \][/tex]
Rearrange this equation to form a quadratic equation:
[tex]\[ -16t^2 + 163t + 13 - 186 = 0 \][/tex]
[tex]\[ -16t^2 + 163t - 173 = 0 \][/tex]
To solve this quadratic equation, you can use the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = -16 \)[/tex], [tex]\( b = 163 \)[/tex], and [tex]\( c = -173 \)[/tex].
After solving this equation, we find two possible times. The relevant positive time when the object reaches 186 feet is approximately:
[tex]\[ t \approx 9.0 \text{ seconds} \][/tex]
### Part 2: Finding when the object reaches the ground
We now need to find when the object hits the ground, which means the height is 0:
[tex]\[ 0 = -16t^2 + 163t + 13 \][/tex]
Using the quadratic formula with [tex]\( a = -16 \)[/tex], [tex]\( b = 163 \)[/tex], and [tex]\( c = 13 \)[/tex], we solve for [tex]\( t \)[/tex].
Calculating this, we find the relevant positive time when the object reaches the ground:
[tex]\[ t \approx 10.3 \text{ seconds} \][/tex]
Therefore, the object will reach the height of 186 feet at approximately 9.0 seconds, and it will hit the ground at approximately 10.3 seconds.
1. When the object's height will reach 186 feet.
2. When the object will reach the ground.
Let's go through this step by step:
### Part 1: Finding when the height is 186 feet
The equation for the height [tex]\( h \)[/tex] of the object is:
[tex]\[ h = -16t^2 + 163t + 13 \][/tex]
We want to find when this height is 186 feet:
[tex]\[ 186 = -16t^2 + 163t + 13 \][/tex]
Rearrange this equation to form a quadratic equation:
[tex]\[ -16t^2 + 163t + 13 - 186 = 0 \][/tex]
[tex]\[ -16t^2 + 163t - 173 = 0 \][/tex]
To solve this quadratic equation, you can use the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = -16 \)[/tex], [tex]\( b = 163 \)[/tex], and [tex]\( c = -173 \)[/tex].
After solving this equation, we find two possible times. The relevant positive time when the object reaches 186 feet is approximately:
[tex]\[ t \approx 9.0 \text{ seconds} \][/tex]
### Part 2: Finding when the object reaches the ground
We now need to find when the object hits the ground, which means the height is 0:
[tex]\[ 0 = -16t^2 + 163t + 13 \][/tex]
Using the quadratic formula with [tex]\( a = -16 \)[/tex], [tex]\( b = 163 \)[/tex], and [tex]\( c = 13 \)[/tex], we solve for [tex]\( t \)[/tex].
Calculating this, we find the relevant positive time when the object reaches the ground:
[tex]\[ t \approx 10.3 \text{ seconds} \][/tex]
Therefore, the object will reach the height of 186 feet at approximately 9.0 seconds, and it will hit the ground at approximately 10.3 seconds.
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