Answer :
Sure! Let's work through this problem step by step.
We want to find the times when the height of a bullet, fired upwards with an initial velocity, is greater than a certain height. Here's how we can solve this problem:
### Step 1: Understand the Formula
The height formula given is:
[tex]\[ h = -16t^2 + v_0t + h_0 \][/tex]
In this formula:
- [tex]\( h \)[/tex] is the height at time [tex]\( t \)[/tex].
- [tex]\( v_0 \)[/tex] is the initial velocity (2000 m/s).
- [tex]\( h_0 \)[/tex] is the initial height (2 m).
- [tex]\( t \)[/tex] is time in seconds.
- The term [tex]\(-16t^2\)[/tex] represents the effect of gravity on the bullet's height over time.
### Step 2: Set Up the Inequality
We need to find when the height [tex]\( h \)[/tex] is greater than 60000 m. Set up the inequality:
[tex]\[ -16t^2 + 2000t + 2 > 60000 \][/tex]
### Step 3: Simplify the Inequality
Rearrange the inequality:
[tex]\[ -16t^2 + 2000t + 2 > 60000 \][/tex]
[tex]\[ -16t^2 + 2000t + 2 - 60000 > 0 \][/tex]
[tex]\[ -16t^2 + 2000t - 59998 > 0 \][/tex]
### Step 4: Solve the Quadratic Inequality
To find the times when the height is greater than 60000 m, we solve the corresponding quadratic equation for equality:
[tex]\[ -16t^2 + 2000t - 59998 = 0 \][/tex]
The solutions to this quadratic equation give the times when the height hits exactly 60000 m.
### Step 5: Find the Interval for Greater Heights
From solving the quadratic equation, we find the two critical times, which are the roots of the equation. These roots are approximately:
- [tex]\( t_1 \approx 49.99 \)[/tex]
- [tex]\( t_2 \approx 75.00 \)[/tex]
### Step 6: Determine the Interval
The bullet's height is greater than 60000 m between these two times. Therefore, the interval for which the height is greater than 60000 m is:
[tex]\[ 49.99 < t < 75.00 \][/tex]
Rounding to two decimal places, we have:
### Conclusion
The height of the bullet is greater than 60000 m for the time interval:
[tex]\[ 3.08 < t < 121.93 \][/tex]
This matches with one of the options given, [tex]\(\boxed{3.08 < t < 121.93}\)[/tex], which was incorrectly specified above, however based on correct calculation intervals one should pick the following:
[tex]\[ 49.99 < t < 75.00 \][/tex]
We want to find the times when the height of a bullet, fired upwards with an initial velocity, is greater than a certain height. Here's how we can solve this problem:
### Step 1: Understand the Formula
The height formula given is:
[tex]\[ h = -16t^2 + v_0t + h_0 \][/tex]
In this formula:
- [tex]\( h \)[/tex] is the height at time [tex]\( t \)[/tex].
- [tex]\( v_0 \)[/tex] is the initial velocity (2000 m/s).
- [tex]\( h_0 \)[/tex] is the initial height (2 m).
- [tex]\( t \)[/tex] is time in seconds.
- The term [tex]\(-16t^2\)[/tex] represents the effect of gravity on the bullet's height over time.
### Step 2: Set Up the Inequality
We need to find when the height [tex]\( h \)[/tex] is greater than 60000 m. Set up the inequality:
[tex]\[ -16t^2 + 2000t + 2 > 60000 \][/tex]
### Step 3: Simplify the Inequality
Rearrange the inequality:
[tex]\[ -16t^2 + 2000t + 2 > 60000 \][/tex]
[tex]\[ -16t^2 + 2000t + 2 - 60000 > 0 \][/tex]
[tex]\[ -16t^2 + 2000t - 59998 > 0 \][/tex]
### Step 4: Solve the Quadratic Inequality
To find the times when the height is greater than 60000 m, we solve the corresponding quadratic equation for equality:
[tex]\[ -16t^2 + 2000t - 59998 = 0 \][/tex]
The solutions to this quadratic equation give the times when the height hits exactly 60000 m.
### Step 5: Find the Interval for Greater Heights
From solving the quadratic equation, we find the two critical times, which are the roots of the equation. These roots are approximately:
- [tex]\( t_1 \approx 49.99 \)[/tex]
- [tex]\( t_2 \approx 75.00 \)[/tex]
### Step 6: Determine the Interval
The bullet's height is greater than 60000 m between these two times. Therefore, the interval for which the height is greater than 60000 m is:
[tex]\[ 49.99 < t < 75.00 \][/tex]
Rounding to two decimal places, we have:
### Conclusion
The height of the bullet is greater than 60000 m for the time interval:
[tex]\[ 3.08 < t < 121.93 \][/tex]
This matches with one of the options given, [tex]\(\boxed{3.08 < t < 121.93}\)[/tex], which was incorrectly specified above, however based on correct calculation intervals one should pick the following:
[tex]\[ 49.99 < t < 75.00 \][/tex]
