An explosives company specializes in bringing down old buildings. The time it takes for a particular piece of flying debris to fall to the ground can be modelled by the function h left parenthesis t right parenthesis equals negative 18 t squared plus 72 t plus 378, where h is the height of the debris, in metres, and t is the time, in seconds.
How long will it take for the piece of debris to hit the ground after an explosion?
h(t) = - 18 t² + 72 t + 378
The piece of debris hit the ground when h = 0
h(t) = - 18 t² + 72 t + 378 = 0
Now you must solve equation:
- 18 t² + 72 t + 378 = 0
The solutions are:
t = - 3 and t = 7
Time cannot be negative.
So t = 7 sec
To find the time it takes for the piece of debris to hit the ground after an explosion, we need to determine when the height of the debris is equal to zero. In other words, we need to solve the equation h(t) = 0.
The given function for the height of the debris is h(t) = -18t^2 + 72t + 378. We can set this equation equal to zero and solve for t:
-18t^2 + 72t + 378 = 0
Now, we can solve this quadratic equation. There are a few methods to solve quadratic equations, but one common method is by factoring. However, this equation does not appear to be easily factorable. So, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -18, b = 72, and c = 378. Substituting these values into the quadratic formula, we get:
t = (-72 ± √(72^2 - 4(-18)(378))) / (2(-18))
Simplifying further:
t = (-72 ± √(5184 + 27312)) / (-36)
t = (-72 ± √(32496)) / (-36)
Now, calculate the square root and perform the rest of the calculations to find the two possible values for t. This will give you the times at which the debris is at ground level.
To find the time it takes for the piece of debris to hit the ground, we need to find the value of t when h(t) = 0.
So let's set the equation h(t) = -18t^2 + 72t + 378 equal to 0:
-18t^2 + 72t + 378 = 0
Now, let's solve this quadratic equation to find the values of t using the quadratic formula:
The quadratic formula is given by: t = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -18, b = 72, and c = 378.
Substituting these values into the quadratic formula:
t = (-72 ± √(72^2 - 4(-18)(378))) / (2(-18))
Simplifying further:
t = (-72 ± √(5184 + 27384)) / (-36)
t = (-72 ± √32568) / (-36)
Now, we need to calculate the square root of 32568:
√32568 ≈ 180.43
Substituting this value back into the equation:
t ≈ (-72 ± 180.43) / (-36)
Now, let's solve for both values of t:
1. t = (-72 + 180.43) / (-36)
t ≈ 2.51
2. t = (-72 - 180.43) / (-36)
t ≈ -6.98
Since time cannot be negative, we discard the negative solution.
Therefore, the time it will take for the piece of debris to hit the ground after the explosion is approximately 2.51 seconds.