a firecracker is fired from the ground. The height of the firecracker at a given time is modeled by the function h(t) =-5t^2+50t, where h(t) is the height in metres and t is the time in seconds. When will the firecracker reach a height of 45m ?
solve for t in
45 = -5t^2 + 50t
5t^2 - 50t + 45 = 0
t^2 - 10t + 9 = 0
(t-9)(t-1) = 0
it will reach 45m on the way up at t=1, and on the way back down at t=9.
Why did the firecracker schedule a meeting with the height of 45m?
Because it wanted to skyrocket its career! Boom, time for some math humor!
To find when the firecracker will reach a height of 45m, we need to solve the equation h(t) = 45:
-5t^2 + 50t = 45
Let's bring the equation to the standard form:
-5t^2 + 50t - 45 = 0
Now, to solve the quadratic equation, we can either factorize it or use the quadratic formula. Let me grab my quadratic formula joke book real quick!
Why did the quadratic equation date the square root?
Because they had real chemistry!
Alright, enough with the jokes. Let's apply the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -5, b = 50, and c = -45. Plugging these values into the formula, we get:
t = (-50 ± √(50^2 - 4(-5)(-45))) / 2(-5)
t = (-50 ± √(2500 - 900)) / -10
t = (-50 ± √1600) / -10
t = (-50 ± 40) / -10
Now, let's solve for t:
t = (-50 + 40) / -10 = -10 / -10 = 1
t = (-50 - 40) / -10 = -90 / -10 = 9
So, the firecracker will reach a height of 45m at t = 1 second and t = 9 seconds.
It looks like the firecracker will make a big bang in the air at both 1 and 9 seconds! Keep a watch out for the sparks!
To find when the firecracker reaches a height of 45 meters, we need to set the height function h(t) equal to 45 and solve for t.
So let's set up the equation: h(t) = -5t^2 + 50t = 45.
Now we need to solve this quadratic equation for t. To do this, we can rearrange the equation to get:
-5t^2 + 50t - 45 = 0.
Now, we have a quadratic equation in the form of at^2 + bt + c = 0, where a = -5, b = 50, and c = -45.
We can solve this equation using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a).
Substituting the values, we get:
t = (-(50) ± √((50)^2 - 4(-5)(-45))) / (2(-5)).
Simplifying further:
t = (-50 ± √(2500 + 900)) / (-10).
t = (-50 ± √3400) / (-10).
Now, we can simplify the square root:
t = (-50 ± 58.31) / (-10).
Now, we have two possible solutions:
t₁ = (-50 + 58.31) / (-10) ≈ 0.831 seconds.
t₂ = (-50 - 58.31) / (-10) ≈ 10.831 seconds.
Since time cannot be negative, the firecracker reaches a height of 45 meters approximately 0.831 seconds after it was fired from the ground.
To find the time when the firecracker reaches a height of 45m, we need to set h(t) equal to 45 and solve for t.
The given equation is h(t) = -5t^2 + 50t.
Setting h(t) equal to 45, we have:
-5t^2 + 50t = 45
We can rearrange the equation to put it in standard quadratic form:
-5t^2 + 50t - 45 = 0
Now, we can solve this quadratic equation using any suitable method, such as factoring, completing the square, or using the quadratic formula.
Let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = -5, b = 50, and c = -45. Substituting the values into the formula, we get:
t = (-50 ± √((50)^2 - 4(-5)(-45))) / (2(-5))
Simplifying further:
t = (-50 ± √(2500 - 900)) / (-10)
t = (-50 ± √(1600)) / (-10)
t = (-50 ± 40) / (-10)
Now, we have two possible solutions:
1. t = (-50 + 40) / (-10) = -10 / (-10) = 1
2. t = (-50 - 40) / (-10) = -90 / (-10) = 9
Since time (t) cannot be negative, we discard the negative solution.
Therefore, the firecracker will reach a height of 45m after 1 second.