If h= -16t^2 + 112t represents the height of a rocket, in feet, t seconds after it was fired, when will the rocket hit the ground? (Hint: The rocket is on the ground when h equals=0).
Well, well, well, it seems like we have a rocket in a bit of a pickle! Let's solve this riddle, shall we?
So, according to our given equation h = -16t^2 + 112t, we need to find the time when the height (h) equals zero, which means the rocket is kissing the ground. How romantic!
To solve for t, we set the equation to zero:
0 = -16t^2 + 112t
Now, personally, I wouldn't mind if math was as simple as a clown blowing bubbles, but we'll have to take a more logical approach here.
To make this equation a bit friendlier, we'll start by factoring out a common term:
0 = 16t(t - 7)
Now, we have two factors, 16t and (t - 7), and we know that at least one of them must equal zero for the whole equation to be zero. In other words, either 16t = 0 or (t - 7) = 0.
If 16t = 0, then t = 0. Simple enough, right?
And if (t - 7) = 0, then t = 7. It seems like the rocket might meet the ground after 7 seconds. A good time for a crash landing!
So, my friend, the rocket will hit the ground either at t = 0 or t = 7. Keep your eyes on the sky and get ready for some excitement (and maybe a few fireworks) in 7 seconds!
Hope that puts a smile on your face! Let me know if you need any more clown-approved assistance.
To find when the rocket hits the ground, we set the height equation to zero and solve for the value of t.
Given the equation h = -16t^2 + 112t, we can substitute h with zero.
0 = -16t^2 + 112t
Now we can solve this quadratic equation. Let's rearrange it:
16t^2 - 112t = 0
Now we can factor out common terms, which is 16t:
16t(t - 7) = 0
Now we have two factors: 16t = 0 and t - 7 = 0.
For 16t = 0, t must be 0.
For t - 7 = 0, we solve for t:
t = 7
Therefore, the rocket will hit the ground at two different times: at t = 0 seconds (immediately after it was fired) and at t = 7 seconds.
To find when the rocket hits the ground, we need to find the value of t when the height, h, is equal to 0.
Given the equation h = -16t^2 + 112t, we can set h equal to 0 and solve for t.
0 = -16t^2 + 112t
To solve this quadratic equation, we can either factor it or use 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 (in our case, t) are given by:
t = (-b ± √(b^2 - 4ac)) / 2a
Comparing our equation to the standard quadratic form, we have a = -16, b = 112, and c = 0.
Plugging these values into the quadratic formula:
t = (-112 ± √(112^2 - 4 * -16 * 0)) / (2 * -16)
Simplifying further:
t = (-112 ± √(12544)) / (-32)
t = (-112 ± 112) / (-32)
Now, we have two possible solutions:
t₁ = (-112 + 112) / (-32) = 0 / (-32) = 0
t₂ = (-112 - 112) / (-32) = (-224) / (-32) = 7
Since time cannot be negative in this context, we discard the t₁ = 0 solution.
Therefore, the rocket will hit the ground after 7 seconds.
0 = -16 t^2 + 112 t
factoring ... 0 = -16 t (t - 7)
0 = -16 t ... t = 0 (launch)
0 = t - 7 ... t = 7 (return)