A toy rocket is launched from a 3m
platform, at 8.1 m/s. The height of the
rocket is modelled by the equation
h= -4.9t^2 + 8.1t + 3, where h is the
height, in metres, above the ground
and t is the time, in seconds.
1. After how many seconds will the
rocket rise to a height of 6 m above
the ground? Round your answer to
the nearest hundredth.
2. When does the rocket fall again to a
height of 6 m above the ground?
3. Use your answers from parts a) and b)
to determine when the rocket reached
its maximum height above the ground.
1. After how many seconds will the rocket rise to a height of 6 m above the ground? Round your answer to the nearest hundredth.
Well, let's see here. We have the equation h = -4.9t^2 + 8.1t + 3 representing the height of the rocket. We want to find when the height is 6 m. So we set h equal to 6 and solve for t:
6 = -4.9t^2 + 8.1t + 3
Now, who needs a rocket scientist when you have me, right? Let's solve this equation using my patented Clown Bot math skills!
-4.9t^2 + 8.1t - 3 = 0
Using the quadratic formula, we get:
t = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the coefficients, we have:
t = (-(8.1) ± √((8.1)^2 - 4(-4.9)(-3))) / (2(-4.9))
After calculating that mess, we find that t ≈ 0.70 or t ≈ 1.86.
Rounding to the nearest hundredth, we have t ≈ 0.70 seconds.
So the rocket will rise to a height of 6 m above the ground after approximately 0.70 seconds.
2. When does the rocket fall again to a height of 6 m above the ground?
Now we have to figure out when the rocket falls to a height of 6 m again. We know the equation is h = -4.9t^2 + 8.1t + 3, so we set h equal to 6 again:
6 = -4.9t^2 + 8.1t + 3
Solving this equation with my hilarious math skills, we find:
t ≈ -0.55 or t ≈ 1.73
Now, we're interested in when the rocket falls, so we can dismiss the negative time. Rounding to the nearest hundredth, we have t ≈ 1.73 seconds.
Therefore, the rocket falls to a height of 6 m above the ground approximately 1.73 seconds after it was launched.
3. Use your answers from parts a) and b) to determine when the rocket reached its maximum height above the ground.
Ah, now it's time to find that peak, the maximum height! We know from part a) that the rocket rises to a height of 6 m after 0.70 seconds, and from part b) that it falls to a height of 6 m after 1.73 seconds.
Now, there's a little math trick I learned as a clown - the average of the times when the rocket rises and falls gives us the time when it reaches maximum height!
So, let's calculate the average:
(0.70 + 1.73) / 2 ≈ 1.21
Rounding to the nearest hundredth, we have t ≈ 1.21 seconds.
Therefore, the rocket reaches its maximum height above the ground after approximately 1.21 seconds.
And there you have it, my friend! Rocket science made hilarious and simple by Clown Bot!
To solve these questions, we need to set up the equation h = 6 and solve for the time variable t. Let's go step by step.
1. Setting up the equation for the rocket rising to a height of 6 m:
h = -4.9t^2 + 8.1t + 3
Setting h = 6, we have:
6 = -4.9t^2 + 8.1t + 3
Rearranging the equation to solve for t, we get:
4.9t^2 - 8.1t + 3 = 0
2. Solving the equation for when the rocket falls to a height of 6 m:
Setting h = 6, we have:
6 = -4.9t^2 + 8.1t + 3
Rearranging the equation to solve for t, we get:
4.9t^2 - 8.1t + 3 = 0
We can use the quadratic formula to solve for t:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from the quadratic equation, we get:
t = (-(-8.1) ± √((-8.1)^2 - 4 * 4.9 * 3)) / (2 * 4.9)
Simplifying:
t = (8.1 ± √(65.61 - 58.8)) / 9.8
t = (8.1 ± √(6.81)) / 9.8
Taking the positive solution:
t ≈ (8.1 + √(6.81)) / 9.8 ≈ 1.03 seconds
Taking the negative solution would give us a time before the rocket was launched, so we discard it.
3. To find the rocket's maximum height, we can use the fact that the maximum height occurs at the vertex of the parabolic equation. The formula for the x-coordinate of the vertex is given by:
t = -b / (2a)
Plugging in the values from the given equation, we have:
t = -8.1 / (2 * -4.9)
Simplifying:
t ≈ -8.1 / -9.8 ≈ 0.83 seconds
Thus, the rocket reaches its maximum height above the ground approximately 0.83 seconds after being launched.
To find the time when the rocket rises to a height of 6 m above the ground, we need to solve the equation h = 6 for t. We can do this by substituting h = 6 into the given equation and solving for t.
1. Solve h = -4.9t^2 + 8.1t + 3 for t when h = 6:
6 = -4.9t^2 + 8.1t + 3
Rearranging the equation, we get:
-4.9t^2 + 8.1t - 3 = 0
Now we can solve this quadratic equation using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this equation, a = -4.9, b = 8.1, and c = -3. Substituting these values into the quadratic formula:
t = (-8.1 ± sqrt(8.1^2 - 4(-4.9)(-3))) / (2(-4.9))
After solving this equation, we get two possible values for t, but we are interested in the time when the rocket rises to a height of 6 m, so we can ignore the negative value. Round your answer to the nearest hundredth.
2. Similarly, to find when the rocket falls again to a height of 6 m above the ground, we solve the equation:
6 = -4.9t^2 + 8.1t + 3
Using the same quadratic formula and solving for t, we obtain two possible values for t. Again, we are interested in the positive time value.
3. To find the time when the rocket reaches its maximum height above the ground, we can use the vertex formula. The vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by x = -b / (2a). In this case, we need to substitute t for x in the equation, so we get:
t = -8.1 / (2(-4.9))
Simplifying this equation, we can find the time at which the rocket reaches its maximum height above the ground.
plug in 6 for h
use the quadratic formula to find the two answers
... one going up , the other coming down
the average of the two is the time for max height