Lukalu is rappelling off a cliff. The parametric equations that describe her horizontal and vertical position as a function of time are x(t)=8t and y(t) = -16t^2+100 and . How long does it take her to reach the ground? How far away from the cliff is she when she lands? Remember to show all of the steps that you use to solve the problem.
For Y, since the "ground" Is assumed to be when y(t) = 0, you want to solve for t.
0=-16t^2+100
16t^2=100
t^2=100/16
t^2=6.25
t=2.5seconds
Now that you know t, which is how long it takes to get to the ground, you can plug it into x(t) to find out the horizontal distance traveled
x(t)=8t
x(2.5)=8*2.5=20 feet
Actually Correction Man, the t on the other side of the equation has nothing to do with solving it, because the equation can just as easily be just y=16t^2+100 or x=8t. So Kasey is correct. The answer is x=8(2.5) which is x=20
Thank you so much!!
Actually the it is NOT 20 feet traveled kasey, you forgot to cancel out 2.5 on other side of the equation by dividing 20 by 2.5 and then you will get the CORRECT answer of x=8 feet traveled horizontally before hitting the ground safely which was t=2.5 seconds. Those are your actual answers guys.
Ahh, rappelling off a cliff, what a thrilling adventure! Let's figure out how long it takes for our dear Lukalu to reach the ground and how far she lands from the cliff.
To find the time it takes for her to reach the ground, we need to determine when her vertical position, y(t), becomes zero.
Let's set y(t) = 0 and solve for t:
-16t^2 + 100 = 0
Dividing by -16 gives us:
t^2 - 100/16 = 0
Simplifying further:
t^2 - 6.25 = 0
Now, we can solve this quadratic equation. But hey, don't worry, I've got a joke for you while we're at it:
Why don't scientists trust atoms? Because they make up everything!
Back to business now! Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = 0, and c = -6.25, we can find the solutions. Taking the positive root since we're dealing with time, we have:
t = (√(0^2 - 4(1)(-6.25))) / (2(1))
t = (√(0 + 25)) / 2
t = 5/2 = 2.5
So it takes Lukalu 2.5 seconds to reach the ground. Quite the speedy descent!
Now, let's find out how far away from the cliff she lands. We can use her horizontal position, x(t), at that time:
x(t) = 8t
Plugging in t = 2.5:
x(2.5) = 8 * 2.5
x(2.5) = 20
Lukalu lands 20 units away from the cliff. She must be an expert at aiming for the perfect landing spot!
Hope that answers your question while adding a little humor to your day. If you need any more assistance or jokes, feel free to ask!
To find the time it takes for Lukalu to reach the ground, we need to find the value of t when her vertical position, y(t), equals 0. This is because when her vertical position is 0, she has reached the ground.
Given that the vertical position as a function of time is y(t) = -16t^2 + 100, we can set y(t) equal to 0 and solve for t:
-16t^2 + 100 = 0
Let's solve this equation step by step:
1. Subtract 100 from both sides of the equation:
-16t^2 = -100
2. Divide both sides of the equation by -16 to isolate t^2:
t^2 = (-100) / (-16)
3. Simplify the right side of the equation:
t^2 = 6.25
4. Take the square root of both sides of the equation to solve for t:
t = ± √6.25
Since time cannot be negative in this context, we can discard the negative solution. Therefore, t = √6.25.
Calculating the square root of 6.25 gives us t ≈ 2.5.
So, it takes Lukalu approximately 2.5 units of time to reach the ground.
To find the distance Lukalu is away from the cliff when she lands, we can substitute the value of t we just found into the horizontal position equation x(t) = 8t:
x(2.5) = 8(2.5)
x(2.5) = 20
Therefore, when Lukalu lands, she is 20 units away from the cliff.
In summary:
- It takes Lukalu approximately 2.5 units of time to reach the ground.
- When she lands, she is 20 units away from the cliff.