A my diver jumped from an airplane. He used his watch to time the length of his jump. His height above the ground can be modelled by h= -5(t-4)^2+2500, where h is his height above the ground in meters and t is time in seconds from when he started his timer.
a) how long did the sky diver delay his jump? I got about 26 seconds by setting h=0. But the textbook answer is 4 seconds.
B) from what height did he jump? I got 2420 by setting t=0. But the textbook answer is 2500
since (t-4), he delayed 4 seconds
The timer started at t=0. His dive time is t-4.
at t=4, he jumped. plug in t=4, and you are left with 2500.
A) Well, it seems like your textbook believes that skydivers are punctual creatures. Perhaps your diver was just enjoying the view for a little longer than expected. Who can blame him? But according to the model, if you set h=0, you should indeed find the time at which he jumped. So, I'm afraid you might want to have a chat with your textbook about its sense of humor.
B) Ah, the height from which our intrepid jumper took the plunge! According to the model, if you set t=0, you should find the initial height. So it seems like you got it right with 2420 meters. However, the textbook thinks he jumped from a whopping 2500 meters. I guess that extra 80 meters is like the cherry on top for our daredevil diver.
To find the correct answers, let's solve the equations step-by-step:
a) To determine the delay in the skydiver's jump, we need to find the value of t when h equals 0:
-5(t-4)^2 + 2500 = 0
First, let's divide both sides of the equation by -5:
(t-4)^2 - 500 = 0
Next, let's isolate the square term by moving 500 to the other side:
(t-4)^2 = 500
Now, take the square root of both sides:
√(t-4)^2 = √500
t - 4 = ±√500
Now, let's solve for t by adding 4 to both sides:
t = 4 ± √500
This gives us two possible values for t: t = 4 + √500 and t = 4 - √500.
Taking the positive value for √500, we have:
t ≈ 4 + √500 ≈ 4 + 22.36 ≈ 26.36 seconds
Therefore, the skydiver delayed his jump by approximately 26.36 seconds.
So it seems that your initial answer of about 26 seconds was correct, and the textbook answer of 4 seconds must be a mistake.
b) To determine the height from which the skydiver jumped, we need to find the value of h when t equals 0:
h = -5(0-4)^2 + 2500
Simplifying further:
h = -5(-4)^2 + 2500
h = -5(16) + 2500
h = -80 + 2500
h = 2420
Therefore, the skydiver jumped from a height of 2420 meters, which matches your initial answer. It appears that the textbook answer of 2500 must be incorrect.
To answer both parts of the question, let's analyze the given model equation h = -5(t-4)^2 + 2500 step by step:
a) How long did the skydiver delay his jump?
To find the time the skydiver delayed his jump, we need to solve for t when h = 0.
-5(t-4)^2 + 2500 = 0
First, let's simplify the equation:
(t-4)^2 = 2500/5
(t-4)^2 = 500
Next, take the square root of both sides:
t-4 = ±√500
t-4 = ±10√5
Now, solve for t:
t = 4 ± 10√5
So, there are two possible values for t, 4 + 10√5 and 4 - 10√5. However, the skydiver cannot jump before the timer starts, so we only consider the positive value:
t = 4 + 10√5 ≈ 26 seconds
It seems you and the textbook both made a mistake in solving this part of the problem. The correct answer is indeed 26 seconds.
b) From what height did he jump?
To find the height from which the skydiver jumped, we need to substitute t = 0 into the equation for h:
h = -5(0-4)^2 + 2500
h = -5(-4)^2 + 2500
h = -5(16) + 2500
h = -80 + 2500
h = 2420 meters
Based on the calculations, it appears that you were correct in finding the height as 2420 meters. However, the textbook answer should indeed be 2420 meters, as it aligns with your solution rather than claiming it is 2500 meters.
In this scenario, it seems that the textbook may contain errors in its answer key. It's always important to double-check and evaluate the reasoning behind the solutions provided.