Question 202884: Could one of the Calculus tutors please help me with the following Advanced Calculus question. PLEASEE!!!!
A stunt diver leaps off a bridge 50m above a river with an initial upward velocity of 10m/s. Her altitude is given by h= (-5t^2+10t+50, where t is in seconds and h is in metres.
a)Construct a table of values to illustrate her height in 1-second intervals until she hits the water.(2 marks)
b)What does a negative value for height represent?(1 mark)
c)Construct a table showing her velocity during 1-second intervals starting at time 0.(2 marks)
d)When does her velocity change from positive to negative?(1 mark)
e)What does this change represent?(1 mark)
f)When does she hit the water?(1 mark)
g)What is her average velocity in the half-second before she hits the river?(1 mark)
h)Why is her average velocity from t=0 to t=2 equal to 0?(1 mark)
Answer by dyakobovitch(40)   (Show Source):
You can put this solution on YOUR website! A stunt diver leaps off a bridge 50m above a river with an initial upward velocity of 10m/s. Her altitude is given by h= (-5t^2+10t+50, where t is in seconds and h is in metres.
a)Construct a table of values to illustrate her height in 1-second intervals
until she hits the water.(2 marks)
We assume that the time function starts at t=0. So, we choose (h,t)= (0,50) (1,55), (2,50) (3,35) (4,10)...
The function reduces to -5(t^2-2t-10) =0, and so you hit the water at time= 1 + sqrt(11) seconds (by solving the quadratic formula).
b)What does a negative value for height represent?(1 mark)
A negative value for height means that you are under water in the river. =)
c)Construct a table showing her velocity during 1-second intervals starting at time 0.(2 marks)
To find the velocity, you need to take the derivative of the height function. so h'(t)=-10t+10. Inserting the time values from above, we have our velocity as (0,10), (1,0), (2,-10), (3,-20), (4,-30), etc... Notice how the velocity becomes negative. This indicates that we are taking a downward trajectory and are accelerating downward from gravity.
d)When does her velocity change from positive to negative?(1 mark)
This is the critical value for the h'(t), which occurs at t=1 or when h'(t)=0.
e)What does this change represent?(1 mark)
That our concavity is changing, we are now accelerating downward, that we have negative velocity, that we have reached our maximum height and are now falling. (When you jump off a bridge you first increase in height ever-so-slightly, before reaching that maximum height and then falling as you dive downward.
f)When does she hit the water?(1 mark)
Review part a) at time= 1 + sqrt(11) seconds.
g)What is her average velocity in the half-second before she hits the river?(1 mark)
Since she hits the water at 1 + sqrt(11) seconds, I assume you are looking for the specific velocity at the half second before, not the average velocity. Therefore, using the time input of 0.5 + sqrt(11) seconds in the h'(t) function, we get her velocity as -10(0.5 + sqrt(11)) + 10 = 5 -10sqrt(11) as her velocity. Notice this is a check for the problem since the velocity is negative, indicating that she is moving downward.
h)Why is her average velocity from t=0 to t=2 equal to 0?(1 mark)
Average velocity is the sum of velocity divided by the time interval. Her velocity at t=2 is -10. Her velocity at 0=10. So average velocity is (-10 + 10)/2=0.
Feel free to comment if you have any more questions or concerns. - David
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