Question

Casey is at a bat, and hits home run ball. The height of the ball in feet can be modeled by the function h(t) = -9t^2 + 72t, where t is time in seconds. But wait! Jim the center field jumper leaps into the air according to the function g(t) = -15t^2 + 240t -951 a. use a graphing calculator to sketch the two graphs and find the intersection. b. did Jim time his jump right to catch the ball? c. if the fence is 8 ft high, did Jim jump high enough to catch the ball and prevent the home run? d.what assumptions are we making if we concluded that Jim caught the ball?

Answer 1

"Using a graphing calculator"? How can we help you with that?

Answer 2

the intersection is (7.9,8.77) ,how to work out the last three questions

Answer 3

B: Were they at the same height at the same time?
C: Did he catch the ball at a height greater than 8'?
D: You'll have to answer this one. It needs to come out of your brain.

Answer 4

But how do you know when JIm jumped?

Answer 5

I don't know,either.But this is the question.

Answer 6

The definition is clear. The ball is hit and Jim jumps at t = 0. It's a little silly. Just go with it.

Answer 7

so we just sub t=0 into g(t) = -15t^2 + 240t -951?but g=-951 in that way.Actually, I don't know what is g ?

Answer 8

It is VERY UNLIKELY that an outfielder catches a ball at the moment it is hit. Why woudl t = 0 be of particular importance. You need to know if the two functions are EVER at the same height, some time after t = 0 and before the ball hits the ground.

Here's the ball...

h(t) = -9t^2 + 72

-9t^2 + 72 = 0 ==> t^2 = -72/(-9) = 8 ==> t = sqrt(8) = 2.8284

Okay, so we now know that Jim has 2.8284 seconds to figure out where the ball it. Does his function do that?

Answer 9

So the jim didn't time his jump right to catch the ball. He jumped when t=0,but actually he should jump after 2.8284s right?

Answer 10

No. That's too late. The ball is already on the ground.

-9t^2 + 72t = -15t^2 + 240t -951

Solve for t.

Answer 11

Hold on! I wrote down h(t) incorrectly.

h(t) = -9t^2 + 72t

-9t^2 + 72t = 0 ==> t = 0 or t = 8

Okay, now Jim has 8 seconds to figure it out. Sorry about that. What a difference a typo makes!

Answer 12

-9t^2 + 72t = -15t^2 + 240t -951 t=7.9

Answer 13

Okay, here's the scoop.

The Ball
h(t) = -9t^2 + 72t
This is a little wrong as there should be some small constant at the end. Maybe 1.5. We'll just have to let that go. This function is applicable ONLY for t in [0,8]

Jim
g(t) = -15t^2 + 240t -951
This is a little harder to explain. As-is, it is quite confusing. I'm pretty sure he didn't start jumping at 951 ft below the ground. This function is applicable ONLY for t in [7.225,8.775]. This is the only time g(t) > 0! Essentially, Jim jumps at t = 7.225 and lands at 8.775. The ball had better be in the air at that time. Since the ball lands at t = 8, we have hope.

Some conclusions.

The fact that the ball is subject to one force of gravity and Jim quite another force of gravity is very, very odd, but we'll just have to let that go.

If the ball had hit the ground BEFORE 7.225, it would have been silly of Jim to jump at all.

As you already surmised in the answer to part A, Jim and the Ball get together at (7.867,8.770).

t = 7.867 and the ball should be caught 8.770 ft above the ground (unless Jim chokes and drops the ball.)

a. use a graphing calculator to sketch the two graphs and find the intersection.

Got that: (7.867,8.770)

b. Did Jim time his jump right to catch the ball?

It appears so. There is an intersection of the two models when both models are positive.

c. if the fence is 8 ft high, did Jim jump high enough to catch the ball and prevent the home run?

Absolutely. He caught the ball 0.770 ft above the height of the fence.

d. what assumptions are we making if we concluded that Jim caught the ball?

We've talked about a few.

He didn't choke and have the ball cone out of his glove.
That crazy right fielder didn't smack into him.
The ball didn't bounce off his head and go over the fence, anyway.
Jim was actually tracking the ball -- if he was running the wrong direction, jumping will not be beneficial.
The fence didn't stop him and he actually was able to reach the ball.
There may be others.

Answer 14

I got it .Thanks a lot.

Answer 15

Sure, Jim can play defense, but can he hit?!

Answer 16

A couple of more assumptions: i) The trajectory of the ball and the trajectory of the jump is in the same plane. ii) g(t) represents the trajectory of the hand that catches the ball...

lol

Answer 17

@aum Fair Enough. Either those or we can give Jim a little allowance for the length of his arms. This expands his range a little beyond the specific plane. Catching baseballs is not an exact science. We should let Jim adjust to the situation a little while in the heat of the moment. :-)