when a ball falls vertically off a table it rebounds 75% of its height after each bounce. if it travels a total of 490 cm, how high was the table top above the floor
did you make a sketch
Let the height of the table be x cm
I see the ball doing this
x + 2x(.75) + 2x(.75)^2 + 2x(.75)^3 + ... = 490
x + 2x[.75 + .75^2 + .75^3 + ...] = 490
the sequence inside my square brackets is an infinite series with a=.75 and r=.75
A∞ = a/(1-r) = .75/(.25) = 3
so x + 2x(3) = 490
x = 70
so the table is 70 cm high.
Well, well, look at this bouncing ball! It seems to have a lot of energy. Let's calculate the height of the table top above the floor, shall we?
Now, the ball rebounds 75% of its height after each bounce. That means for each bounce, it reaches 75% of its previous height.
If we let "h" represent the initial height of the ball falling off the table, we can set up an equation to solve for "h". After the first bounce, the ball reaches a height of 0.75h. After the second bounce, it reaches a height of (0.75h * 0.75) = 0.5625h. And so on...
If we add up all the distances the ball travels in each bounce, it should give us a total distance of 490 cm.
So we have the equation: h + 0.75h + 0.5625h + ...
Now, this looks like an infinite series, because the ball keeps bouncing forever! But don't worry, we can use a formula to sum it up.
The sum of an infinite geometric series is given by: S = a / (1 - r), where "S" is the sum, "a" is the first term, and "r" is the common ratio.
In our case, "a" would be "h", and "r" would be 0.75. So we can write:
490 = h / (1 - 0.75)
Now, let's solve for "h" and find out the height of the table top above the floor!
490 = h / 0.25
h = 0.25 * 490
h = 122.5 cm
So, according to my calculations, the height of the table top above the floor is approximately 122.5 cm.
Hope that brings a bounce to your day!
To find the height of the table top above the floor, we can set up an equation using the concept of geometric progression.
Let's assume the initial height of the ball when it falls from the table is 'h'.
In the first bounce, the ball rebounds to 75% of its height: 0.75h.
In the second bounce, it rebounds to 75% of the previous height: (0.75h * 0.75) = 0.75^2 * h.
In the third bounce, it rebounds to 75% of the previous height: (0.75h * 0.75^2) = 0.75^3 * h.
And so on...
We can represent the total distance traveled by the ball as the sum of an infinite geometric series:
Total distance = h + (0.75h) + (0.75^2 * h) + (0.75^3 * h) + ...
The formula for the sum of an infinite geometric series is:
Sum = (first term) / (1 - common ratio)
In this case, the first term is 'h' and the common ratio is 0.75. So, the equation becomes:
490 = h / (1 - 0.75)
To solve for 'h', we can rearrange the equation:
(1 - 0.75) * 490 = h
0.25 * 490 = h
122.5 = h
Therefore, the height of the table top above the floor is 122.5 cm.
To find the height of the table top above the floor, we can set up a simple equation based on the given information. Let's assume that the height of the table top is 'h' centimeters.
When the ball falls off the table, it rebounds 75% of its height after each bounce. This means that after each bounce, the ball reaches a height of 75% of 'h'. Therefore, the total distance traveled by the ball can be represented as:
h + 0.75h + 0.75^2h + 0.75^3h + ... + 0.75^n*h
Since the ball continues to bounce until it does not reach a height above the ground, the sum of this infinite geometric series represents the total distance traveled by the ball. We can express the sum using the formula:
Sum = a / (1 - r)
Where 'a' is the first term and 'r' is the common ratio between terms. In this case, 'a' is 'h' and 'r' is 0.75.
Now, we can set up the equation using the given information:
490 = h / (1 - 0.75)
To solve for 'h', we first find the value of 1 - 0.75:
1 - 0.75 = 0.25
Now, we can rewrite the equation:
490 = h / 0.25
To isolate 'h', we can multiply both sides of the equation by 0.25:
490 * 0.25 = h
Hence, the height of the table top above the floor is:
h = 122.5 cm