1.4 How Fast Does a Penny Fall from the Empire State Building?

A penny is dropped from the top of the Empire State Building, from a height of 1,250 feet. The height of the penny, in feet, t seconds after it is dropped is given by the function H(t)=1250-16t^2.

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Find the exact time t when the penny reaches the ground.

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Find the average rate of change in the penny’s height during the total length of its drop.

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Determine how many feet the penny fell during each two second interval.
What do you notice?

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Is the penny speeding up, slowing down, or falling at a constant speed? How do you know?

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Graph y=H(t).

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Is the graph of H(t) concave up or concave down? What does this mean in the context of the problem?

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Here's the chart from earlier.

What do you notice about the rate of change of these average rate of changes?

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How do those differences compare to the linear function from earlier?
What does that tell us about this function?

Traveling back in time for a moment...
Last year you had a lesson called Finite Differences...

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Match the function name to the table of values.

Draggable item		Corresponding Item
Exponential
Linear
Cubic
Quadratic

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When are finding differences useful? How do we need to be careful?

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What can concavity tell us about our average rate of change?

Quadratic functions have a constant second difference, meaning the change over each interval grows or declines linearly.

Find the exact time t when the penny reaches the ground.

Find the average rate of change in the penny’s height during the total length of its drop.

Determine how many feet the penny fell during each two second interval. What do you notice?

Is the penny speeding up, slowing down, or falling at a constant speed? How do you know?

Graph y=H(t).

Is the graph of H(t) concave up or concave down? What does this mean in the context of the problem?

Here's the chart from earlier.What do you notice about the rate of change of these average rate of changes?

How do those differences compare to the linear function from earlier?What does that tell us about this function?

Traveling back in time for a moment...

Last year you had a lesson called Finite Differences...

Match the function name to the table of values.

When are finding differences useful? How do we need to be careful?

What can concavity tell us about our average rate of change?

Quadratic functions have a constant second difference, meaning the change over each interval grows or declines linearly.

Determine how many feet the penny fell during each two second interval.
What do you notice?

Here's the chart from earlier.

What do you notice about the rate of change of these average rate of changes?

How do those differences compare to the linear function from earlier?
What does that tell us about this function?