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A child shot a toy rocket into the air from their playhouse which stood 5 ft above the ground. The rocket rose to 50 ft above the ground and landed on the ground 3.4 seconds later. The path of the toy rocket could be modeled by a quadratic function, h(t), where h is the height of the toy rocket in feet after t seconds.

(1) What is the practical domain of the function?

(2) What is the practical range of the function?

(3) How would the theoretical domain and range of the function differ from the practical domain and range of the situation?

A child shot a toy rocket into the air from their playhouse which stood 5 ft above the ground The rocket rose to 50 ft above the ground and landed on the ground class=

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Answer:

Part A)

All real numbers between 0 and 3.4 (including 0 and 3.4).

Part B)

All real numbers between 5 and 50 (including 5 and 50).

Part C)

See below.

Step-by-step explanation:

We know that the rocket rose to a maximum of 50 feet and was in the air for 3.4 seconds. The rocket started at a height of 5 feet.

Part A)

The practical domain of a function is the domain with the context. Here, our domain is the number of seconds that had passed. Time can't be negative, so our domain must be greater than or equal to 0. Also, since we know that our rocket landed after 3.4 seconds, this means that our practical domain is all real numbers between 0 and 3.4 including 0 and 3.4.

Part B)

The practical range of a function is the range with the context. Here, our range is the height the rocket reaches. We know that the rocket started at a height of 5 feet, so our practical range should be greater than or equal to 5. We also know that the maximum height the rocket reached was 50 feet. So, our practical range is all real numbers between 5 and 50 including 5 and 50.

Part C)

The theoretical domain and range differs from the practical domain and range in that the theoretical counterparts don't consider the context. For the theoretical domain and range, we will analyze this as a normal quadratic. Since it's a quadratic, the theoretical domain is all real numbers. Since our maximum value is y=50, our theoretical range is all numbers less than or equal to 50.

However, this can't be used since negative time (e.g. height after -2 seconds) is inapplicable and negative height doesn't make sense in this context either (well, technically negative height can indicate that the rocket goes underground, but this is not mentioned here).

So, it is important to determine when to use practical domain/range and theoretical domain/range!