The Height, H H H , Of A Falling Object T T T Seconds After It Is Dropped From A Platform 300 Feet Above The Ground Is Modeled By The Function H ( T ) = 300 − 16 T 2 H(t) = 300 - 16t^2 H ( T ) = 300 − 16 T 2 . Which Expression Could Be Used To Determine The Average Rate At

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Introduction

The height, hh, of a falling object tt seconds after it is dropped from a platform 300 feet above the ground is modeled by the function h(t)=300−16t2h(t) = 300 - 16t^2. This function represents the height of the object at any given time tt. In this article, we will explore how to determine the average rate of change of the height of the object over a given time interval.

Understanding Average Rate of Change

The average rate of change of a function over a given interval is a measure of how much the function changes over that interval. In the context of the height of a falling object, the average rate of change represents the average rate at which the height of the object changes over a given time interval.

Calculating Average Rate of Change

To calculate the average rate of change of the height of the object over a given time interval, we can use the following formula:

Average Rate of Change=Change in HeightChange in Time\text{Average Rate of Change} = \frac{\text{Change in Height}}{\text{Change in Time}}

In mathematical terms, this can be represented as:

Average Rate of Change=h(t2)−h(t1)t2−t1\text{Average Rate of Change} = \frac{h(t_2) - h(t_1)}{t_2 - t_1}

where h(t2)h(t_2) and h(t1)h(t_1) are the heights of the object at times t2t_2 and t1t_1, respectively.

Applying the Formula to the Given Function

To determine the average rate of change of the height of the object over a given time interval, we can apply the formula to the given function h(t)=300−16t2h(t) = 300 - 16t^2.

Let's say we want to find the average rate of change of the height of the object over the time interval from t=0t = 0 to t=2t = 2. We can plug in the values of t1=0t_1 = 0 and t2=2t_2 = 2 into the formula:

Average Rate of Change=h(2)−h(0)2−0\text{Average Rate of Change} = \frac{h(2) - h(0)}{2 - 0}

First, we need to find the values of h(2)h(2) and h(0)h(0) by plugging in the values of tt into the given function:

h(2)=300−16(2)2=300−64=236h(2) = 300 - 16(2)^2 = 300 - 64 = 236

h(0)=300−16(0)2=300h(0) = 300 - 16(0)^2 = 300

Now, we can plug in these values into the formula:

Average Rate of Change=236−3002−0=−642=−32\text{Average Rate of Change} = \frac{236 - 300}{2 - 0} = \frac{-64}{2} = -32

Interpretation of the Results

The average rate of change of the height of the object over the time interval from t=0t = 0 to t=2t = 2 is −32-32 feet per second. This means that, on average, the height of the object decreases by 32 feet every second over this time interval.

Conclusion

In conclusion, the average rate of change of the height of a falling object can be calculated using the formula:

Average Rate of Change=h(t2)−h(t1)t2−t1\text{Average Rate of Change} = \frac{h(t_2) - h(t_1)}{t_2 - t_1}

By applying this formula to the given function h(t)=300−16t2h(t) = 300 - 16t^2, we can determine the average rate of change of the height of the object over a given time interval.

Example Use Cases

The concept of average rate of change can be applied to various real-world scenarios, such as:

  • Calculating the average rate of change of the height of a projectile over a given time interval
  • Determining the average rate of change of the temperature of a substance over a given time interval
  • Calculating the average rate of change of the distance traveled by an object over a given time interval

Tips and Tricks

When calculating the average rate of change of a function, make sure to:

  • Use the correct formula: Average Rate of Change=h(t2)−h(t1)t2−t1\text{Average Rate of Change} = \frac{h(t_2) - h(t_1)}{t_2 - t_1}
  • Plug in the correct values of t1t_1 and t2t_2
  • Simplify the expression to obtain the final answer

Q: What is the average rate of change of the height of a falling object?

A: The average rate of change of the height of a falling object is a measure of how much the height of the object changes over a given time interval. It can be calculated using the formula:

Average Rate of Change=h(t2)−h(t1)t2−t1\text{Average Rate of Change} = \frac{h(t_2) - h(t_1)}{t_2 - t_1}

Q: How do I calculate the average rate of change of the height of a falling object?

A: To calculate the average rate of change of the height of a falling object, you need to:

  1. Identify the function that models the height of the object, such as h(t)=300−16t2h(t) = 300 - 16t^2
  2. Determine the time interval over which you want to calculate the average rate of change
  3. Plug in the values of t1t_1 and t2t_2 into the formula
  4. Simplify the expression to obtain the final answer

Q: What is the significance of the average rate of change of the height of a falling object?

A: The average rate of change of the height of a falling object represents the average rate at which the height of the object changes over a given time interval. This can be useful in various real-world scenarios, such as:

  • Calculating the average rate of change of the height of a projectile over a given time interval
  • Determining the average rate of change of the temperature of a substance over a given time interval
  • Calculating the average rate of change of the distance traveled by an object over a given time interval

Q: Can I use the average rate of change of the height of a falling object to predict the future height of the object?

A: No, the average rate of change of the height of a falling object is a measure of the average rate of change of the height of the object over a given time interval, not a prediction of the future height of the object. To predict the future height of the object, you would need to use a different mathematical model, such as a differential equation.

Q: How do I determine the time interval over which I want to calculate the average rate of change of the height of a falling object?

A: The time interval over which you want to calculate the average rate of change of the height of a falling object will depend on the specific problem you are trying to solve. For example, if you are trying to calculate the average rate of change of the height of a projectile over a given time interval, you may want to choose a time interval that is relevant to the problem, such as the time it takes for the projectile to reach its maximum height.

Q: Can I use the average rate of change of the height of a falling object to calculate the velocity of the object?

A: Yes, the average rate of change of the height of a falling object can be used to calculate the velocity of the object. The velocity of the object is the rate of change of its position with respect to time, and the average rate of change of the height of the object is a measure of the average rate of change of its position with respect to time.

Q: How do I calculate the velocity of the object using the average rate of change of the height of a falling object?

A: To calculate the velocity of the object using the average rate of change of the height of a falling object, you can use the following formula:

Velocity=Average Rate of Change of HeightTime Interval\text{Velocity} = \frac{\text{Average Rate of Change of Height}}{\text{Time Interval}}

For example, if the average rate of change of the height of the object is −32-32 feet per second and the time interval is 22 seconds, the velocity of the object would be:

Velocity=−322=−16\text{Velocity} = \frac{-32}{2} = -16

Q: Can I use the average rate of change of the height of a falling object to calculate the acceleration of the object?

A: Yes, the average rate of change of the height of a falling object can be used to calculate the acceleration of the object. The acceleration of the object is the rate of change of its velocity with respect to time, and the average rate of change of the height of the object is a measure of the average rate of change of its velocity with respect to time.

Q: How do I calculate the acceleration of the object using the average rate of change of the height of a falling object?

A: To calculate the acceleration of the object using the average rate of change of the height of a falling object, you can use the following formula:

Acceleration=Average Rate of Change of VelocityTime Interval\text{Acceleration} = \frac{\text{Average Rate of Change of Velocity}}{\text{Time Interval}}

For example, if the average rate of change of the velocity of the object is −16-16 feet per second per second and the time interval is 22 seconds, the acceleration of the object would be:

Acceleration=−162=−8\text{Acceleration} = \frac{-16}{2} = -8

Conclusion

In conclusion, the average rate of change of the height of a falling object is a measure of how much the height of the object changes over a given time interval. It can be calculated using the formula:

Average Rate of Change=h(t2)−h(t1)t2−t1\text{Average Rate of Change} = \frac{h(t_2) - h(t_1)}{t_2 - t_1}

The average rate of change of the height of a falling object can be used to calculate the velocity and acceleration of the object, and it can be useful in various real-world scenarios.