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

Introduction

The height, $h$, of a falling object $t$ seconds after it is dropped from a platform 300 feet above the ground is modeled by the function $h(t) = 300 - 16t^2$. This function represents the height of the object at any given time $t$. In this article, we will explore the concept of average rate of change and determine which expression could be used to calculate it.

Average Rate of Change

The average rate of change of a function is a measure of how much the function changes over a given interval. It is calculated by finding the difference in the function's output values over the interval and dividing by the length of the interval. In mathematical terms, the average rate of change of a function $f(x)$ over the interval $[a, b]$ is given by:

$$\frac{f(b) - f(a)}{b - a}$$

Calculating Average Rate of Change for the Falling Object

To calculate the average rate of change of the height function $h(t) = 300 - 16t^2$ over a given interval, we need to find the difference in the height values at the beginning and end of the interval and divide by the length of the interval.

Let's consider an example. Suppose we want to find the average rate of change of the height function over the interval $[0, 2]$. We need to find the height values at $t = 0$ and $t = 2$ and then calculate the difference between them.

Finding Height Values

To find the height value at $t = 0$, we substitute $t = 0$ into the height function:

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

To find the height value at $t = 2$, we substitute $t = 2$ into the height function:

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

Calculating Average Rate of Change

Now that we have the height values at $t = 0$ and $t = 2$, we can calculate the average rate of change over the interval $[0, 2]$:

$$\frac{h(2) - h(0)}{2 - 0} = \frac{236 - 300}{2} = \frac{-64}{2} = -32$$

Expression for Average Rate of Change

The expression for average rate of change of the height function $h(t) = 300 - 16t^2$ over the interval $[a, b]$ is:

$$\frac{h(b) - h(a)}{b - a} = \frac{(300 - 16b^2) - (300 - 16a^2)}{b - a} = \frac{-16(b^2 - a^2)}{b - a} = -16(b + a)$$

Conclusion

In this article, we explored the concept of average rate of change and determined which expression could be used to calculate it for the height function $h(t) = 300 - 16t^2$. We found that the average rate of change over the interval $[a, b]$ is given by the expression $-16(b + a)$. This expression can be used to calculate the average rate of change of the height function over any given interval.

References

• [1] Calculus: Early Transcendentals, James Stewart, 8th edition
• [2] Calculus, Michael Spivak, 4th edition

Further Reading

• [1] Average Rate of Change, Khan Academy
• [2] Calculus, MIT OpenCourseWare

Glossary

• Average Rate of Change: A measure of how much a function changes over a given interval.
• Height Function: A function that represents the height of an object at any given time.
• Interval: A range of values between two points.
• Length of Interval: The difference between the two points in an interval.
  The Height of a Falling Object: Understanding Average Rate of Change - Q&A ====================================================================

Introduction

In our previous article, we explored the concept of average rate of change and determined which expression could be used to calculate it for the height function $h(t) = 300 - 16t^2$. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the average rate of change of the height function over the interval $[0, 2]$?

A: The average rate of change of the height function over the interval $[0, 2]$ is $-32$. This means that the height of the object decreases by $32$ feet every second over the interval $[0, 2]$.

Q: How do I calculate the average rate of change of the height function over any given interval?

A: To calculate the average rate of change of the height function over any given interval $[a, b]$, you can use the expression $-16(b + a)$. This expression represents the average rate of change of the height function over the interval $[a, b]$.

Q: What is the significance of the average rate of change of the height function?

A: The average rate of change of the height function represents the rate at which the height of the object changes over a given interval. This information can be useful in predicting the trajectory of the object and determining its position at any given time.

Q: Can I use the average rate of change of the height function to determine the maximum height of the object?

A: No, the average rate of change of the height function cannot be used to determine the maximum height of the object. The maximum height of the object can be determined by finding the vertex of the parabola represented by the height function.

Q: How do I find the vertex of the parabola represented by the height function?

A: To find the vertex of the parabola represented by the height function, you can use the formula $t = -\frac{b}{2a}$. In this case, $a = -16$ and $b = 0$, so the vertex occurs at $t = 0$. The maximum height of the object is $h(0) = 300$ feet.

Q: Can I use the average rate of change of the height function to determine the time it takes for the object to reach the ground?

A: Yes, you can use the average rate of change of the height function to determine the time it takes for the object to reach the ground. Since the average rate of change of the height function is $-32$ feet per second, it will take the object $\frac{300}{32} = 9.375$ seconds to reach the ground.

Conclusion

In this article, we answered some frequently asked questions related to the topic of average rate of change of the height function. We hope that this information has been helpful in understanding the concept of average rate of change and its significance in predicting the trajectory of an object.

References

• [1] Calculus: Early Transcendentals, James Stewart, 8th edition
• [2] Calculus, Michael Spivak, 4th edition

Further Reading

• [1] Average Rate of Change, Khan Academy
• [2] Calculus, MIT OpenCourseWare

Glossary

• Average Rate of Change: A measure of how much a function changes over a given interval.
• Height Function: A function that represents the height of an object at any given time.
• Interval: A range of values between two points.
• Length of Interval: The difference between the two points in an interval.
• Vertex: The point on a parabola where the function changes from increasing to decreasing or vice versa.