The Function $h(t)=-16t^2+96t+10$ Models The Path Of A Projectile. Select All The True Statements.Select All That Apply:A. The Projectile Reaches A Maximum Height At Time 3 S.B. The Projectile Reaches A Maximum Height At Time 6 S.C. The

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Introduction

In mathematics, functions are used to model real-world phenomena, and the function $h(t)=-16t^2+96t+10$ is a great example of this. This function models the path of a projectile, where h(t)h(t) represents the height of the projectile at time tt. In this article, we will explore the function and determine which of the given statements are true.

Understanding the Function

The function $h(t)=-16t^2+96t+10$ is a quadratic function, which means it has a parabolic shape. The general form of a quadratic function is f(x)=ax2+bx+cf(x)=ax^2+bx+c, where aa, bb, and cc are constants. In this case, a=−16a=-16, b=96b=96, and c=10c=10.

Finding the Vertex

To find the maximum height of the projectile, we need to find the vertex of the parabola. The vertex of a parabola is the point where the parabola changes direction, and it is the maximum or minimum point of the parabola. To find the vertex, we can use the formula $x=-\frac{b}{2a}$.

Plugging in the values of aa and bb, we get $x=-\frac{96}{2(-16)}=\frac{96}{32}=3$.

Finding the Maximum Height

Now that we have found the time at which the projectile reaches its maximum height, we can find the maximum height itself. To do this, we can plug the value of tt into the function $h(t)=-16t^2+96t+10$.

Plugging in t=3t=3, we get $h(3)=-16(3)^2+96(3)+10=-16(9)+288+10=-144+288+10=154$.

Conclusion

In conclusion, the projectile reaches a maximum height of 154 meters at time t=3t=3 seconds.

Selecting the True Statements

Now that we have found the maximum height and the time at which it occurs, we can select the true statements from the given options.

  • A. The projectile reaches a maximum height at time 3 s.: This statement is true, as we have found that the projectile reaches its maximum height at time t=3t=3 seconds.
  • B. The projectile reaches a maximum height at time 6 s.: This statement is false, as we have found that the projectile reaches its maximum height at time t=3t=3 seconds, not 6 seconds.

The Final Answer

In conclusion, the true statements are:

  • The projectile reaches a maximum height at time 3 s.
  • The function $h(t)=-16t^2+96t+10$ models the path of a projectile.

Discussion

The function $h(t)=-16t^2+96t+10$ is a great example of how mathematics can be used to model real-world phenomena. By understanding the function and its properties, we can gain insights into the behavior of the projectile and make predictions about its path.

Mathematical Concepts

This problem involves several mathematical concepts, including:

  • Quadratic functions: The function $h(t)=-16t^2+96t+10$ is a quadratic function, which means it has a parabolic shape.
  • Vertex: The vertex of a parabola is the point where the parabola changes direction, and it is the maximum or minimum point of the parabola.
  • Maximum height: The maximum height of the projectile is the highest point it reaches, and it occurs at the vertex of the parabola.

Real-World Applications

This problem has several real-world applications, including:

  • Projectile motion: The function $h(t)=-16t^2+96t+10$ models the path of a projectile, which is an important concept in physics and engineering.
  • Optimization: The problem of finding the maximum height of the projectile is an optimization problem, which is an important concept in mathematics and computer science.

Conclusion

Q&A: The Function of a Projectile's Path

Q: What is the function $h(t)=-16t^2+96t+10$ used to model?

A: The function $h(t)=-16t^2+96t+10$ is used to model the path of a projectile.

Q: What is the significance of the vertex of the parabola?

A: The vertex of the parabola is the point where the parabola changes direction, and it is the maximum or minimum point of the parabola. In this case, the vertex represents the maximum height of the projectile.

Q: How do you find the vertex of the parabola?

A: To find the vertex of the parabola, you can use the formula $x=-\frac{b}{2a}$, where aa and bb are the coefficients of the quadratic function.

Q: What is the maximum height of the projectile?

A: The maximum height of the projectile is 154 meters, which occurs at time t=3t=3 seconds.

Q: How do you find the maximum height of the projectile?

A: To find the maximum height of the projectile, you can plug the value of tt into the function $h(t)=-16t^2+96t+10$.

Q: What are some real-world applications of the function $h(t)=-16t^2+96t+10$?

A: Some real-world applications of the function $h(t)=-16t^2+96t+10$ include:

  • Projectile motion: The function models the path of a projectile, which is an important concept in physics and engineering.
  • Optimization: The problem of finding the maximum height of the projectile is an optimization problem, which is an important concept in mathematics and computer science.

Q: What mathematical concepts are involved in understanding the function $h(t)=-16t^2+96t+10$?

A: Some mathematical concepts involved in understanding the function $h(t)=-16t^2+96t+10$ include:

  • Quadratic functions: The function $h(t)=-16t^2+96t+10$ is a quadratic function, which means it has a parabolic shape.
  • Vertex: The vertex of a parabola is the point where the parabola changes direction, and it is the maximum or minimum point of the parabola.
  • Maximum height: The maximum height of the projectile is the highest point it reaches, and it occurs at the vertex of the parabola.

Q: How can the function $h(t)=-16t^2+96t+10$ be used in real-world scenarios?

A: The function $h(t)=-16t^2+96t+10$ can be used in real-world scenarios such as:

  • Designing projectile systems: The function can be used to design and optimize projectile systems, such as catapults and cannons.
  • Modeling real-world phenomena: The function can be used to model real-world phenomena, such as the motion of a thrown ball or the trajectory of a satellite.

Conclusion

In conclusion, the function $h(t)=-16t^2+96t+10$ is a powerful tool for modeling and understanding the path of a projectile. By understanding the function and its properties, we can gain insights into the behavior of the projectile and make predictions about its path.