The Function $h(t)=-16t^2+28t+500$ Represents The Height Of A Rock $t$ Seconds After It Is Propelled By A Slingshot.What Does $h(3.2$\] Represent?A. The Height Of The Rock 3.2 Seconds Before It Reaches The Ground B. The Time

Feb 26, 2025 by ADMIN 226 views

The Function of a Rock's Height After Being Propelled by a Slingshot

Understanding the Function

The given function $h(t)=-16t^2+28t+500$ represents the height of a rock $t$ seconds after it is propelled by a slingshot. This function is a quadratic equation, which is a polynomial of degree two. The general form of a quadratic equation is $ax^2+bx+c$ , where $a$ , $b$ , and $c$ are constants. In this case, the function is in the form of $h(t)=-16t^2+28t+500$ , where $a=-16$ , $b=28$ , and $c=500$ .

What Does $h(3.2)$ Represent?

To find the value of $h(3.2)$ , we need to substitute $t=3.2$ into the function $h(t)=-16t^2+28t+500$ . This means we need to plug in $3.2$ for $t$ in the equation and solve for $h$ .

Calculating $h(3.2)$

To calculate $h(3.2)$ , we need to follow the order of operations (PEMDAS):

Substitute $t=3.2$ into the function: $h(3.2)=-16(3.2)^2+28(3.2)+500$
Evaluate the exponent: $(3.2)^2=10.24$
Multiply $-16$ by $10.24$ : $-16(10.24)=-163.84$
Multiply $28$ by $3.2$ : $28(3.2)=89.6$
Add $-163.84$ , $89.6$ , and $500$ : $h(3.2)=-163.84+89.6+500=425.76$

What Does $h(3.2)$ Represent?

The value of $h(3.2)$ represents the height of the rock $3.2$ seconds after it is propelled by a slingshot. This means that at $t=3.2$ seconds, the rock is at a height of $425.76$ units above the ground.

Conclusion

Discussion

The discussion category for this problem is mathematics. This problem involves understanding and applying the concept of quadratic equations to a real-world scenario. It requires the ability to substitute values into a function and solve for the output.

Solutions

To find the maximum height of the rock, we need to find the vertex of the parabola represented by the function $h(t)=-16t^2+28t+500$ . The vertex of a parabola is given by the formula $x=-\frac{b}{2a}$ . In this case, $a=-16$ and $b=28$ , so the vertex is at $t=-\frac{28}{2(-16)}=0.875$ seconds. To find the maximum height, we need to substitute $t=0.875$ into the function: $h(0.875)=-16(0.875)^2+28(0.875)+500=441.44$ units above the ground.
To find the time it takes for the rock to reach the ground, we need to set $h(t)=0$ and solve for $t$ . This means we need to find the values of $t$ that make the function $h(t)=-16t^2+28t+500$ equal to zero. We can do this by factoring the function or by using the quadratic formula. Using the quadratic formula, we get: $t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-28\pm\sqrt{28^2-4(-16)(500)}}{2(-16)}=\frac{-28\pm\sqrt{784+32000}}{-32}=\frac{-28\pm\sqrt{32584}}{-32}=\frac{-28\pm 180.4}{-32}$ . Since $t$ must be positive, we take the positive root: $t=\frac{-28+180.4}{-32}=\frac{152.4}{-32}=-4.76$ seconds. However, this is not a valid solution since $t$ must be positive. Therefore, we try the other root: $t=\frac{-28-180.4}{-32}=\frac{-208.4}{-32}=6.53$ seconds. This is the time it takes for the rock to reach the ground.
To find the velocity of the rock at $t=3.2$ seconds, we need to find the derivative of the function $h(t)=-16t^2+28t+500$ . The derivative of a function is given by the formula $f'(x)=\frac{df}{dx}$ . In this case, the derivative is $h'(t)=-32t+28$ . To find the velocity at $t=3.2$ seconds, we need to substitute $t=3.2$ into the derivative: $h'(3.2)=-32(3.2)+28=-102.4+28=-74.4$ units per second.

Conclusion

In conclusion, the function $h(t)=-16t^2+28t+500$ represents the height of a rock $t$ seconds after it is propelled by a slingshot. The value of $h(3.2)$ represents the height of the rock $3.2$ seconds after it is propelled by a slingshot, which is $425.76$ units above the ground. The maximum height of the rock is $441.44$ units above the ground, and the time it takes for the rock to reach the ground is $6.53$ seconds. The velocity of the rock at $t=3.2$ seconds is $-74.4$ units per second.

Frequently Asked Questions

Q: What is the function $h(t)=-16t^2+28t+500$ ?

A: The function $h(t)=-16t^2+28t+500$ represents the height of a rock $t$ seconds after it is propelled by a slingshot.

Q: What does $h(3.2)$ represent?

A: The value of $h(3.2)$ represents the height of the rock $3.2$ seconds after it is propelled by a slingshot.

Q: How do I calculate $h(3.2)$ ?

A: To calculate $h(3.2)$ , you need to substitute $t=3.2$ into the function $h(t)=-16t^2+28t+500$ and solve for $h$ .

Q: What is the maximum height of the rock?

A: The maximum height of the rock is $441.44$ units above the ground.

Q: How long does it take for the rock to reach the ground?

A: It takes $6.53$ seconds for the rock to reach the ground.

Q: What is the velocity of the rock at $t=3.2$ seconds?

A: The velocity of the rock at $t=3.2$ seconds is $-74.4$ units per second.

Q: Can I use this function to model other objects that are propelled by a slingshot?

A: Yes, you can use this function to model other objects that are propelled by a slingshot, as long as the object's height is a quadratic function of time.

Q: How do I find the vertex of the parabola represented by the function $h(t)=-16t^2+28t+500$ ?

A: To find the vertex of the parabola, you need to use the formula $x=-\frac{b}{2a}$ , where $a=-16$ and $b=28$ . This gives you $x=-\frac{28}{2(-16)}=0.875$ seconds.

Q: Can I use the quadratic formula to solve for $t$ in the equation $h(t)=-16t^2+28t+500=0$ ?

A: Yes, you can use the quadratic formula to solve for $t$ in the equation $h(t)=-16t^2+28t+500=0$ . The quadratic formula is given by $t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ .

Q: What is the significance of the negative root in the quadratic formula?

A: The negative root in the quadratic formula represents a time that is not valid, as time cannot be negative.

Q: Can I use the derivative of the function $h(t)=-16t^2+28t+500$ to find the velocity of the rock at $t=3.2$ seconds?

A: Yes, you can use the derivative of the function $h(t)=-16t^2+28t+500$ to find the velocity of the rock at $t=3.2$ seconds. The derivative is given by $h'(t)=-32t+28$ .

Q: How do I find the maximum height of the rock using the derivative of the function $h(t)=-16t^2+28t+500$ ?

A: To find the maximum height of the rock, you need to find the vertex of the parabola represented by the function $h(t)=-16t^2+28t+500$ . The vertex is given by the formula $x=-\frac{b}{2a}$ , where $a=-16$ and $b=28$ . This gives you $x=-\frac{28}{2(-16)}=0.875$ seconds. To find the maximum height, you need to substitute $t=0.875$ into the function: $h(0.875)=-16(0.875)^2+28(0.875)+500=441.44$ units above the ground.

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are not propelled by a slingshot?

A: No, you cannot use the function $h(t)=-16t^2+28t+500$ to model other objects that are not propelled by a slingshot, as the function is specifically designed to model the height of a rock propelled by a slingshot.

Q: How do I find the time it takes for the rock to reach the ground using the function $h(t)=-16t^2+28t+500$ ?

A: To find the time it takes for the rock to reach the ground, you need to set $h(t)=0$ and solve for $t$ . This means you need to find the values of $t$ that make the function $h(t)=-16t^2+28t+500$ equal to zero. You can do this by factoring the function or by using the quadratic formula.

Q: What is the significance of the positive root in the quadratic formula?

A: The positive root in the quadratic formula represents a valid time, as time cannot be negative.

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are propelled by a slingshot, but have a different initial velocity?

A: Yes, you can use the function $h(t)=-16t^2+28t+500$ to model other objects that are propelled by a slingshot, but have a different initial velocity. You will need to adjust the coefficients of the function to reflect the new initial velocity.

Q: How do I find the velocity of the rock at $t=3.2$ seconds using the derivative of the function $h(t)=-16t^2+28t+500$ ?

A: To find the velocity of the rock at $t=3.2$ seconds, you need to substitute $t=3.2$ into the derivative $h'(t)=-32t+28$ : $h'(3.2)=-32(3.2)+28=-102.4+28=-74.4$ units per second.

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are not propelled by a slingshot, but have a different initial height?

A: No, you cannot use the function $h(t)=-16t^2+28t+500$ to model other objects that are not propelled by a slingshot, but have a different initial height, as the function is specifically designed to model the height of a rock propelled by a slingshot.

Q: How do I find the maximum height of the rock using the derivative of the function $h(t)=-16t^2+28t+500$ ?

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are propelled by a slingshot, but have a different initial velocity and initial height?

A: Yes, you can use the function $h(t)=-16t^2+28t+500$ to model other objects that are propelled by a slingshot, but have a different initial velocity and initial height. You will need to adjust the coefficients of the function to reflect the new initial velocity and initial height.

Q: How do I find the time it takes for the rock to reach the ground using the derivative of the function $h(t)=-16t^2+28t+500$ ?

A: To find the time it takes for the rock to reach the ground, you need to set $h'(t)=0$ and solve for $t$ . This means you need to find the values of $t$ that make the derivative $h'(t)=-32t+28$ equal to zero. You can do this by factoring the derivative or by using the quadratic formula.

Q: What is the significance of the negative root in the quadratic formula when solving for $t$ in the equation $h'(t)=-32t+28=0$ ?

A: The negative root in the quadratic formula represents a time that is not valid, as time cannot be negative.

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are not propelled by a slingshot, but have a different initial velocity and initial height?

A: No, you cannot use the function $h(t)=-16t^2+28t+500$ to model other objects that are not propelled by a slingshot, but have a different initial velocity and initial height, as the

Understanding the Function

What Does h(3.2)h(3.2)h(3.2) Represent?

Calculating h(3.2)h(3.2)h(3.2)

What Does h(3.2)h(3.2)h(3.2) Represent?

Conclusion

Discussion

Related Problems

Solutions

Conclusion

Frequently Asked Questions

Q: What is the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500?

Q: What does h(3.2)h(3.2)h(3.2) represent?

Q: How do I calculate h(3.2)h(3.2)h(3.2)?

Q: What is the maximum height of the rock?

Q: How long does it take for the rock to reach the ground?

Q: What is the velocity of the rock at t=3.2t=3.2t=3.2 seconds?

Q: Can I use this function to model other objects that are propelled by a slingshot?

Q: How do I find the vertex of the parabola represented by the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500?

Q: Can I use the quadratic formula to solve for ttt in the equation h(t)=−16t2+28t+500=0h(t)=-16t^2+28t+500=0h(t)=−16t2+28t+500=0?

Q: What is the significance of the negative root in the quadratic formula?

Q: Can I use the derivative of the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500 to find the velocity of the rock at t=3.2t=3.2t=3.2 seconds?

Q: How do I find the maximum height of the rock using the derivative of the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500?

Q: Can I use the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500 to model other objects that are not propelled by a slingshot?

Q: How do I find the time it takes for the rock to reach the ground using the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500?

Q: What is the significance of the positive root in the quadratic formula?

Q: Can I use the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500 to model other objects that are propelled by a slingshot, but have a different initial velocity?

Q: How do I find the velocity of the rock at t=3.2t=3.2t=3.2 seconds using the derivative of the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500?

Q: Can I use the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500 to model other objects that are not propelled by a slingshot, but have a different initial height?

Q: How do I find the maximum height of the rock using the derivative of the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500?

Q: Can I use the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500 to model other objects that are propelled by a slingshot, but have a different initial velocity and initial height?

Q: How do I find the time it takes for the rock to reach the ground using the derivative of the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500?

Q: What is the significance of the negative root in the quadratic formula when solving for ttt in the equation h′(t)=−32t+28=0h'(t)=-32t+28=0h′(t)=−32t+28=0?

Q: Can I use the function h(t)=−16t2+28t+500h(t)=-16t^2+28t+500h(t)=−16t2+28t+500 to model other objects that are not propelled by a slingshot, but have a different initial velocity and initial height?

What Does $h(3.2)$ Represent?

Calculating $h(3.2)$

What Does $h(3.2)$ Represent?

Q: What is the function $h(t)=-16t^2+28t+500$ ?

Q: What does $h(3.2)$ represent?

Q: How do I calculate $h(3.2)$ ?

Q: What is the velocity of the rock at $t=3.2$ seconds?

Q: How do I find the vertex of the parabola represented by the function $h(t)=-16t^2+28t+500$ ?

Q: Can I use the quadratic formula to solve for $t$ in the equation $h(t)=-16t^2+28t+500=0$ ?

Q: Can I use the derivative of the function $h(t)=-16t^2+28t+500$ to find the velocity of the rock at $t=3.2$ seconds?

Q: How do I find the maximum height of the rock using the derivative of the function $h(t)=-16t^2+28t+500$ ?

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are not propelled by a slingshot?

Q: How do I find the time it takes for the rock to reach the ground using the function $h(t)=-16t^2+28t+500$ ?

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are propelled by a slingshot, but have a different initial velocity?

Q: How do I find the velocity of the rock at $t=3.2$ seconds using the derivative of the function $h(t)=-16t^2+28t+500$ ?

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are not propelled by a slingshot, but have a different initial height?

Q: How do I find the maximum height of the rock using the derivative of the function $h(t)=-16t^2+28t+500$ ?

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are propelled by a slingshot, but have a different initial velocity and initial height?

Q: How do I find the time it takes for the rock to reach the ground using the derivative of the function $h(t)=-16t^2+28t+500$ ?

Q: What is the significance of the negative root in the quadratic formula when solving for $t$ in the equation $h'(t)=-32t+28=0$ ?

Q: Can I use the function $h(t)=-16t^2+28t+500$ to model other objects that are not propelled by a slingshot, but have a different initial velocity and initial height?