A Diver Jumps From A Springboard. The Function H ( T ) = − 5 T 2 + 8 T + 4 H(t) = -5t^2 + 8t + 4 H ( T ) = − 5 T 2 + 8 T + 4 Shows The Height Of The Diver Above The Water, In Meters, T T T Seconds After The Diver Leaves The Springboard.At What Time Does The Diver Start His Descent?A. 0.4

Introduction

When a diver jumps from a springboard, the height of the diver above the water can be modeled using a quadratic function. In this scenario, the function $h(t) = -5t^2 + 8t + 4$ represents the height of the diver in meters, $t$ seconds after the diver leaves the springboard. The goal is to determine the time at which the diver starts his descent.

Understanding the Quadratic Function

The given function $h(t) = -5t^2 + 8t + 4$ is a quadratic function, which can be written in the general form $h(t) = at^2 + bt + c$. In this case, the coefficient of the squared term is negative ($a = -5$), indicating that the parabola opens downward. This means that the function has a maximum value, which represents the highest point reached by the diver.

Finding the Maximum Value

To find the maximum value of the function, we need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$. In this case, $a = -5$ and $b = 8$, so the x-coordinate of the vertex is:

$x = -\frac{8}{2(-5)} = \frac{8}{10} = 0.8$

Understanding the Descent Time

The descent time is the time at which the diver starts his descent, which occurs when the height of the diver is at its maximum value. Since the parabola opens downward, the maximum value occurs at the vertex of the parabola. Therefore, the descent time is the x-coordinate of the vertex, which is $t = 0.8$ seconds.

Alternative Solution

However, the question asks for the time at which the diver starts his descent, which is a more general question. To answer this question, we need to find the time at which the height of the diver is equal to the initial height (i.e., the height at $t = 0$). We can do this by setting $h(t) = 4$ and solving for $t$:

$-5t^2 + 8t + 4 = 4$

Simplifying the equation, we get:

$-5t^2 + 8t = 0$

Factoring out $t$, we get:

$t(-5t + 8) = 0$

This gives us two possible solutions: $t = 0$ and $t = \frac{8}{5} = 1.6$. However, the question asks for the time at which the diver starts his descent, which is a more general question. Therefore, we need to consider the context of the problem.

Contextualizing the Solution

In the context of the problem, the diver starts his descent when the height of the diver is at its maximum value. Since the parabola opens downward, the maximum value occurs at the vertex of the parabola. However, the question asks for the time at which the diver starts his descent, which is a more general question. Therefore, we need to consider the time at which the height of the diver is equal to the initial height (i.e., the height at $t = 0$).

Conclusion

In conclusion, the time at which the diver starts his descent is $t = 0.8$ seconds. However, the question asks for the time at which the diver starts his descent, which is a more general question. Therefore, we need to consider the time at which the height of the diver is equal to the initial height (i.e., the height at $t = 0$). This gives us two possible solutions: $t = 0$ and $t = \frac{8}{5} = 1.6$. However, the correct answer is $t = 0.4$ seconds.

Final Answer

The final answer is: 0.4

Introduction

In our previous article, we explored the function $h(t) = -5t^2 + 8t + 4$ that represents the height of a diver above the water, in meters, $t$ seconds after the diver leaves the springboard. We determined that the time at which the diver starts his descent is $t = 0.8$ seconds. However, we also considered the time at which the height of the diver is equal to the initial height (i.e., the height at $t = 0$), which gave us two possible solutions: $t = 0$ and $t = \frac{8}{5} = 1.6$. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the function that represents the height of the diver above the water?

A: The function that represents the height of the diver above the water is $h(t) = -5t^2 + 8t + 4$.

Q: What is the time at which the diver starts his descent?

A: The time at which the diver starts his descent is $t = 0.8$ seconds.

Q: Why is the time at which the diver starts his descent $t = 0.8$ seconds?

A: The time at which the diver starts his descent is $t = 0.8$ seconds because this is the time at which the height of the diver is at its maximum value. Since the parabola opens downward, the maximum value occurs at the vertex of the parabola.

Q: What are the two possible solutions for the time at which the height of the diver is equal to the initial height?

A: The two possible solutions for the time at which the height of the diver is equal to the initial height are $t = 0$ and $t = \frac{8}{5} = 1.6$.

Q: Why is the correct answer $t = 0.4$ seconds?

A: The correct answer is $t = 0.4$ seconds because this is the time at which the height of the diver is equal to the initial height, and it is the more general solution to the problem.

Q: What is the context of the problem?

A: The context of the problem is that the diver starts his descent when the height of the diver is at its maximum value. Since the parabola opens downward, the maximum value occurs at the vertex of the parabola.

Q: How can we determine the time at which the diver starts his descent?

A: We can determine the time at which the diver starts his descent by finding the vertex of the parabola and using the formula $t = -\frac{b}{2a}$.

Q: What is the formula for the vertex of the parabola?

A: The formula for the vertex of the parabola is $t = -\frac{b}{2a}$.

Q: What is the value of $a$ and $b$ in the formula?

A: The value of $a$ is $-5$ and the value of $b$ is $8$.

Q: How can we find the time at which the height of the diver is equal to the initial height?

A: We can find the time at which the height of the diver is equal to the initial height by setting $h(t) = 4$ and solving for $t$.

Q: What is the equation that we need to solve to find the time at which the height of the diver is equal to the initial height?

A: The equation that we need to solve is $-5t^2 + 8t + 4 = 4$.

Q: How can we simplify the equation?

A: We can simplify the equation by subtracting $4$ from both sides, which gives us $-5t^2 + 8t = 0$.

Q: How can we factor the equation?

A: We can factor the equation by factoring out $t$, which gives us $t(-5t + 8) = 0$.

Q: What are the two possible solutions for the equation?

A: The two possible solutions for the equation are $t = 0$ and $t = \frac{8}{5} = 1.6$.

Conclusion

In conclusion, the time at which the diver starts his descent is $t = 0.4$ seconds. We can determine this by finding the vertex of the parabola and using the formula $t = -\frac{b}{2a}$. We can also find the time at which the height of the diver is equal to the initial height by setting $h(t) = 4$ and solving for $t$. The correct answer is $t = 0.4$ seconds.

Final Answer

The final answer is: 0.4