Quadratics are an important concept in mathematics, often used to model various real-life situations. In this article, we will explore how quadratics can be used to represent and analyze the distance traveled by a car.

The given exam question presents us with a quadratic equation that represents the distance traveled by a car over time. The equation is: S = -t^2 + 60t, where S represents the distance traveled (in kilometers) and t represents the time traveled (in hours).

To answer the questions posed, let’s start by understanding the characteristics of this quadratic equation. The general form of a quadratic equation is Ax^2 + Bx + C, where A, B, and C are constants. In our case, the equation is -t^2 + 60t, which matches the general form.

Now, let’s answer the questions one by one:

1. To determine when the car reaches the maximum distance, we can use the vertex formula for a quadratic equation: x = -B/2A. In our equation, A = -1 and B = 60. Plugging in these values, we find that t = -60/(2*-1), which simplifies to t = 30. Therefore, the car reaches its maximum distance after 30 hours of travel.

2. To find the maximum distance traveled, we substitute the value of t = 30 into the equation S = -t^2 + 60t. Evaluating this expression, we get S = -(30)^2 + 60(30), which simplifies to S = -900 + 1800 = 900. Thus, the maximum distance traveled by the car is 900 kilometers.

3. To determine when the car returns to a distance of 0 kilometers, we can set the equation S = -t^2 + 60t equal to 0 and solve for t. Rearranging the equation, we have -t^2 + 60t = 0. Factoring out t, we get t(-t + 60) = 0. From this, we can conclude that either t = 0 or -t + 60 = 0. Solving the second equation, we find -t = -60, which simplifies to t = 60. Therefore, the car returns to a distance of 0 kilometers after 60 hours of travel.

In conclusion, by understanding and applying the concepts of quadratics, we were able to analyze and answer the questions related to the distance traveled by a car. Quadratic equations are a powerful tool for modeling and solving various real-life problems, and their applications extend beyond this simple exam question.
在數學中，二次方程式是一個重要的概念，通常用於模擬各種現實生活中的情況。在本文中，我們將探討如何使用二次方程式來表示和分析汽車的行駛距離。

給定的考試題目給出了一個表示汽車行駛距離隨時間變化的二次方程式。這個方程式是：S = -t^2 + 60t，其中 S 表示行駛距離（以公里為單位），t 表示行駛時間（以小時為單位）。

為了回答這些問題，讓我們先了解一下這個二次方程式的特性。一個二次方程式的一般形式是 Ax^2 + Bx + C，其中 A、B 和 C 是常數。在我們的情況下，方程式是 -t^2 + 60t，它符合一般形式。

現在，讓我們逐一回答這些問題：

1. 為了確定汽車何時達到最大距離，我們可以使用二次方程式的頂點公式：x = -B/2A。在我們的方程式中，A = -1，B = 60。代入這些值，我們可以得出 t = -60/(2*-1)，這簡化為 t = 30。因此，汽車在行駛 30 小時後達到最大距離。

2. 為了找出行駛的最大距離，我們將 t = 30 的值代入方程式 S = -t^2 + 60t。計算此表達式，我們得到 S = -(30)^2 + 60(30)，這簡化為 S = -900 + 1800 = 900。因此，汽車的最大行駛距離為 900 公里。

3. 為了確定汽車何時返回到 0 公里的距離，我們可以把方程式 S = -t^2 + 60t 設為 0，然後求解 t。重新排列方程式，我們得到 -t^2 + 60t = 0。因式分解 t，我們得到 t(-t + 60) = 0。由此，我們可以得出結論，要麼 t = 0，要麼 -t + 60 = 0。解第二個方程式，我們找到 -t = -60，這簡化為 t = 60。因此，汽車在行駛 60 小時後返回到 0 公里的距離。

總之，通過理解和應用二次方程式的概念，我們能夠分析和回答與汽車行駛距離相關的問題。二次方程式是用於模擬和解決各種現實問題的強大工具，其應用不僅限於這個簡單的考試題目中。

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