The height above the ground of a stone thrown upwards is given by s(t), where t is measured in seconds. After 1 second, the height of the stone is 48 feet above the ground, and after 1.5 seconds, the height of the stone is 60 feet above the ground. Evaluate s(1) and s(1.5), and then find the average velocity of the stone over the time interval [1, 1.5].

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First, understand that s(t) represents the height of the stone at time t. We are given s(1) = 48 feet and s(1.5) = 60 feet. These values indicate the height of the stone at 1 second and 1.5 seconds, respectively.

To find the average velocity of the stone over the time interval [1, 1.5], use the formula for average velocity: \( v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \), where \( t_1 = 1 \) and \( t_2 = 1.5 \).

Substitute the given values into the average velocity formula: \( v_{avg} = \frac{s(1.5) - s(1)}{1.5 - 1} \).

Calculate the difference in height: \( s(1.5) - s(1) = 60 - 48 \).

Divide the difference in height by the difference in time: \( v_{avg} = \frac{12}{0.5} \). This will give you the average velocity of the stone over the interval [1, 1.5].

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. In this context, evaluating s(1) and s(1.5) means finding the height of the stone at 1 second and 1.5 seconds, respectively. This is crucial for understanding the stone's position at those specific times.

Average Velocity

Average velocity is defined as the change in position over the change in time. It can be calculated using the formula (s(t2) - s(t1)) / (t2 - t1). In this problem, the average velocity of the stone over the interval [1, 1.5] is determined by finding the difference in height at these two times and dividing by the time interval.

Rate of Change

Rate of change refers to how a quantity changes with respect to another variable, often time. In calculus, this concept is fundamental for understanding motion, as it relates to how quickly the height of the stone changes as time progresses. The average velocity calculated in this problem is a specific instance of the rate of change of the stone's height with respect to time.