Problem 51 - Calculus Early Transcendentals

Question

A stone is launched vertically upward from a cliff 192 feet above the ground at a speed of \(64 \mathrm{ft} / \mathrm{s}\). Its height above the ground \(t\) seconds after the launch is given by \(s=-16 t^{2}+64 t+192,\) for \(0 \leq t \leq 6 .\) When does the stone reach its maximum height?

Step-by-Step Solution

Answer

Answer: The stone reaches its maximum height at 2 seconds.

Step 1: Identify the quadratic function

The given height function is a quadratic function: \(s(t) = -16t^2 + 64t + 192\).

Step 2: Find the vertex of the parabola

The vertex of a parabola with the equation \(s(t) = at^2 + bt + c\) can be found using the formula \(t_v = \frac{-b}{2a}\). In this case, we have \(t_v = \frac{-64}{2(-16)}\).

Step 3: Calculate the time of the maximum height

Compute the value of \(t_v\) to find the time when the stone reaches its maximum height: \(t_v = \frac{-64}{2(-16)} = \frac{-64}{-32} = 2\) seconds. So, the stone reaches its maximum height at \(t = 2\) seconds.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point because it represents either the maximum or the minimum point of the quadratic function. For a function in the form of \(f(t) = at^2 + bt + c\), the vertex can be found using the formula \(t_v = \frac{-b}{2a}\).
This formula gives us the time when the maximum or minimum height occurs. In the given problem, the coefficient \(a\) is \(-16\) and \(b\) is \(64\).

• This makes \(t_v = \frac{-64}{2(-16)} = 2\) seconds.

This calculation indicates that the stone reaches its maximum height 2 seconds after it is launched. Identifying the vertex of a parabola is especially useful in real-life scenarios like trajectory calculations. The vertex encompasses all the information about the direction and the extremum (maximum/minimum) of the motion.

Parabola

A parabola is a U-shaped curve that can open either upwards or downwards. In the context of quadratic functions, its shape is dictated by the leading coefficient \(a\).

• If \(a\) is negative, as in this exercise (\(-16\)), the parabola opens downwards, indicating a maximum point.
• If \(a\) were positive, it would open upwards, suggesting a minimum point.

Parabolas are essential in mathematics because they model many natural phenomena, like projectile motion.
When dealing with problems like this exercise, understanding the basic properties of parabolas and their opening direction can anticipate the nature and location of their extremum - in this case, the maximum height of the stone.

Maximum Height

The maximum height in a quadratic function context is an important characteristic because it signifies the highest point reached in the parabola's curve.
When dealing with physical situations, like the trajectory of a stone, the maximum height corresponds to the highest point above ground reached by the object.
To find the maximum height:

• First, determine the time \(t\) when the maximum height is reached using the vertex formula \(t_v = \frac{-b}{2a}\).
• Compute the maximum height by substituting \(t_v\) back into the original equation.

For our stone, the maximum height occurs at \(t = 2\). Plugging this back into the height equation, \(s(2) = -16(2)^2 + 64(2) + 192\) gives the stone's maximum height of 256 feet. This simplifies complex motion calculations by breaking them into comprehensible steps using algebraic methods.