Problem 45 - Beginning and Intermediate Algebra

Question

Two physics students from American River College find that when a bottle of California sparkling wine is shaken several times, held upright, and uncorked, its cork travels according to the function$$s(t)=-16 t^{2}+64 t+1$$where \(s\) is its height in feet above the ground \(t\) seconds after being released. After how many seconds will it reach its maximum height? What is the maximum height?

Step-by-Step Solution

Answer

The cork reaches its maximum height after 2 seconds, and the maximum height is 65 feet.

Step 1: Identify the function and its parameters

The given function is \[s(t)=-16 t^{2}+64 t+1\]. This is a quadratic function of the form \[s(t) = at^2 + bt + c\]. Here, \[a = -16\], \[b = 64\], and \[c = 1\].

Step 2: Determine the time to reach maximum height

For a quadratic function \[s(t) = at^2 + bt + c\], the time \(t\) at which the maximum height is achieved can be found using the vertex formula \[t = -\frac{b}{2a}\]. Substitute \[a = -16\] and \[b = 64\] into the formula: \[t = -\frac{64}{2(-16)} = 2\] seconds.

Step 3: Calculate the maximum height

Substitute \[t = 2\] back into the function \[s(t)=-16 t^{2}+64 t+1\] to find the maximum height: \[s(2) = -16(2)^2 + 64(2) + 1\]. Calculate the values step-by-step: \(-16(4) = -64\), \(64(2) = 128\), so \[s(2) = -64 + 128 + 1 = 65\]. The maximum height is 65 feet.

Key Concepts

maximum height

Understanding the maximum height of a projectile, like a cork from a wine bottle, is crucial in many physics applications. The maximum height is the highest point that the cork reaches above the ground after it is launched. In the given function, \(s(t)=-16 t^{2}+64 t+1\), it represents the height of the cork at any time \(t\) seconds. Quadratic functions like this one form a parabola. When the parabola opens downwards (as indicated by a negative coefficient of \(t^2\)), the vertex represents the maximum point. This vertex is where the maximum height occurs for the cork in question.

vertex formula

To find the time when the cork reaches its maximum height, we use the vertex formula. For any quadratic function \(at^2 + bt + c\), the time at which the maximum or minimum value is reached is given by \(t = -\frac{b}{2a}\). This formula comes from calculus and represents the axis of symmetry of the parabola. In the given function, \(a = -16\) and \(b = 64\), so substituting into the formula gives us:
\[t = -\frac{64}{2(-16)} = 2\]
This means the cork reaches its maximum height at 2 seconds after being released.

physics applications

Understanding the vertex formula and maximum height has many practical physics applications. For example, it helps in determining the peak point of any projectile, including sports balls, fireworks, or even engineering contexts. In our case, calculating the maximum height \(s(2)\) confirms that the cork reaches 65 feet above the ground. By substituting 2 seconds back into the height function \(s(t)\), the equation is:
\[s(2) = -16(2)^2 + 64(2) + 1\]
Breaking it down we calculate:
\(-16(4) = -64\)
\(64(2) = 128\)
Then adding these results and the constant, \(65\):
The maximum height is critical for understanding and predicting the behavior of objects in motion. So, whether you're studying simple trajectories in physics class or working on complex engineering projects, these fundamental principles remain the same.