Problem 72 - College Algebra

Question

A rocket is launched in the air. Its height, in meters, above sea level, as a function of time, in seconds, is given by \(h(t)=-4.9 t^{2}+229 t+234\) . Find the maximum height the rocket attains.

Step-by-Step Solution

Answer

The maximum height the rocket attains is approximately 2912.45 meters.

Step 1: Understand the Quadratic Function

The function given is \(h(t) = -4.9t^2 + 229t + 234\), which is a quadratic equation in the form \(ax^2 + bx + c\). The coefficient \(a = -4.9\) indicates the parabola opens downwards, meaning the vertex represents the maximum point of the function.

Step 2: Find the Vertex of the Parabola

To find the maximum height, calculate the vertex of the parabola. The time \(t\) at the vertex is given by the formula \(t = -\frac{b}{2a}\). Substitute \(b = 229\) and \(a = -4.9\) into the formula to find \(t\).

Step 3: Calculate \(t\) for Maximum Height

Using the formula \(t = -\frac{229}{2 \times -4.9}\), compute \(t = -\frac{229}{-9.8} \approx 23.37\) seconds. This is the time at which the rocket reaches its maximum height.

Step 4: Determine the Maximum Height

Substitute \(t = 23.37\) back into the original height function \(h(t) = -4.9t^2 + 229t + 234\) to find the maximum height. Calculate \(h(23.37) = -4.9(23.37)^2 + 229(23.37) + 234\).

Step 5: Solve for \(h(23.37)\)

Calculate \(-4.9(23.37)^2 + 229(23.37) + 234\) to find the maximum height. Simplify it to get \( \approx 2912.445 \) meters.

Key Concepts

Maximum Height

When dealing with quadratic functions in the form of \( ax^2 + bx + c \), identifying the maximum height is one of the primary tasks when analyzing the graph of this equation in problems like motion in physics or other scenarios involving vertical motion.
For the rocket's flight, the maximum height corresponds to the vertex of the parabola. Since the coefficient \( a = -4.9 \) is negative, the parabolic curve opens downwards, indicating the maximum point at the top of the curve. This maximum point represents the highest position the rocket reaches before descending.
To find this specific height, you must first compute the time \( t \) at which it happens, then substitute that value back into the quadratic equation. This process will give you the exact height above sea level at the peak of the rocket's flight.

Vertex of a Parabola

The vertex of a parabola is a crucial concept when dealing with quadratic equations, especially when you want to find optimal values like maximum or minimum heights. For any quadratic equation in the form \( ax^2 + bx + c \), the vertex can be used to identify these critical points.

• Formula: The vertex \( t \) in time is found using the formula \( t = -\frac{b}{2a} \),\ where \( b \) and \( a \) are the coefficients from the equation.

In our function \( h(t) = -4.9t^2 + 229t + 234 \), substituting \( b = 229 \) and \( a = -4.9 \) into the vertex formula gives \( t \approx 23.37 \) seconds. This time value represents the moment when the vertex reaches the top of its path, or equivalently, when the rocket reaches its highest altitude.

Solving Quadratic Equations

Solving quadratic equations involves several methods, but to find specific values like maximum or minimum heights, the vertex formula is most directly useful. For our rocket's height equation, substituting the calculated vertex time \( t \) into the original equation \( h(t) \) yields the maximum height.

• Substitution: Insert \( t = 23.37 \) into \( h(t) = -4.9t^2 + 229t + 234 \).
• Calculation: Compute \(-4.9(23.37)^2 + 229(23.37) + 234 \) which simplifies to approximately \( 2912.445 \) meters.

This result represents the peak of the rocket's flight and offers a practical example of solving quadratic equations to find real-world results. Understanding and mastering these concepts of quadratic functions enable students to tackle more complex physics and mathematics problems efficiently.