Problem 72 - College Algebra with Corequisite Support

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 attained by the rocket is approximately 2911.59 meters.

Step 1: Identify the Function Type

The function given is a quadratic function in the form \( h(t) = -4.9t^2 + 229t + 234 \). Quadratic functions have the standard form \( ax^2 + bx + c \), where here \( a = -4.9 \), \( b = 229 \), and \( c = 234 \).

Step 2: Determine the Vertex Formula

The maximum or minimum point of a quadratic function is at its vertex. For a function in the form \( at^2 + bt + c \), the time \( t \) at which the vertex occurs is given by the formula \( t = \frac{-b}{2a} \).

Step 3: Calculate the Time at Vertex

Substitute \( a = -4.9 \) and \( b = 229 \) into the vertex formula: \[ t = \frac{-229}{2(-4.9)} = \frac{229}{9.8} \approx 23.37 \]. The time at which maximum height is reached is approximately 23.37 seconds.

Step 4: Calculate the Maximum Height

Now, substitute \( t = 23.37 \) back into the original height function to find the maximum height: \[ h(23.37) = -4.9(23.37)^2 + 229(23.37) + 234 \]. Compute each term: \( -4.9(23.37)^2 \approx -2679.14 \), \( 229(23.37) \approx 5356.73 \), and \( 234 \). Add these to get \( h(23.37) \approx 2911.59 \).

Step 5: Interpret the Result

The calculation tells us that the maximum height the rocket reaches is approximately 2911.59 meters above sea level.

Key Concepts

Vertex of Quadratic Function

In a quadratic function like the one given for the rocket's height, the vertex represents either the maximum or minimum point of the parabola formed by plotting the equation. A quadratic function has the general form

• \( ax^2 + bx + c \)
• where \(a\), \(b\), and \(c\) are constants. For our function, \(a = -4.9\), \(b = 229\), and \(c = 234\).

Because the coefficient \(a\) is negative in this equation, the parabola opens downwards, indicating a maximum point at the vertex. To find the vertex, you apply the vertex formula, which gives you the time \(t\) when the function reaches this extreme height or depth.

The specific vertex formula to find the time \(t\) of the vertex is:

• \( t = \frac{-b}{2a} \)

This formula is crucial for identifying the time at which the maximum height is reached. Once you figure out \(t\) using this method, you can substitute it back into the original quadratic equation to find the height at this time.

Maximum Height Calculation

After determining the time \(t\) using the vertex formula, the next step involves calculating the maximum height at this specific time. For our quadratic function

• \( h(t) = -4.9t^2 + 229t + 234 \)

we substitute the calculated time \( t = 23.37 \) seconds back into the function. This substitution allows us to solve for \(h(t)\), which gives us the maximum height that the rocket will achieve.

Breaking this calculation down, you will perform the following computations:

• Square the time: \( (23.37)^2 \)
• Multiply by \( -4.9 \)
• Multiply \( 229 \times 23.37 \)
• Add \( 234 \) to the result of the previous step

When these are completed, you sum the results—yielding the maximum height of approximately 2911.59 meters. This height represents how far above sea level the rocket reaches at its peak.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation given by the standard form \( ax^2 + bx + c = 0 \). This formula is essential when you need to calculate where the quadratic equation crosses the x-axis, but it can also have applications in other contexts involving quadratics.

For such calculations, the quadratic formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)

This formula effectively finds the values of \(x\) that satisfy the equation \( ax^2 + bx + c = 0 \). In the rocket height problem, while the quadratic formula is not directly employed to find the maximum height, understanding how to solve for \(x\) is crucial, especially if you're finding where the height becomes zero (i.e., when the rocket hits the ground) or predicting times for ascent and descent.

While in this exercise, we primarily focus on the vertex to determine maximum height, grasping the utility of the quadratic formula can significantly enhance problem-solving skills around quadratic functions.