Problem 73 - College Algebra

Question

A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is is iven by \(h(t)=-4.9 t^{2}+24 t+8 .\) How long does it take to reach maximum height?

Step-by-Step Solution

Answer

2.45 seconds

Step 1: Understand the Problem

We need to find the time, \(t\), at which the ball reaches its maximum height given the height function \(h(t) = -4.9t^2 + 24t + 8\). This is a parabolic equation in the standard form \(at^2 + bt + c\). The maximum height occurs at the vertex of the parabola.

Step 2: Identify the Formula for Time of Maximum Height

For a quadratic equation of the form \(ax^2 + bx + c\), the time at which the maximum (vertex) occurs is given by the formula \(t = \frac{-b}{2a}\). Here, \(a = -4.9\) and \(b = 24\).

Step 3: Substitute Values into the Vertex Formula

Substitute \(a = -4.9\) and \(b = 24\) into the formula: \[ t = \frac{-b}{2a} = \frac{-24}{2(-4.9)} \]

Step 4: Perform the Calculation

Calculate the expression: \[ t = \frac{-24}{-9.8} = \frac{24}{9.8} \]This simplifies to approximately \(t \approx 2.45\) seconds when we perform the division.

Key Concepts

maximum height

When a ball is thrown into the air, it follows a curved path and at its highest point, it reaches what we call the "maximum height". In the context of a quadratic function representing the motion, this maximum occurs at the vertex of the parabola. The height equation is usually given in the form of a parabolic equation, such as \( h(t) = -4.9t^2 + 24t + 8 \).

To find the maximum height, we are actually looking for the vertex of the parabola. Why? Because the vertex represents the peak or the highest point in the trajectory of the ball.

In practical applications, calculating the maximum height involves determining the time it takes to reach that point and then simply evaluating the height function at this specific time. This gives you the highest point in the air above the ground that the ball will reach.

vertex formula

The vertex formula is a handy piece of math that helps you find the most critical point on a parabola - its vertex.

When dealing with quadratic functions, especially in physics problems involving projectiles or other motions, knowing the vertex is crucial. The formula to find the time \( t \) at which the maximum height or vertex occurs is

\[ t = \frac{-b}{2a} \]

Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation in the form \( ax^2 + bx + c \).

• The \( a \) value indicates the direction of the parabola (upwards or downwards).
• The \( b \) represents the linear coefficient that affects the slope.
• \( c \) is the constant term.

Using this formula, you can swiftly find out how long it will take for your ball, or any other projectile, to reach its highest point in flight.

parabolic equations

Parabolic equations are a type of quadratic function that forms a U-shaped graph known as a parabola.

The general form of a parabolic equation is \( ax^2 + bx + c \), and it holds significant importance in representing physical phenomena like projectile motion.

The coefficients in the equation influence the shape and position of the parabola:

• \( a \) determines if the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)). If it's negative like our exercise, the parabola opens downwards, indicating a maximum point.
• \( b \) affects the tilt of the parabola along the x-axis.
• \( c \) moves the parabola up or down along the y-axis.

In real-world scenarios, parabolas help us predict and understand the behavior of moving objects, like determining the trajectory a ball will take when thrown from a certain height.