Problem 65 - Algebra 1

Question

In Exercises 65 and 66 use the following information. A bottlenose dolphin jumps out of the water. The path the dolphin travels can be modeled by \(h=-0.2 d^{2}+2 d,\) where \(h\) represents the height of the dolphin and \(d\) represents horizontal distance. What is the maximum height the dolphin reaches?

Step-by-Step Solution

Answer

Therefore, the maximum height that the dolphin reaches is 5 units.

Step 1: Determine the Values of a and b in the Given Function

From the quadratic function \( h=-0.2d^{2}+2d \), we can see that \( a = -0.2 \) and \( b = 2 \). We don't require the value of \( c \) as it won't be used in our calculation to find the maximum height.

Step 2: Calculate the Value of d at the Vertex of the Parabola

Substitute the values of \( a \) and \( b \) into the formula \( d = -b/{2a} \). \nSo, \( d = -2/{-0.4} = 5 \)

Step 3: Substitute the Value of d into the Function

The maximum height is obtained by substituting \( d = 5 \) into the quadratic function \( h = -0.2*(5)^{2}+2*5 \), which gives \( h = 5 \).

Key Concepts

Vertex of a Parabola

When analyzing quadratic functions like the one representing the dolphin's path, the vertex of the parabola is crucial. In a quadratic function of the form \( ax^2 + bx + c \), the vertex is either the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards.
For a downward-opening parabola, which occurs when the coefficient \( a \) is negative, the vertex is the maximum point.
The formula to find the vertex is \( x = -\frac{b}{2a} \). This gives us the x-coordinate (or \( d \), in the dolphin's case) of the vertex. By substituting this value back into the equation, we find the corresponding y-coordinate, or height.

• The vertex provides the maximum height the dolphin reaches during its jump.
• It succinctly represents the peak of the parabolic motion.

Understanding the vertex helps in determining critical points in parabolas, particularly in real-world contexts like physical motions.

Maximum Height

Every jump or throw that follows a parabolic path has a maximum height. This height is the highest point reached during the trajectory. For the dolphin's jump described by the quadratic function \( h = -0.2d^2 + 2d \), finding the maximum height involves determining where the peak of the parabola occurs.
We calculated the distance \( d \) at the vertex using \( d = -\frac{b}{2a} \) which was found to be 5. Substituting this value back into the equation gives us the height: \( h = -0.2(5)^2 + 2 \times 5 \), resulting in a height of 5 meters.

• The maximum height is reached at the vertex of the parabola.
• This value is significant as it showcases the peak performance in physical motion.

Identifying the maximum height can help predict outcomes and plan actions accordingly, essential in areas like physics and engineering.

Parabolic Motion

Parabolic motion is a fascinating concept observed in physical phenomena, such as the flight of projectiles or a dolphin's jump out of the water. This motion follows a curved path due to gravity, forming a parabola. In the equation \( h = -0.2d^2 + 2d \), the path of the dolphin's jump is plotted as a parabola in which the horizontal distance is transformed through the function to determine height.
A few attributes of parabolic motion include:

• The symmetry around the vertex, indicating even distance on both sides of the peak.
• The speed and angle of launch affect the initial shape and reach of the parabola.

Understanding parabolic motion is vital in sports, engineering, and various applications where predicting path trajectories is necessary.

Quadratic Equation

A quadratic equation is an expression that includes terms up to the second degree, typically represented as \( ax^2 + bx + c = 0 \). In our specific exercise, it is expressed as a function modeling the height of a dolphin jump: \( h = -0.2d^2 + 2d \). Quadratic equations are instrumental in depicting parabolic curves.
The importance of quadratic equations lies in their versatility:

• They model real-world situations, such as projectile motion and architectural designs.
• Solve for zeros or roots, showing when a projectile hits the ground.

In the dolphin problem, understanding the quadratic equation enabled us to predict not only the maximum height but also how different launches or changes in initial motion might affect the jump's trajectory.