Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Water is flowing from a horizontal pipe 48 feet above the ground. The falling stream of water has the shape of a parabola whose vertex (0,48) is at the end of the pipe (see figure). The stream of water strikes the ocean at the point \((10 \sqrt{3}, 0) .\) Find the equation of the path taken by the Water.

Step-by-Step Solution

Answer

The equation of the path taken by the water being drained from the pipe is \(y = -0.16x^{2} + 48\)

Step 1: Understanding the Parabola

The equation of a parabola with vertex at (h, k) is given by \(y = a(x-h)^{2} + k\), where 'a' is a coefficient that determines the width of the parabola. When 'a' > 0, the parabola opens upwards, and when 'a' < 0, the parabola opens downwards. Here, since the direction of the water stream is downwards, 'a' should be less than zero.

Step 2: Replacing the Vertex

Replace the h and k in the parabola equation with the given vertex (0, 48). Thus our equation becomes \(y = a(x-0)^{2} + 48\) or \(y = ax^{2} + 48\).

Step 3: Finding the Value of 'a'

Now, let's replace the x and y in the equation with the coordinates of the point where the water hits the ocean, which is (10√3, 0). Now our equation is \(0 = a(10√3)^{2} + 48\). Solving the equation gives us a = -48/(300), or a = -8/50, or a = -0.16.

Step 4: Forming the Final Equation

Substitute the value of 'a' back into equation from Step 2 to get the final equation of the parabolic path of the water stream, which is \(y = -0.16x^{2} + 48\)

Key Concepts

Parabola in Physics

The concept of a parabola in physics plays a crucial role in understanding the trajectory of objects under the influence of gravity. Physically, a parabola represents the path followed by an object when it is propelled into the air and is acted upon by gravity, assuming there is no air resistance.

A parabola is defined by the general quadratic equation in the form of \(y = a(x-h)^{2} + k\), where the vertex of the parabola is at the point (h, k), and the coefficient 'a' signifies the parabola's width and direction. In the scenario of our exercise, the parabola represents the path of water flowing from a pipe. Since the water is falling downward due to gravity, we can infer that our parabola opens downwards and, hence 'a' should be negative.

This understanding allows us to model the situation accurately, and in this instance, we identified that when the water stream strikes the ground, its trajectory is described by an equation that correlates to the description of a parabolic curve. Given the vertex as the starting point of the stream and the point where it impacts the ground, the parabolic motion equation is a precise mathematical representation of the physical event.

Projectile Motion

Projectile motion is a form of motion experienced by an object that is thrown near the Earth’s surface and moves along a curved path under the action of gravity alone. The path that a projectile follows is a parabola. Understanding projectile motion is essential for solving problems that involve objects being launched or thrown.

In our scenario, the water from the pipe is essentially a projectile once it exits the pipe. The factors governing the projectile’s trajectory are its initial velocity, angle of projection, and the acceleration due to gravity. While these factors are not explicitly stated in our exercise, they are inherent to the shape of the parabolic path defined by the equation we derived.

Components of Projectile Motion

Two independent components make up projectile motion:

• Horizontal motion: Consistent and unvarying, as it is not affected by gravity.
• Vertical motion: Influenced by gravity, causing the projectile to accelerate downwards.

By mathematically combining these two motions, we obtain the classic parabolic shape of a projectile's trajectory, characterized by our quadratic equation.

Quadratic Equations

Quadratic equations are a category of polynomial equations that generally take the form of \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are constants. These equations are pivotal in various areas of algebra and play an essential role in physics, especially when analyzing motions like those of projectiles.

In the form of our exercise, the quadratic equation doesn't have the 'b' and 'c' terms but is a simplified version, tailored to represent a parabola with only the parameter 'a' and constant term determining its shape. The process of solving for 'a' involves substituting known points on the parabola into this equation and solving for the unknown coefficients.

Solving Quadratic Equations

• The quadratic formula, which can find 'a' when a parabola intersects a certain point.
• Factoring, which we use if the quadratic equation can be easily decomposed into binomial factors.
• Completing the square, used to derive the quadratic formula itself or to turn an equation into vertex form.

The final equation \(y = -0.16x^2 + 48\) that we derived in our exercise is the answer to the quadratic equation that describes the path of water as it streams out of the pipe and falls to the ground.