Problem 47 - Introductory Algebra for College Students

Question

When the shot is released at an angle of \(35^{\circ},\) its height, \(y,\) in feet, can be modeled by $$y=-0.01 x^{2}+0.7 x+6.1$$ where \(x\) is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through(c) and verify your answers using the blue graph. a. What is the maximum height of the shot and how far from its point of release does this occur? b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw? c. From what height was the shot released?

Step-by-Step Solution

Answer

The shot reaches a maximum height of \(18.85\) feet at a horizontal distance of \(35\) feet from its release point. The shot's maximum horizontal distance is \(70.0\) feet and it was released from a height of \(6.1\) feet.

Step 1: Find the maximum height of the shot

The maximum height corresponds to the highest point of the parabola, or the vertex. The x-coordinate of the vertex of a parabola given by \(y=ax^2+bx+c\) can be found using the formula \(-\frac{b}{2a}\). Using this formula with \(-0.01\) for \(a\) and \(0.7\) for \(b\), we find that the x-coordinate of the vertex is \(35\) feet. Substituting \(35\) into the equation for \(y\), we find that the maximum height attained is \( \(-0.01 * 35^2 + 0.7 * 35 + 6.1\) or approximately \(18.85\) feet.

Step 2: Find the maximum horizontal distance

The maximum horizontal distance will be where the parabola intersects the x-axis, making \(y = 0\). We can find the roots of a quadratic function using the quadratic formula \(\frac{-b ± \sqrt{b^2 - 4ac}}{2a}\). So, solving for \(x\) when \(y = 0\) using the values of \(a = -0.01\), \(b = 0.7\), and \(c = 6.1\), we find that the smaller root is negative (which is not possible in this context) and the larger root is approximately \(70.0\) feet. Therefore, the maximum horizontal distance or the throw distance is \(70.0\) feet.

Step 3: Find the release height of the shot

The height from which the shot was released would be the value of \(y\) when \(x = 0\), which is the y-intercept. By substituting \(x = 0\) into the given equation, we find that the value of \(y\) or \(y = -0.01 * 0^2 + 0.7 * 0 + 6.1\) which shows that the shot was released from a height of \(6.1\) feet.

Key Concepts

Parabolic Motion

Parabolic motion is a fascinating concept in mathematics and physics that describes the path of objects thrown or propelled into the air, such as a shot put or a thrown ball. This type of motion is characterized by a curved trajectory, known as a parabola.

The beauty of parabolic motion lies in its predictability, as it follows a specific mathematical model. In this case, the height of an object, say a shot, can be expressed as a quadratic function:

• The general form of the quadratic equation is: \( y = ax^2 + bx + c \)
• The curved trajectory is due to the acceleration caused by gravity, represented by the coefficient \(a\).
• The coefficients \(b\) and \(c\) describe the initial velocity and release height, respectively.

By understanding these parameters, we can predict the object's path and determine key points, such as maximum height and distance. This is particularly useful in activities that involve trajectory analysis, like sports or engineering.

Vertex of a Parabola

The vertex of a parabola is a crucial point on a curve that represents either the maximum or minimum value of the quadratic function, essentially the highest or lowest point of the trajectory.

In mathematics, the importance of the vertex can be seen in:

• Determining the maximum height that an object reaches when modeled by a parabolic equation.
• Identifying the point at which the direction of the motion changes.

To find the x-coordinate of the vertex in the general equation \(y = ax^2 + bx + c\), use:

• \( x = -\frac{b}{2a} \)

Substituting the given values from the original problem, a value of 35 feet for the x-coordinate indicates the horizontal distance where the shot reaches its maximum height.
By inserting this value back into the quadratic equation, one calculates the maximum height the shot attains, which in the original task was found to be approximately 18.85 feet.

Quadratic Formula

The quadratic formula is a robust tool used to find the roots of a quadratic equation, which are the values of \(x\) where the function equals zero. These roots are also known as the solutions or x-intercepts of the equation.

The formula is given by:

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

This expression allows us to solve for \(x\) when the height \(y\) becomes zero, determining the horizontal distance where an object returns to ground level.

In practice:

• Applying the quadratic formula involves identifying the coefficients \(a\), \(b\), and \(c\), then plugging them into the formula to solve for \(x\).
• From the original problem, using \(a = -0.01\), \(b = 0.7\), and \(c = 6.1\), it was determined that the larger root, 70 feet, represented the distance where the shot landed.

This method ensures that you consider all potential values, even discarding non-physical ones to find meaningful solutions for real-life problems.