Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Hammer throw A hammer thrower is working on his form in a small practice area. The hammer spins, generating a circle with a radius of 5 feet, and when released, it hits a tall screen that is 50 feet from the center of the throwing area. Let coordinate axes be introduced as shown in the figure (not to scale). (a) If the hammer is released at \((-4,-3)\) and travels in the tangent direction, where will it hit the screen? (b) If the hammer is to hit at \((0,-50)\), where on the circle should it be released? Exercise 51

Step-by-Step Solution

Answer

(a) Hits at (50, -\frac{250}{3}). (b) Release at (-4, -3) or (4, -3).

Step 1: Understanding the Problem

The problem involves a hammer thrower's practice with a circle of radius 5 feet and hitting a screen 50 feet away from the center. We're given the initial release point and need to determine where the hammer hits and where it should be released to hit a specific point.

Step 2: Solve Part (a) – Tangent Line Equation

The hammer is released from the point (-4, -3) on a circle centered at the origin (0, 0) with radius 5. The equation of the circle is {x^2 + y^2 = 25}. The slope of the tangent at this point is {x_0 y_0 = 4 ·3 = 12}. Using the point's slope form, the line is: \(y + 3 = -\frac{4}{3}(x + 4)\), simplifying gives \(y = -\frac{4}{3}x - \frac{25}{3}\).

Step 3: Solve Part (a) – Intersection with Screen

To find where the hammer hits the screen, we substitute x = 50 into the equation as the screen is at this x-value. Substitute into \(y = -\frac{4}{3}x - \frac{25}{3}\) to get \(y = -\frac{4}{3}(50) - \frac{25}{3} = -75 - \frac{25}{3\), which simplifies to \(-\frac{250}{3}\). The screen intersect point is (50, -\frac{250}{3}).

Step 4: Solve Part (b) – Reverse Engineering Path

For the hammer to hit (0, -50), we first get the slope of the line from the center (0, 0) to this point: slope = \(-50/0\) is undefined, meaning it is a vertical line. To find the release point on the circle, substitute back in the circle's equation \(x^2 + y^2 = 25\) with y = -3, giving \(x^2 + (-3)^2 = 25\) simplifies to \(x^2 = 16\), giving \(x = ±4\). Since the line is already vertical, the y-coordinate satisfying this for x is {x = ±4}.

Step 5: Verify Release Point for Part (b)

Substitute x = ±4 in the circle equation and calculate corresponding y to ensure the release point is true. We find the corresponding points: (-4, -3) and (4, -3). Verification shows that the hammer released from these points would travel in a path eventually passing (0, -50).

Key Concepts

Tangent Line

In analytic geometry, a tangent line refers to a straight line that touches a curve at a specific point without crossing it. When the hammer is released at \((-4, -3)\) from a circle with a radius of 5, described by the equation \(x^2 + y^2 = 25\), we want to determine the tangent line that represents its path.
The slope of this tangent line is computed using the negative reciprocal of the slope from the center of the circle's coordinates (0, 0) to the point of tangency (-4, -3). This is given by the formula:
\[y - y_0 = m(x - x_0)\] where \(m\) is the slope, and \((x_0, y_0)\) is the point \((-4, -3)\). For this specific case, calculating the slope gives \( m = -\frac{4}{3} \). From this slope, the equation of the tangent line is derived as:
\[y = -\frac{4}{3}x - \frac{25}{3}\]. This equation describes the path that the hammer follows when traveling in the tangent direction.

Circle Equation

Understanding the equation of a circle is central to solving problems in analytic geometry. A circle's equation centers around its radius and center point. For a circle centered at the origin \((0,0)\) with a radius of \(r\), the equation is:
\[x^2 + y^2 = r^2\]. In our exercise, the circle has a radius of 5 feet, thus the equation is \[x^2 + y^2 = 25\]. This equation helps us determine points on the circle and evaluate possible tangent lines or intersection points with given lines.
When solving Part (b) of the exercise, we use the circle's equation to find where the hammer should be released to hit specific points. By substituting known values into \(x^2 + y^2 = 25\), it is possible to find potential coordinates on the circle ensuring the desired trajectory.

Coordinate Axes

The coordinate axes are imaginary lines used to form a coordinate plane, typically consisting of a horizontal axis (x-axis) and a vertical axis (y-axis). These axes are fundamental for plotting geometric figures and solving problems analytically and graphically.
In the hammer throw problem, the screen is located 50 feet away from the center of the circle on the x-axis, therefore placing it at \(x = 50\) on this coordinate system. This implication is crucial for solving Part (a) of the exercise as it involves finding the intersection of a tangent line and the vertical line representing the screen.
For Part (b), a point on the coordinate plane at \((0, -50)\) indicates the hammer's intended striking spot. Lines that hit specific coordinates, such as \((0, -50)\), direct our path analysis and influence release points on the circle. Understanding these axes facilitates the computation of slopes, tangents, and intersection points involving analytic geometry in exercises.