Problem 254 - Precalculus

Question

The height of a piston, \(h,\) in inches, can be modeled by the equation \(y=2 \cos x+5,\) where \(x\) represents the crank angle. Find the height of the piston when the crank angle is \(55^{\circ} .\)

Step-by-Step Solution

Answer

The height is approximately 6.1472 inches when the crank angle is 55°.

Step 1: Understand the Problem

We need to find the height of the piston when the crank angle \( x \) is \( 55^{\circ} \). The formula given is \( y = 2\cos x + 5 \). Here, \( y \) represents the height of the piston.

Step 2: Substitute the Crank Angle into the Equation

Substitute \( x = 55^{\circ} \) into the equation: \( y = 2\cos(55^{\circ}) + 5 \).

Step 3: Calculate the Cosine Value

First, find the cosine of \( 55^{\circ} \):\[ \cos(55^{\circ}) \approx 0.5736 \]This value can be found using a calculator or trigonometry table.

Step 4: Calculate the Height

Substitute the cosine value back into the equation:\[ y = 2 \times 0.5736 + 5 \]Simplify the equation:\[ y = 1.1472 + 5 = 6.1472 \]

Step 5: Interpret the Result

The height of the piston when the crank angle is \( 55^{\circ} \) is approximately \( 6.1472 \) inches.

Key Concepts

Cosine function

The cosine function is a fundamental concept in trigonometry. It relates the angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse. In the unit circle, which is a circle of radius one centered at the origin of a coordinate plane, the cosine of an angle is the horizontal coordinate of a point on the circle's circumference. This means the cosine function varies between -1 and 1 as the angle changes from 0 to 360 degrees.
The mathematical representation of the cosine of an angle, \( x \), is written as \( \cos(x) \). This function is periodic, with a period of 360 degrees or \( 2\pi \) radians. With every complete cycle, it starts to repeat its values.
Understanding the cosine function helps in applications involving oscillatory behaviors, such as waves, circular motion, and here, in modeling the movement of a piston with changing crank angle. It allows us to predict how the height of the piston changes as the crank rotates.

Crank angle

The crank angle is a critical parameter in mechanical engineering, particularly when dealing with internal combustion engines or any mechanism involving rotational motion.

• The crank angle refers to the angular position of the crankshaft.
• It is measured in degrees, with most engines having a total crank angle range from 0 to 720 degrees (covering two 360-degree rotations).

In the context of the piston movement modeled by our cosine function, the crank angle directly affects the piston's height. This is because the piston's motion in a cylinder is typically not linear but follows a sinusoidal path. Hence, specifying the crank angle allows us to determine the exact vertical position of the piston at any given time.

Piston height

Piston height is another essential concept in the study of engines and rotational mechanisms. It refers to the vertical distance between the piston and a predefined reference point, often the base of the cylinder. The piston's height changes dynamically as the crankshaft rotates, a motion often optimized for efficiency and power in engine design.

• In our example, the piston height is represented by the equation \( y = 2\cos(x) + 5 \), where \( y \) is the piston height and \( x \) is the crank angle.
• The constant 2 adjusts the amplitude of the piston's movement, meaning the total vertical distance traveled is 4 inches (since 2 is doubled due to cosine's range from 1 to -1).
• The constant 5 raises the baseline height of the piston, ensuring it doesn't reach a negative height value.

The piston height equation helps engineers model and predict the real-world functioning of engines, providing insights into how changes in design parameters can affect overall performance.