Problem 83 - Algebra and Trigonometry

Question

A force of magnitude 700 pounds is required to hold a boat and its trailer in place on a ramp whose incline is \(10^{\circ}\) to the horizontal. What is the combined weight of the boat and its trailer?

Step-by-Step Solution

Answer

The combined weight of the boat and its trailer is approximately 4032.69 pounds.

Step 1: Identify the relationship

To solve for the combined weight, recognize that the force required to hold the boat and trailer in place is the component of the weight parallel to the incline.

Step 2: Set up the equation

The weight component parallel to the incline is given by: where: - W is the combined weight of the boat and trailer. This equation can be rearranged to solve for W: \[ F = W \times \sin(\theta) \]

Step 3: Substitute the given values

Given: \( F = 700 \) pounds and \( \theta = 10^{\circ} \) Substitute these values into the equation: \[ 700 = W \times \sin(10^{\circ}) \]

Step 4: Solve for the weight

Solving for W: \[ W = \frac{700}{\sin(10^{\circ})} \] Use a calculator to find \( \sin(10^{\circ}) \approx 0.1736 \). Then, \[ W = \frac{700}{0.1736} \approx 4032.69 \] Hence, the combined weight is approximately 4032.69 pounds.

Key Concepts

Force Resolution

When dealing with forces in physics, it's important to break them down into components. This process is known as force resolution. In our exercise, the force holding the boat and trailer is 700 pounds. Imagine this force as an arrow pointing up the inclined plane. Now, this force can be split into two parts: one perpendicular to the slope and another parallel to the slope.

The force parallel to the incline is what we're focusing on. This component directly opposes the weight's component that tends to move the boat down the slope. By breaking down forces, we can more easily analyze and solve problems involving different directions.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent help us relate angles to side lengths in a right triangle. In our problem, we use the sine function. Imagine a right triangle formed by:
- the incline of the ramp
- the force parallel to the incline
- and the weight acting downward.

Here's the relationship in our context: \( \text{Force parallel} = W \times \text{sin}( \theta ) \).
The sine function allows us to find the component of a force along an incline using the angle and the total weight. This simplification makes our calculations clearer and more manageable.

Inclined Planes

An inclined plane is a flat surface tilted at an angle to the horizontal. This setup is common in physics problems to study the forces and motion objects experience on these surfaces. Think of a ramp or a hill.

When objects rest on an inclined plane, gravity's force on them can be resolved into two components: one perpendicular to the plane and one parallel. The parallel component tries to slide the object down. The question asks us to find out how much the incline affects the weight's force parallel to the plane, needed to prevent the boat and trailer from moving.

Weight Calculation in Physics

Weight calculation in physics often involves decomposing forces using angles and trigonometric functions. In our example, the weight of the boat and trailer is the force due to gravity acting straight down. To balance this weight on a slope, we use the weight's component along the slope.

By recognizing that \( F = W \times \text{sin}( \theta ) \), we substitute known values to find: \[ \text{Weight} = \frac{\text{Force}}{\text{sin}( \theta )} = \frac{700 \text{ pounds}}{\text{sin}(10^\text{\textdegree})} \approx 4032.69 \text{ pounds} \]. This calculation shows us the effective weight holding the boat in place on the incline.