(Modeling) Grade Resistance Solve each problem. See Example 3. A car traveling on a -3° downhill grade has a grade resistance of -145 lb. Determine the weight of the car to the nearest hundred pounds.

Verified step by step guidance

1. Understand that grade resistance (R) is related to the weight (W) of the car and the grade angle (θ) by the formula: \(R = W \sin(\theta)\).
2. Identify the given values: grade resistance \(R = -145\) lb and grade angle \(\theta = -3^\circ\) (negative because it is downhill).
3. Rewrite the formula to solve for the weight \(W\): \(W = \frac{R}{\sin(\theta)}\).
4. Calculate \(\sin(-3^\circ)\) using a calculator or trigonometric tables, remembering that sine of a negative angle is negative.
5. Substitute the values of \(R\) and \(\sin(\theta)\) into the formula to find \(W\), then round the result to the nearest hundred pounds.

Key Concepts

Grade Resistance

Grade resistance is the force opposing the motion of a vehicle due to the slope of the road. It is calculated as the component of the vehicle's weight acting along the incline, typically using the sine of the grade angle. A negative grade resistance indicates a downhill slope aiding the vehicle's motion.

Relationship Between Weight and Grade Resistance

The grade resistance force is directly proportional to the vehicle's weight and the sine of the grade angle. This relationship allows us to find the vehicle's weight by rearranging the formula: Grade Resistance = Weight × sin(grade angle). Knowing the grade resistance and angle, we can solve for the weight.

Trigonometric Functions in Inclined Planes

Trigonometric functions, especially sine, are used to resolve forces on inclined planes. The sine of the grade angle gives the ratio of the vertical component of the weight to the hypotenuse (weight). Understanding how to apply sine to find components of forces is essential in problems involving slopes.