(Modeling) Grade Resistance Solve each problem. See Example 3. Find the grade resistance, to the nearest ten pounds, for a 2400-lb car traveling on a -2.4° downhill grade.

Verified step by step guidance

1. Identify the weight of the car, which is 2400 lbs, and the angle of the downhill grade, which is -2.4°.
2. Understand that grade resistance is the component of the car's weight acting parallel to the slope of the hill. This can be calculated using the formula: \( R = W \cdot \sin(\theta) \), where \( R \) is the grade resistance, \( W \) is the weight of the car, and \( \theta \) is the angle of the slope.
3. Substitute the given values into the formula: \( R = 2400 \cdot \sin(-2.4°) \).
4. Calculate \( \sin(-2.4°) \). Remember that the sine of a negative angle is the negative of the sine of the positive angle, so \( \sin(-2.4°) = -\sin(2.4°) \).
5. Multiply the weight of the car by the sine of the angle to find the grade resistance: \( R = 2400 \cdot (-\sin(2.4°)) \). Round the result to the nearest ten pounds.

Key Concepts

Grade Resistance

Grade resistance refers to the force opposing the motion of a vehicle due to the incline or decline of the road. It is calculated based on the weight of the vehicle and the angle of the grade. A negative grade indicates a downhill slope, which reduces the effective weight of the vehicle acting against gravity.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In the context of grade resistance, these functions help determine the vertical and horizontal components of the weight of the vehicle based on the angle of the grade. Understanding these functions is essential for calculating forces acting on the vehicle.

Force Calculation

Force calculation involves determining the net force acting on an object, which is essential for understanding motion. In this scenario, the force due to gravity acting on the car must be resolved into components to find the grade resistance. This requires applying Newton's laws of motion and understanding how forces interact on inclined planes.