(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. Understand that grade resistance is the component of the car's weight acting along the slope due to gravity. It can be found using the formula: \(\text{Grade Resistance} = W \times \sin(\theta)\), where \(W\) is the weight of the car and \(\theta\) is the angle of the slope.
2. Identify the given values: the weight of the car \(W = 2400\) pounds, and the slope angle \(\theta = -2.4^\circ\). The negative sign indicates a downhill grade.
3. Calculate the sine of the angle \(\theta\). Since the angle is negative, \(\sin(-2.4^\circ) = -\sin(2.4^\circ)\), which means the grade resistance will be negative, representing a force aiding the motion downhill.
4. Multiply the weight \(W\) by \(\sin(\theta)\) to find the grade resistance: \(2400 \times \sin(-2.4^\circ)\).
5. Interpret the result: the magnitude gives the force in pounds, and the negative sign indicates the direction is downhill. Round the magnitude to the nearest ten pounds as requested.

Key Concepts

Grade Resistance

Grade resistance is the force opposing the motion of a vehicle due to the slope of the road. It depends on the vehicle's weight and the angle of the incline or decline, calculated as the component of gravitational force parallel to the slope.

Trigonometric Functions and Angles

Understanding how to use trigonometric functions, especially sine, is essential to relate the angle of the slope to the forces involved. The sine of the grade angle gives the ratio of the vertical height change to the hypotenuse, which helps find the force component along the slope.

Force Calculation Using Weight and Angle

The grade resistance force is found by multiplying the vehicle's weight by the sine of the slope angle. For a downhill grade, this force acts in the direction of motion, and its magnitude is proportional to both the weight and the sine of the negative angle.