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QuestionFind the grade resistance for a 2000-pound car on a 2.4∘2.4^{\circ} uphill grade using F=Wsin⁡θF=W \sin \theta. Answer in pounds.

Studdy Solution

STEP 1

Assumptions1. The weight of the car (W) is2000 pounds. The grade of the hill (θ) is.4 degrees3. The force due to gravity (grade resistance, F) is calculated using the formula =Wsin⁡θ=W \sin \theta

STEP 2

First, we need to convert the grade of the hill from degrees to radians, because the sin function in the formula uses radians. We can do this using the conversion factor π180\frac{\pi}{180}.
θradians=θdegrees×π180\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}

STEP 3

Now, plug in the given value for the grade of the hill to calculate the grade in radians.
θradians=2.∘×π180\theta_{radians} =2.^{\circ} \times \frac{\pi}{180}

STEP 4

Calculate the grade in radians.
θradians=2.4∘×π180≈0.0419 radians\theta_{radians} =2.4^{\circ} \times \frac{\pi}{180} \approx0.0419 \, radians

STEP 5

Now that we have the grade in radians, we can find the grade resistance () using the formula =Wsin⁡θ=W \sin \theta.=Wsin⁡θradians = W \sin \theta_{radians}

STEP 6

Plug in the values for the weight of the car and the grade in radians to calculate the grade resistance.
=2000 pounds×sin⁡(0.0419 radians) =2000 \, pounds \times \sin(0.0419 \, radians)

STEP 7

Calculate the grade resistance.
=2000 pounds×sin⁡(0.0419 radians)≈83. pounds =2000 \, pounds \times \sin(0.0419 \, radians) \approx83. \, pounds

STEP 8

Since the problem asks for the grade resistance to be rounded to the nearest pound, we round83.8 pounds to84 pounds.
The grade resistance is84 pounds.

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