Find the force required to keep a 3000-lb car parked on a hill...

Question

Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizontal.

Accepted Answer

Hey, everyone here we are given a garbage truck is parked on a hilly road. If the weight of the truck is £7200 and the angle between the road and the horizontal is 21 degrees. What is the force required to keep the truck on its position? We have four answer choice options. A £2580.2. B £1997.9. C £2463.8 and D £2512.4. To begin this problem, we just want to create a sketch containing our given information. So to start the sketch, we need to draw in our horizontal line. And from there, we can draw in our hill which our, our garbage truck is parked on so we can draw this hill by just extending a line upward from our horizontal. And we know that the angle between the road and the horizontal is 21 degrees. So we can go ahead and label this angle. And now we can represent our garbage truck as just a rectangle on our hill. Next, we also know that the truck weighs £7200. So we need to represent this weight force as an arrow which extends from our truck directly downward. And we can label this arrow as £7200. And we must note that this arrow is perpendicular with the horizontal. And now we are asked to find the force required to keep our truck on this hill. So we need to represent this unknown force as an arrow which extends from our truck upwards along our hill. And now we just want to label this arrow as our unknown force. F. And with this, we have finished sketching our given information. So now we just need to start adding to this sketch so that we have enough information to solve four F. So our first step is to complete the triangle underneath our truck by just extending a line downward from the hill to the horizontal. And this line must be perpendicular to the horizontal so that it forms a 90 degree angle and thus completes our triangle. However, this triangle alone does not provide us enough information. So we just want to recreate this triangle utilizing our weight force. So next, we want to extend a line downward from the truck, except this time, the line must be perpendicular to the hill. And so now we have a line perpendicular to the hill and our weight forest which is perpendicular to the horizontal So this tells us that the angle in between these two lines is our 21 degree angle. And now to complete our triangle, we need to recall that our unknown force F is parallel with the hill. So we know it forms a 90 degree angle with our newly added line. So we just want to include this force, this known force on our triangle by extending an arrow from our new line to the weight force. And so we can now label our 90 degree angle which formed as well as label this side as force F. And with our new triangle, we have one angle known as well as the length of the hypotenuse known. So we have enough information to solve four F. So first, we want to recall that sine of theta is equal to the opposite side from the angle divided by the hypotenuse. So now utilizing all of our information and referencing our new triangle, we have that sign of 21 degrees is equivalent to F divided by £7200. And with this equation, we can begin solving for F by first multiplying both sides of the equation by the denominator on the right hand side which is £7200. And now multiplying and rewriting, we have that F is equal to £7200 multiplied by sine of 21 degrees. And now just evaluating this expression, we find that F is equal to £2580.2492. And now our last step is to just round this value to the nearest 10th. And so we find that F is approximately equal to £2580.2. And with this final rounded answer, we are left with answer choice. A where again, we found that our unknown force is £2580.2. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizontal.

Key Concepts

Resolving Forces on an Inclined Plane

Trigonometric Functions in Force Analysis

Equilibrium and Static Friction

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