A force of 25 lb is required to hold an 80-lb crate on a hill....

Question

A force of 25 lb is required to hold an 80-lb crate on a hill. What angle does the hill make with the horizontal?

Accepted Answer

Hey, everyone here we are given a £14 force is needed to keep the box filled with trigonometry books stable on an inclined ramp. If the weight of the box is £46 what is the angle between the ramp and the horizontal? Here, we have four answer choice options. A 20 degrees b 18 degrees C 12 degrees and D 16 degrees. So to begin this problem, we want to create a sketch containing our given information. So to start our sketch, we just want to draw a horizontal line and from there, we can draw in our ramp by just extending a line upward from the horizontal at some angle. And now we just want to label this angle, the angle in between the ramp and the horizontal as our unknown angle. Theta next, we can represent our box filled with books as just a rectangle sitting on the ramp. Next, we are given that we need a £14 force so that our box is stable on the ramp. So we just want to represent this force as an arrow going from our rectangle upward along the ramp. And so now we just want to label this arrow as £14. Lastly, we know that the box weighs £46. So this tells us we have a weight force which we need to show on our sketch. So we can represent this force as another arrow going directly downward from our box. And this arrow must be perpendicular with the horizontal. And so now we just want to label this arrow as £46. And now we have finished sketching all of our given information. So we want to start adding to the sketch so we can eventually evaluate theta. So our next step is to just finish the triangle forming underneath our box. And so we just want to extend a line downward from our ramp to the horizontal and this line must be perpendicular with the horizontal so that it forms a 90 degree angle. And so we have finished our triangle. However, looking at this triangle, we can notice that we do not have any values for any of the sites. So, so far we have no way to solve for theta. So to get more information, we want to recreate this triangle utilizing our weight force. So first, we just want to extend a line from our rectangle downwards, except this line must be perpendicular with the ramp. And so now we have a line perpendicular to the ramp as well as our weight force, which we know is perpendicular to the horizontal. So this tells us that the angle in between these two lines is our unknown angle theta. And so now completing the triangle, we just want to recall our £14 force, which we know is parallel with the ramp. So we know it forms a 90 degree angle with our new line. So we just want to take this force and represent it on our new triangle by extending an arrow from our new line to the weight force. And now we know we form a 90 degree angle with our new line. And we just want to label this arrow as £14. Now looking at our new triangle, we notice we know the value of the hypotenuse as well as one of the lengths. And so we have enough information to solve for theta. So first, we just want to recall that sign of theta is equivalent to the opposite side from the angle divided by the hypotenuse. So now using this information, along with our new triangle, we can begin solving for theta by first recognizing that sin of theta is equivalent to £14 divided by £46. And now we just want to start simplifying. So we can first notice that the pound units cancel each other out. And we can now notice that the numerator and the denominator both share a factor of two. So dividing the numerator and the denominator by two, the numerator becomes seven and the denominator becomes 23. So now just rewriting, we have that sin of theta is equal to seven divided by 23. And now just solving for theta, all we have to do is take the sine inverse of both sides. And so we have that theta is equal to sine inverse of seven divided by 23. And now just evaluating this expression, we know that theta is equal to 17.7189 degrees. And now our final step is to just round our value to the nearest whole degree. So we find that the is approximately equal to 18 degrees. And with this final rounded answer, we are left with answer choice B where again, we found that the angle in between the ramp and the horizontal is 18 degrees. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

A force of 25 lb is required to hold an 80-lb crate on a hill. What angle does the hill make with the horizontal?

Key Concepts

Resolving Forces on an Inclined Plane

Trigonometric Relationships in Right Triangles

Equilibrium of Forces

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