Two people are carrying a box. One person exerts a force of 15...

Question

Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box.

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Accepted Answer

Welcome back. Everyone in this problem. Two porters are carrying a luggage box of a passenger. The first porter exerts a force of £180 at an angle of 53.2 degrees. While the second porter exerts a force of £111 at an angle of 72.8 degrees. We want to calculate the weight of the box for our answer choices. A is £182. B says it's £146. C £156 and D £261. Now first, let's try to visualize what's really going on here. So we have a luggage box, ok? For a passenger and two porters are carrying it where it says the first porter exerts a force of £180 at an angle of 53.2 degrees. Probably looks something like that. Ok? And I think I could push that over a bit more. Uh Let me bring it right here. So this looks a bit better. I don't know. Uh my apologies. One, my, my, my stencil is giving me a bit of trouble today. But yes, this is what it looks like, ok. So our first force 100 and £80 at 53.2 degrees, ok? And our second porter is exerting a force of £111 ok, at an angle of 72.8 degrees. So that's £111. And we want to calculate the weight of the box. We know the weight of the box is acting downwards. Now, how are we going to figure that out? Well, here, for this weight of the box, remember that it's acting vertically. So we could also think of it as acting from here. OK. So this is essentially the weight of the box. No, for our box, for our box, we have or rather for our forces, we have the both of them acting in these directions. And if we were to think about it, when we are thinking about forces, we could label our forces on a parallelogram, we could imagine that they form a parallelogram here that looks something like this where the weight of the box, OK. The weight of the box is the diagonal that runs through the center of the parallelogram. Now this is the parallelogram rule of vectors which tells us that the resultant forces of two vectors is the diagonal that is formed in the parallelogram of both of those vectors. Now here, if we were to find that weight, we need to have an idea of some of the angles or the anger between the two vectors. OK? Because we have the angles outside, but we don't know the angles inside. And even if we do find the angle here, the question is how are we going to relate it to figure all the weight? Well, we also know that for a parallelogram, the diagonals are equal. So the weight which is this vertical diagonal would be the same thing as are of the same magnitude as this horizontal diagonal that we've drawn. And with this diagonal, if we can find the angle between both of those vectors, then we should be able to use the law of cosines because we have two sides and an angle between them. So let's try to figure out the angle between both of those sides using that to find the magnitude of our diagonal, let's call our diagonal D which will thus give us the weight of the box. So let's start by finding our angle. Now, we have a straight line here at the box and the angle at which the porters are carrying the luggage box. And we know that angles on a straight line sum to 180 degrees. So what does that tell us then? Well here? OK. And we're assuming that the box is perfectly balanced, then four angles on a straight line. And let me put that in red. OK. Recall that for angles on a straight line, we know there's up to 180 degrees. So our angle here, let's call this angle D angle D is going to be equal to 180 degrees minus the sum of 53.2 degrees and 72.8 degrees. No, 53.2 degrees plus 72.8 degrees equals 126 degrees and 180 minus 126 equals 54 degrees. So the angle between our force, both of our forces is 54 degrees. Now let's isolate this as a triangle here because as we said, eventually, we want to use the law of cosines to solve. So we have our unknown magnitude D, OK. We have our forces being exerted, OK? Which are £180 and £111 respectively. OK? And we have the anger between them, which is 54 degrees. Now because we have two sides and an angle between them, we can use the law of cosines and in the law of cosines, it basically tells us that the square of one side is going to be equal to the square of the other two sides minus two multiplied by those sides and the cosine of the angle between them. So it's a little bit like the Pythagorean theorem, but just with a little extra at the end. So if we recall the cosine rule, recall that in the cosine rule, the square of one side, in this case, D squared would be equal to, or let me say, recalled by the law of cosines. Let's make sure we write out exactly what we're doing here. Then by the law of cosines, the square of D is going to be equal to the square of the other two sides. That's 180 squared plus 111 squared minus two multiplied by those two sides, 180 multiplied by 111 multiplied by the cosine of the angle between those two sides, which in this case is 54 degrees. Here, we have all the information we need. So we can solve for D because now if D squared is equal to this expression, then D is going to be equal to the square root of the expression. In other words, the square root of 180 squared plus 111 squared minus two, multiplied by 180 multiplied by 111 multiplied by the cosine of 54 degrees. Now, to get the exact value of the weight of our box here, the exact value of D, we can go ahead and plug it into our calculators. And if we remember all of our results were written to the nearest pound, so we can write our magnitude to the nearest pound as well. Now, when we put it into our calculators, we'll find that the magnitude D is going to be equal to £146. So if our horizontal length here is £146 that's our force here. That means the width of the box is going to be £146. If we look back on our answer choices, that would have been answer choice. B thanks a lot for watching everyone. I hope this video helped.

Struggling with Trigonometry?

Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box.

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Struggling with Trigonometry?

Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box. <IMAGE>

Watch next

Geometric Vectors practice set

Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box.

<IMAGE>