A hot-air balloon rising straight up from a level field is tra...

Question

A hot-air balloon rising straight up from a level field is tracked by a range finder located 500 ft from the point of liftoff. Express the balloon’s height as a function of the angle the line from the range finder to the balloon makes with the ground.

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. An observer standing on top of a cliff that is 200 m high spots a ship at sea below the cliff. The observer uses a device to measure the angle of depression theta to the ship. Express the distanced from the base of the cliff to the ship as a function of the angle of depression. Awesome. So it appears for this particular problem, the final answer we're ultimately trying to solve for is we're trying to express the distanced from the base of the cliff from where our observer is standing to where the ship is located below the cliff at C. As a function of the angle of depression. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice sensors to see what our final answer might be. Note that they all state that D of theta is equal to some expression. So A is 200 multiplied by the tangent of theta, B is 200 multiplied by the cotangent of theta, C is 200 multiplied by the sine of theta, and finally D is 200 multiplied by the cosine of theta. Awesome. So first off, in order to express the distanced from the base of the cliff to where the ship is located as a function of the angle of depression theta, we must first focus our attention on all the pieces of information that we know. So we know that the height of the cliff is 200 m, and the angle of depression theta will form a right angle for where the where our ship is located at sea. So if we use trigonometry, as you should recall, we should be able to write that tangent of theta is going to be equal to 200 divided by D, where once again 200 will be the height. Of our cliff, once again, and D is the distance D from the base of the cliff to where the ship is located at C. So now, our next The step that we need to do is we need to rearrange our equation to isolate and solve for D because we do not know what the value of D is. So when we rearrange our equation to isolate and solve for D, using a little bit of algebra, we will find that D is equal to 200 divided by the tangent of theta. So now, our next step is we need to Note that without a doubt, when we solve for D of theta as a function of D of theta, we will find that D of theta is going to be equal to 200 multiplied by the cotangent of theta. Fantastic, because as you should recall, cotangenta theta is essentially one divided by tangent of theta. So you could write, instead of saying one divided by tangent of theta, you just say cotangent of theta. And that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer without a doubt has to be multiple choice answer letter B, which once again states that D of theta is equal to 200 multiplied by cotangent of theta. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

A hot-air balloon rising straight up from a level field is tracked by a range finder located 500 ft from the point of liftoff. Express the balloon’s height as a function of the angle the line from the range finder to the balloon makes with the ground.

Key Concepts

Trigonometric Functions

Right Triangle Relationships

Function Representation

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