{Use of Tech} Angle of elevation A small plane, moving at 70 m...

Question

{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure).

b. Graph dθ/dx as a function of x and determine the point at which θ changes most rapidly.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says a lighthouse stands 200 m tall on a straight coastline, and a ship sails directly towards the lighthouse. Let theta measured in degrees, be the angle of elevation of the lighthouse from the ship, lit the graph of the derivative of theta with respect to X, and use it to find the X coordinate where the rate of change of theta is maximum. Below the problem, we're given a diagram that shows our ship sailing towards the lighthouse, and we have a right triangle drawn on this diagram, where the angle theta is 1 quarter of that right triangle, and the adjacent lake to theta is labeled as X. And the opposite leg is labeled as having a height of 200 m. Now we're asked to plot the graph of the derivative of theta with respect to X. So first we need to calculate that derivative. So, here we're actually going to use geometry to come up with a relationship between theta and X. So, if I look at our diagram, we have a right triangle. Where the angle theta is in one corner of the right triangle next to the ship, and that is going to be our angle of elevation. And the adjacent leg to that angle theta is labeled as X, and the opposite leg is labeled as 200 m. So we can use our tangent function in order to relate these quantities. So we will have the tangent of data is equal to recall that tangent of data is equal to the opposite leg of the triangle divided by the adjacent leg of the triangle or that angle. And so in this case, that is going to be 200. Divided by X So to solve for theta, we need to just take the inverse tangent of both sides of this equation. So we will have theta is equal to the inverse tangent. Of the quantity of 200 divided by X. So now we need to calculate the derivative of theta with respect to X. We'll have the derivative of theta with respect to X. is equal to the derivative with respect to X. Of the inverse tangent. Of the quantity of 200 divided by x. In quantity So taking this derivative, this is going to be equal to. Here I'm gonna be taking the derivative of the arc tangent function. And so recall that the derivative of your inverse tangent function is going to be 1 divided by 1 plus the argument squared. So that'll be 200 divided by X, and that quantity is going to be squared. But we'll also need to apply our chain rule and take the derivative of the argument. So we're going to be taking the derivative with respect to X of 200 divided by X. And when we take this derivative, that will be minus 200 divided by X squared in quantity. So, we can simplify this expression, and this is going to be equal to minus 200. Divided by the quantity of X squared. Plus 200 squad, which is 40,000. So this here is the function that we need to graph. And so, we're going to graph this on an empty um graph. And so we will have the derivative of data with respect to X, along the Y axis, and X along the X-axis. So, we're just going to find some points to actually plot on this graph. So, we'll make a table. And in this table, we will have, in the first column, we'll have X, and in the second column, we will have the derivative of theta with respect to X. So the first value that we will look at for X is going to be 0, and the derivative of data with respect to X, when we use our calculator to evaluate the expression when we substitute in 0 for X into our expression, this turns out to be -0.005, and we're actually going to plot that point on our graph. That's our first point. The next point that we'll look at is when X is equal to 50. And when we evaluate our derivative of data with respect to X, when X is equal to 50, we get minus 0.0047. So we'll plot that point on our graph as well. The next point that we're going to look at is when X is equal to 100. And in this case, we uh we evaluate our derivative of data with respect to X, we get minus 0.004. This will apply to that point on our graph as well. Now we'll make the jump up to X equal to 200. And evaluating our derivative when X is equal to 200, we have minus 0.0025. And so we will plot that point also on our graph. The next point that we're going to look at is when X is equal to 300. And again, evaluating our derivative when X is equal to 300, we have minus 0.0015. And again, we will add that point to our graph. And then the final point that we're going to look at is when X is equal to 500. And when we evaluate our derivative of data with respect to X, When X is equal to 500, we get minus 0.0007. And so we'll place that point on our graph as well. And if we look at our function for the derivative of data with respect to X, we notice that as X approaches infinity, our derivative there is going to approach 0. So we have a horizontal asymptote at Y equal to 0. So we're going to use that information to draw our graph, draw our function now. So we're going to connect our points. Of this graph, using a smooth curve. And making sure that our function here is approaching Y equal to 0 as X approaches infinity. So, this is the graph that we were actually looking for. And if we use our graph here, we noticed that the maximum rate of change, which is going to be the absolute value of the derivative of data with respect to X, is actually going to occur when X equal to 0. That means that we will have a maximum. Rate of change. At x equal to 0. That is the last half of this question answered. So, I hope that this explanation has been useful, and I'll see you in the next video.

{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>

b. Graph dθ/dx as a function of x and determine the point at which θ changes most rapidly.

Key Concepts

Angle of Elevation

Derivative and Rate of Change

Graphing Functions

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