Calculus 1 question: position of drone in respect to missile. Uses tangent lines and derivative

[sharkbomby]

A surface-to-air missile is attempting to down a Russian drone over Ukraine. The drone is moving along the x-axis in the negative direction. The surface-to-air missile is pursuing the drone down along the x-axis along the exponential curve y = 2x*exp(kx), where k is a constant specific to the missile’s guidance system. The missile’s inertial guidance system makes it to where it is always facing the drone. It is noon and the sun is directly over-head. When the missile’s shadow is at x0, where is the drone? (0 is a subscript)

The teacher has since given us a hint in which he says “let m be drone location and find the tangent line of the missile.”
He also wrote:
2xe^kx/x0-m0 = d/dx 2xe^kx when missile is at m0. (0 is a subscript)
“Should get m0 in terms of x0+k”


I am very confused on this question and honesty don’t know where to start. I have drawn a graph and plotted the drone on the x-axis moving towards negative infinity, and I have plugged the function into desmos to understand the basic shape of the graph. If anyone has any ideas on what this question is asking or can guide me in the right direction I’d really appreciate it. Thank you!

 

[Dr.Peterson]

A surface-to-air missile is attempting to down a Russian drone over Ukraine. The drone is moving along the x-axis in the negative direction. The surface-to-air missile is pursuing the drone down along the x-axis along the exponential curve y = 2x*exp(kx), where k is a constant specific to the missile’s guidance system. The missile’s inertial guidance system makes it to where it is always facing the drone. It is noon and the sun is directly over-head. When the missile’s shadow is at x₀, where is the drone? (0 is a subscript)

The teacher has since given us a hint in which he says “let m be drone location and find the tangent line of the missile.”
He also wrote:
2xe^kx/x₀-m₀ = d/dx 2xe^kx when missile is at m₀. (0 is a subscript)
“Should get m₀ in terms of x₀+k”


I am very confused on this question and honesty don’t know where to start. I have drawn a graph and plotted the drone on the x-axis moving towards negative infinity, and I have plugged the function into desmos to understand the basic shape of the graph. If anyone has any ideas on what this question is asking or can guide me in the right direction I’d really appreciate it. Thank you!

I have to say, I had no idea, at first, what this problem means. I assume you quoted it exactly, and checked for errors? (By the way, the edit menu has a subscript button in the second "three vertical dot" submenu; I used it in copying your question above.)

As I read it, we have no information about the location of the drone; we know only that it is moving toward the left. Apparently what it is meant to say is that the path the missile follows is what it is because it is tracking the drone, and is always moving directly toward it (though that is not what I would have understood by "makes it to where it is always facing the drone"). In effect, we seem to be working backward from the missile's path to the actual motion of the drone.

So you want to find the x-intercept of the tangent line at a point (x₀, y₀) on the missile curve:

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Here the missile is the red dot and the drone is the green dot. The dotted line is the tangent that shows the missile is moving toward the drone.

The hint is one way to find that x-intercept; there are other ways to think of it, so you don't have to do it just that way.

By the way, the path of the missile presumably doesn't really include the part that is underground!