It wouldn't look good if the national Christmas tree in Washington DC toppled over, so they've got it rigged up nice and tight. (As festive as it may seem, being impaled by a falling spruce is quite an unpleasant way to die.) A series of 10-foot-high support posts surrounds the tree, each 25 feet away from the tree's stump.
Each support wire runs from the ground to the top of the support pole to the top of the tree in one, continuous, straight line. Knowing what you do, find the minimum length of the wire you'd need if you have no idea what the height of the national tree is.
Difficulty: 
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I will label the wire w and imagine that the tree is just a vertical line extending from the trunk and ultimately connecting to the wire at the treetop. I will call that length t, for tree. Therefore, we have two right triangles in the diagram. Both share a portion of the ground as a side, and both share the angle formed by the ground and the wire. Because the brown triangle side (having length 10) and t are perpendicular to the ground, the two triangles are similar according to Angle-Angle Similarity (from your days in geometry class). Because those two triangles are similar, all of their corresponding sides must be in proportion. Before I set up that proportion, I want to set up one last variable, called x. It will represent the length of ground beginning at the shared vertex and ending at the measurement of 25 in the diagram. Here is a simplified diagram and my proportionality statement:
Cross-multiply and solve for t, because we're going to need this information later. In fact, once I solve for t, I'm also going to find the derivative of t with respect to x, because we'll also need that. You probably won't think to do this if you don't know the information is needed ahead of time, and that's okay...you'd just circle around and do this later.
Time to figure out what our primary equation is; it should be solved for the thing we're trying to minimize. Since we're trying to make w as small, as possible, we need an equation in the form of "w = ". Since we're dealing with right triangles, the easiest plan is to use the Pythagorean Theorem and solve it for w:
(There's no need to make the right-hand side of the equation both plus and minus the radical, since w has to be a positive number, or the answer wouldn't make sense.) Okay, now that we've got our primary equation, we need to derive it with respect to x. You might object, saying "Hold on a minute! Shouldn't we substitute in the value of t, and get our equation down to one variable before we take the derivative?" Sure, you can, but since t is a binomial, squaring is a little ugly. For now, I'll allow the t's to hang around and take care of them in a bit.
That looks ugly, no doubt. However, we need to determine where that derivative either equals 0 or is undefined. It'll be undefined whenever the denominator equals 0 or the interior of the radical is negative. Neither of those things can possibly happen, so that's good news. Therefore, we only need to determine where the derivative equals 0. When is a fraction equal to 0? When its numerator equals 0, so we can effectively ignore the denominator for the rest of the problem. So, I'll set w' equal to 0 and plug in the values for t and dt/dx we already found early on. (Before we can solve the equation, it has to be in terms of a single variable.)
By ye not afraid to solve this ugly equation with your calculator to get the answer, x = 13.572088 feet. Danger! This is not the answer the problem asks for! We need to find w. One way to do so is to first find t using the equation we found oh, so long ago:
We're not done yet, but now we can finally find w using our primary equation:
Therefore, you'd need just under 48 feet of wire for each support post around the national tree.
