If we've viewed as an angle, if we're kind of above it, you can kind of start to see how this figure would look. To help us visualize this shape here, I've kind of drawn a picture Now obviously that changesĪs we change our X value. It's a right triangle, and then this distance this distance between that point and this point is the same as the distance between F of X and G of X. Isosceles right triangle sits along the base. Isosceles right triangle with a hypotenuse of the If you were to actually flatten it out, the cross sections would look like this. This cross section is going to look like this, if you were going to flatten it out. Sections of this figure, that our vertical, I should say our perpendicular to the X axis, those cross sections are going to be isosceles right triangles. Sections of the figure, that's what this yellow line is. What I've drawn here in blue, you could view this kind of the top ridge of the figure. Lets see if we can imagine a three-dimensional shape whose base could be viewed as this shaded in region between the graphs of Y is equal to F of X and Y is equal G of X. Your bounds should obviously be the least and greatest x-values that lie on the circle. You should have the base length from the previous step, which is all you need to find the cross-sectional area.Ĥ. The cross-section is an equilateral triangle, and you probably learned how to calculate the area for one of those long ago. Remember that to express a circle in terms of a single variable, you need two functions (one for above the x-axis and one for below it, in this case).ģ. A width dx, then, should given you a cross-section with volume, and you can integrate dx and still be able to compute the area for the cross-section. You know the cross-section is perpendicular to the x-axis. Integrate along the axis using the relevant bounds.Ī couple of hints for this particular problem:ġ. Find an expression for the area of the cross-section in terms of the base and/or the variable of integration.Ĥ. Find an expression in terms of that variable for the width of the base at a given point along the axis.ģ. Figure out which axis (and thus which variable) you'll be using for integration.Ģ. And we use that information and the Pythagorean Theorem to solve for x.I won't give you the answer, but I'll offer a general strategy for questions of that variety:ġ. So this is x over two and this is x over two. Two congruent right triangles and so it also splits this base into two. So the key of realization here is isosceles triangle, the altitudes splits it into So this length right over here, that's going to be five and indeed, five squared plus 12 squared, that's 25 plus 144 is 169, 13 squared. This distance right here, the whole thing, the whole thing is So x is equal to the principle root of 100 which is equal to positive 10. But since we're dealing with distances, we know that we want the This purely mathematically and say, x could be Is equal to 25 times four is equal to 100. We can multiply both sides by four to isolate the x squared. So subtracting 144 from both sides and what do we get? On the left hand side, we have x squared over four is equal to 169 minus 144. That's just x squared over two squared plus 144 144 is equal to 13 squared is 169. This is just the Pythagorean Theorem now. We can write that x over two squared plus the other side plus 12 squared is going to be equal to We can say that x over two squared that's the base right over here this side right over here. Let's use the Pythagorean Theorem on this right triangle on the right hand side. And so now we can use that information and the fact and the Pythagorean Theorem to solve for x. So this is going to be x over two and this is going to be x over two. So they're both going to have 13 they're going to have one side that's 13, one side that is 12 and so this and this side are going to be the same. And since you have twoĪngles that are the same and you have a side between them that is the same this altitude of 12 is on both triangles, we know that both of these So that is going to be the same as that right over there. Because it's an isosceles triangle, this 90 degrees is the Is an isosceles triangle, we're going to have twoĪngles that are the same. Well the key realization to solve this is to realize that thisĪltitude that they dropped, this is going to form a right angle here and a right angle here and notice, both of these triangles, because this whole thing To find the value of x in the isosceles triangle shown below.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |