Question 69010This question is from textbook
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While you are in Washington, D.C., you visit the Washington Monument. In the scale diagram of a cross section of the monument you are given that AB= 8.4 meters, BC= 152.5 meters, ED= 17.6 meters, and CE= 5.25 meters.
NOTE: The diagram (Washington Monument) is a long rectangle with a triangle place on top of the rectangle. There is a dotted lined down the middle (or bisects)of triangle and goes through the rectangle. The corner bottom right of the rectangle is A. The middle of A and the left side of the rectangle, which is not lettered, is B. The triangle is (Not lettered)DE. D being the point of the monument. The middle of Not lettered and E is C. D, C, and B are all on the dotted line.
The volume of a pyramid is 1/3hB, where h is the height and B is the area of the base. If the bottom portion of the Washington Monument continued to rise at the same steepness to form a single pyramid, how tall would it be?
This question is from textbook
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! A way to do this problem using geometry is to apply the features of similar triangles.
Similar triangles have two major features:
(1) The measures of the corresponding angles of each triangle are equal.
(2) The lengths of the corresponding sides of each triangle are in the same ratio.
To get ready to do this problem, make a sketch of the cross-sectional area of the Washington Monument, just as was done in your textbook diagram. Then erase line DE and point D.
Next, place a straight edge along the dotted line BC and extend the dotted line upward above the monument for some distance. Then place the straight edge along along line AE and extend it further above the monument until it intersects with the extension of the dashed line. Label the point of intersection with the letter F. (Keep points A, B, C, and E just as you described them in the problem. Lengths AB, BC, and CE are the same as you described in the problem.)
Now look at two triangles: triangle ABF and triangle ECF. They are similar triangles, and we can state this as a fact if we can show that their corresponding angles are equal. Note that line EC is perpendicular to the dashed line BCF. Also line AB is perpendicular to the dashed line. And since they are both perpendicular to the same line, lines EC and AB are parallel. Line BCE is a transverse to parallel lines EC and AB. Therefore, angle ABC in triangle ABF and angle ECF in triangle ECF have equal measures because they are corresponding angles of parallel lines that are cut by a transverse.
Notice that angle F is an angle common to both triangle ECF and triangle ABF.
Since two angles of each of the two triangles have equal measures, the third angles of each triangle must have equal measures also (because the sum of the three angles of a triangle is 180 degrees.
Now we can use the fact that corresponding sides of each triangle are in the same ratio. That ratio can be found by noting that side EC in triangle ECF corresponds to side AB in triangle ABF (they are both opposite to the angle F so they are corresponding sides).
The ratio of the long side (side AB) to the short side (side EC) is:
8.4 meters/5.25 meters = 1.6
Now look at the corresponding sides formed by the dashed line. In the big triangle (triangle ABF) this side is length BC plus length CF. However the problem says that BC is 152.5 meters. So the side BF in the big triangle is 152.5 plus the length CF. Write this as (152.5 + CF). In the small triangle (triangle ECF the corresponding side is just CF. Now, set the ratio of these two sides -- big side to small side:
(152.5 + CF)/(CF)
but this ratio must be 1.6 as we determined above. Therefore, you can write:
(152.5 + CF)/CF = 1.6
Multiply both sides by the length CF to get rid of the denominator on the left side:
152.5 + CF = 1.6*CF
Subtract CF from both sides:
152.5 = 1.6CF - CF = 0.6CF
Divide both sides by 0.6 to find that CF = 152.5/0.6 = 254.1667
Recognize now that the total height BF equals BC + CF = 152.5 + 254.1667
So the total height is 406.6667 meters which rounds to 406.7 meters.
Hope this helps extend your knowledge of geometry.
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