Lesson: Intermediate Problem Solving - 25t07

Practice: Using A Sketch

[Page 25 of 27]

 

Answer each question that follows. Draw a diagram or sketch if you feel that it would help your answer to be more accurate or quicker. Click Continue after you've selected your answer.

1. Routes X, Y, and Z intersect to form a triangle. If route X and route Y intersect at a 20 degree angle and if Z is perpendicular to route X, at what angle do streets Z and Y intersect?
  50 degrees 60 degrees 70 degrees
2. A length of string is knotted at and intervals. If the string is laid flat and horizontal, approximately what fraction of the string's length lies between the sixth and seventh knots from the left?
 
3. A circle with radius 4 is intersected by a single line at points X and Y. The maximum possible distance between X and Y is
  8 4 4
4. A rectangular box is 6 inches wide, 8 inches long, and 5 inches high. What is the greatest possible (straight-line) distance, in inches, between any two points on the box?
 


This problem , like the last one, asks us to find a maximum distance, but this time we're dealing with a solid figure that is a bit more difficult to visualize. Drawing a solid figure with the given dimensions is a good idea:

Now, what is the question asking us to find? The longest possible straight line between any two points on the box. Using our diagram as a guide, we can see that the diagonal of the 8 by 6 side of the rectangle might be the longest distance, but remember that we are now dealing with a solid; the inside of the rectangle must now be considered as well. By sketching on our diagram, we find that the longest distance is actually the inside, 3-D diagonal as shown below:


This line is the hypotenuse of the height of the box and of the diagonal of the 8 by 6 box side

Since the 8 by 6 diagonal is the hypotenuse of a triangle with legs of 6 and 8, then the diagonal must equal 10 (common right triangle 6:8:10). We can then use the Pythagorean theorem to find the length of the blue line, which is the maximum distance:


Choice (B) is correct. As you can see, it would be very difficult to keep all of that straight without a sketch.

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