**Intersection of Lines and Planes Problems and Solutions**

**PROBLEM 1 **

Suppose a communications cable is strung above a fresh-water lake. Imagine that the cable does not sag at all, and is attached at the tops of a set of utility poles. Suppose the engineering literature recommends that the cable be suspended 10 meters above “effective ground,” and that “effective ground” is, on average, 2 meters below the average level of the surface of a body of fresh water. How tall should the poles be? Assume they are all perfectly vertical.

**SOLUTION 1**

Because the poles are perfectly vertical, they are perpendicular to the surface of the lake. This means that the poles should each be tall enough so their tops are 10 meters above “effective ground,” so they should each extend 10 − 2 meters, or 8 meters, above the water surface. The actual height of each pole depends on the depth of the lake at the point where it is placed. It is assumed that the lake is small enough, and/or weather conditions reasonable enough, so the lake does not acquire waves so high that they inundate the cable!

**PROBLEM 2**

Imagine that you are flying a kite over a perfectly flat field. The kite is of a design that flies at a “high angle.” Suppose the kite line does not sag, and the kite flies only 10° away from the vertical. (Some kites can actually fly straight overhead!) Imagine that it is a sunny day, and the sun is shining down from exactly the zenith. What is the angle between the kite string (also called the kite line) and its shadow on the flat field?

**SOLUTION 2**

Suppose you stand at point *S* on the surface of the field, which we call plane *X* . The kite line and its shadow lie along lines *SR* and *ST* , as shown in Fig. 7-13. The sun shines down along a line *QS* that is normal to plane *X* . Lines *SQ, SR* , and *ST* all lie in a common plane *Y* , which is perpendicular to plane *X* . We know that the measure of *∠QSR* is 10°, because we are given this information. We also know that the measure of *∠QST* is 90°, because line *QS* is normal to plane *X* , and line *ST* lies in plane *X* . Because lines *SR, ST* , and *SQ* all lie in the same plane *Y* , we know that the measures of the angles among them add as follows:

*∠QSR* + *∠RST* = *∠QST*

and therefore

*∠RST* = *∠QST* − *∠QSR*

The measure of *∠RST* , which represents the angle between the kite line and its shadow, is equal to 90° − 10°, or 80°.

Practice problems for these concepts can be found at: Points, Lines, and Planes Practice Test.

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