3 questions

3:  Let ABCDEFGH be a cube of side length 5, as shown. Let P and Q be points on line AB and line AE, respectively, such that AP = 2 and AQ = 1. The plane through C, P, and Q intersects line DH at R. Find DR.

see answer: https://web2.0calc.com/questions/cube-question-help

2:  A rectangular box is 8 cm thick, and its square bases measure 32 cm by 32 cm. What is the distance, in centimeters, from the center point P of one square base to corner Q of the opposite base? Express your answer in simplest terms.

\(\begin{array}{|rcll|} \hline PQ &=& \sqrt{(\frac{32}{2})^2+ (\frac{32}{2})^2 + 8^2 } \\ &=& \sqrt{16^2+ 16^2 + 8^2 } \\ &=& \sqrt{576 } \\ &=& 24\ cm \\ \hline \end{array}\)

1:  Let triangle PQR be a triangle in the plane, and let S be a point outside the plane of triangle PQR, so that SPQR is a pyramid whose faces are all triangles. Suppose that every edge of SPQR has length 18 or 41, but no face of SPQR is equilateral. Then what is the surface area of SPQR?

Heron:

\(\begin{array}{|rcll|} \hline A &=& 4\times \sqrt{s(s-41)(s-41)(s-18)} \quad & | \quad s=\frac{41+41+18}{2}=50 \\ &=& 4\times \sqrt{50\cdot(50-41)^2\cdot (50-18)} \\ &=& 4\times \sqrt{50\cdot 9^2\cdot 32} \\ &=& 4\cdot9 \sqrt{50\cdot 32} \\ &=& 36\sqrt{1600} \\ &=& 36\cdot 40 \\ &=& 1440 \\ \hline \end{array}\)