Questions: ABCDE is a rectangular-based pyramid with a height of 21 cm. Find the angle between the line DE and the plane ABCD. Give your answer correct to 1 decimal place.

Solution Steps

• The pyramid \(ABCDE\) has a rectangular base \(ABCD\).
• The height of the pyramid from the apex \(E\) to the base \(ABCD\) is 21 cm.
• The base \(AB\) is 21 cm and \(AD\) is 17 cm.

• Place the base \(ABCD\) on the xy-plane with \(A\) at the origin \((0,0,0)\).
• \(B\) at \((21,0,0)\), \(D\) at \((0,17,0)\), and \(C\) at \((21,17,0)\).
• The apex \(E\) is directly above the center of the base. The center of the base is \((10.5, 8.5, 0)\).
• Therefore, \(E\) is at \((10.5, 8.5, 21)\).

• \(D\) is at \((0, 17, 0)\) and \(E\) is at \((10.5, 8.5, 21)\).
• The vector \(\overrightarrow{DE}\) is calculated as: \[ \overrightarrow{DE} = (10.5 - 0, 8.5 - 17, 21 - 0) = (10.5, -8.5, 21) \]

• The plane \(ABCD\) is the xy-plane, so its normal vector is \(\overrightarrow{n} = (0, 0, 1)\).

• The angle \(\theta\) between the line \(\overrightarrow{DE}\) and the plane \(ABCD\) can be found using the dot product formula: \[ \cos \theta = \frac{\overrightarrow{DE} \cdot \overrightarrow{n}}{|\overrightarrow{DE}| \cdot |\overrightarrow{n}|} \]
• Calculate the dot product \(\overrightarrow{DE} \cdot \overrightarrow{n}\): \[ \overrightarrow{DE} \cdot \overrightarrow{n} = (10.5, -8.5, 21) \cdot (0, 0, 1) = 21 \]
• Calculate the magnitudes: \[ |\overrightarrow{DE}| = \sqrt{10.5^2 + (-8.5)^2 + 21^2} = \sqrt{110.25 + 72.25 + 441} = \sqrt{623.5} \] \[ |\overrightarrow{n}| = \sqrt{0^2 + 0^2 + 1^2} = 1 \]
• Therefore: \[ \cos \theta = \frac{21}{\sqrt{623.5}} \] \[ \theta = \cos^{-1} \left( \frac{21}{\sqrt{623.5}} \right) \]

Final Answer