If the sides of a rectangular solid are as shown, and , determine the angle, θ, between the diagonal of the base of the solid and the diagonal of the solid.Round answer to two decimal places. A)21.91^o B)24.09^o C)17.21^o D)26.28^o E)19.86^o

Question

Quizplus · Accepted Answer

The diagonal of the base can be found using the Pythagorean theorem: $d_1^2=10^2+8^2=164 \Rightarrow d_1=\sqrt{164}=2\sqrt{41}$.

The diagonal of the solid can be found using the Pythagorean theorem again: $d_2^2=10^2+8^2+6^2=180 \Rightarrow d_2=\sqrt{180}=6\sqrt{5}$.

Now, we can use the dot product formula to find the angle $	heta$ between the two diagonals: $$\cos 	heta = \frac{\mathbf{d_1} \cdot \mathbf{d_2}}{|\mathbf{d_1}||\mathbf{d_2}|}=\frac{(0)(10)+(\sqrt{164})(8)+(0)(6)}{(2\sqrt{41})(6\sqrt{5})}=\frac{4\sqrt{41}}{12\sqrt{5}}=\frac{\sqrt{41}}{3\sqrt{5}}$$

Taking the inverse cosine, we get $	heta \approx \boxed{24.09}$ degrees. Therefore, the best choice is B.

If the Sides of a Rectangular Solid Are as Shown