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Problem 829

In the rectangular solid shown, \(\underline{D A}=4\) in. \(\underline{D C}=3\), and \(\underline{\mathrm{GC}}=12\). (a) Find the length of \(\underline{\mathrm{CA}}\), a diagonal of the base \(\mathrm{ABCD}\). (b) Using the result found in part (a), find the length of \(\underline{G A}\). a diagonal of the solid. (c) If \(\underline{\mathrm{DA}}=\ell, \underline{\mathrm{DC}}=\mathrm{w}\), and \(\underline{\mathrm{GC}}=\mathrm{h}\), represent the length of

Short Answer

Expert verified
In summary, we found: (a) The length of the diagonal of the base, \(\underline{\mathrm{CA}} = 5\). (b) The length of the diagonal of the solid, \(\underline{G A} = 13\). (c) The length of the diagonal of the solid, \(\underline{G A}\), can be represented in terms of \(\ell, \mathrm{w}\), and \(\mathrm{h}\) as: \(\underline{G A} = \sqrt{h^2 + \ell^2 + w^2}\).
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Step 1: Finding the length of \(\underline{\mathrm{CA}}\) (diagonal of the base)

Using the Pythagorean Theorem, we can find the length of the diagonal of the base, \(\underline{\mathrm{CA}}\), on triangle \(\triangle DAC\). \[\mathrm{CA^2 = DA^2 + DC^2}\] Substitute values \(\underline{D A}=4\) and \(\underline{D C}=3\): \[\mathrm{CA^2 = 4^2 + 3^2 = 16 + 9 = 25}\] So, taking the square root of both sides: \[\underline{\mathrm{CA}} = \sqrt{25} = 5\] The length of the diagonal of the base, \(\underline{\mathrm{CA}}\), is 5 inches.

Step 2: Finding the length of \(\underline{G A}\) (diagonal of the solid)

We can find the length of the diagonal of the solid, \(\underline{G A}\), by using the Pythagorean Theorem again on triangle \(\triangle GCA\). \[GA^2 = GC^2 + CA^2\] Substitute values \(\underline{\mathrm{GC}}=12\) and the result from part (a), \(\underline{\mathrm{CA}}=5\): \[GA^2 = 12^2 + 5^2 = 144 + 25 = 169\] So, taking the square root of both sides: \[\underline{G A} = \sqrt{169} = 13\] The length of the diagonal of the solid, \(\underline{G A}\), is 13 inches.

Step 3: Generalizing the length of \(\underline{G A}\) in terms of \(\ell, \mathrm{w}\), and \(\mathrm{h}\)

Using our work from step 2, we can generalize our expression for the diagonal of the solid, \(\underline{G A}\): \[GA^2 = GC^2 + CA^2\] Now, we go back to step 1 and generalize our expression for \(\underline{\mathrm{CA}}^2\) in terms of \(\ell\) and \(w\): \[CA^2 = DA^2 + DC^2 \Rightarrow CA^2 = \ell^2 + w^2\] Substituting the values \(\underline{D A} = \ell\), \(\underline{D C}=w\), and \(\underline{\mathrm{GC}}=h\) back into our expression, we get: \[GA^2 = h^2 + (\ell^2 + w^2)\] Taking the square root of both sides, we find: \[\underline{G A} = \sqrt{h^2 + \ell^2 + w^2}\] The length of the diagonal of the solid, \(\underline{G A}\), can be represented in terms of \(\ell, \mathrm{w}\), and \(\mathrm{h}\) as: \[\underline{G A} = \sqrt{h^2 + \ell^2 + w^2}\]

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