# A parallelogram has sides with lengths of #21 # and #12 #. If the parallelogram's area is #42 #, what is the length of its longest diagonal?

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The length of the longest diagonal of a parallelogram can be found using the formula:

[ \text{Longest diagonal} = \sqrt{a^2 + b^2 + 2ab \cos(\theta)} ]

where ( a ) and ( b ) are the lengths of the sides of the parallelogram and ( \theta ) is the angle between them.

Given that the sides of the parallelogram have lengths of 21 and 12, and the area of the parallelogram is 42, we can use the formula for the area of a parallelogram:

[ \text{Area} = ab \sin(\theta) ]

Substituting the given values:

[ 42 = 21 \times 12 \times \sin(\theta) ]

Solving for ( \sin(\theta) ):

[ \sin(\theta) = \frac{42}{252} = \frac{1}{6} ]

Since ( \sin(\theta) = \frac{1}{6} ), the angle ( \theta ) is ( \sin^{-1}\left(\frac{1}{6}\right) ).

Then, we can use the law of cosines to find the length of the longest diagonal:

[ \text{Longest diagonal} = \sqrt{21^2 + 12^2 + 2 \times 21 \times 12 \times \cos(\sin^{-1}(1/6))} ]

Calculating:

[ \text{Longest diagonal} = \sqrt{441 + 144 + 2 \times 21 \times 12 \times \sqrt{1 - \left(\frac{1}{6}\right)^2}} ] [ \text{Longest diagonal} = \sqrt{585 + 504 \times \sqrt{\frac{35}{36}}} ] [ \text{Longest diagonal} = \sqrt{585 + 504 \times \frac{\sqrt{35}}{6}} ] [ \text{Longest diagonal} = \sqrt{585 + 84\sqrt{35}} ]

Thus, the length of the longest diagonal of the parallelogram is ( \sqrt{585 + 84\sqrt{35}} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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