# The area of a parallelogram is 486 square cm. The sum of its bases is 54 cm. Each slanted side measures 14 cm. What is the height?

The height is

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To find the height of the parallelogram, you can use the formula for the area of a parallelogram:

[ \text{Area} = \text{base} \times \text{height} ]

Given that the area of the parallelogram is 486 square cm and the sum of its bases is 54 cm, you can express one of the bases in terms of the other using the fact that the sum of the bases is 54 cm. Let's denote the length of one base as ( b_1 ) and the other as ( b_2 ).

Since the sum of the bases is 54 cm, you have:

[ b_1 + b_2 = 54 ]

You also know that the slanted sides of the parallelogram are both 14 cm long. In a parallelogram, the height is the perpendicular distance between the two bases. So, you can form a right triangle with one leg being the height (( h )), the other leg being half the difference of the bases, and the slanted side being the hypotenuse.

Using the Pythagorean theorem:

[ h^2 = (\text{slanted side})^2 - (\text{half of the base difference})^2 ] [ h^2 = 14^2 - \left(\frac{b_1 - b_2}{2}\right)^2 ]

Since ( b_1 + b_2 = 54 ), you can rewrite the above equation in terms of one variable:

[ h^2 = 14^2 - \left(\frac{54 - b_1}{2}\right)^2 ]

Given that the area of the parallelogram is ( 486 , \text{cm}^2 ), you know:

[ \text{Area} = b_1 \times h ] [ 486 = b_1 \times h ]

Solve for ( b_1 ) in terms of ( h ):

[ b_1 = \frac{486}{h} ]

Now, substitute ( b_1 ) in terms of ( h ) into the equation for ( h^2 ):

[ h^2 = 14^2 - \left(\frac{54 - \frac{486}{h}}{2}\right)^2 ]

Now, solve for ( h ) using this equation.

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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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