A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #3 # and #9 # and the pyramid's height is #2 #. If one of the base's corners has an angle of #(3pi)/8#, what is the pyramid's surface area?

Answer 1

#53#

The base has a surface area of #3 xx 9# (it is as if one corner of the #3 xx 9# rectangle has been cut off and attached to the other side).
There are #4# triangles that must be added to this. The basic formula for area is
#"area" = 0.5 xx "base" xx "height"#

There are two triangle sizes and two copies of each.

We need to find the sloped length of the pyramid, which requires the use of Pythagoras' theorem (please look this up on google images as it is relatively simple however is difficult to explain in words).

The two smaller lengths are #2# (the height of the peak) and the distance from the centre of the pyramid to one of the edges - which is half of #9# or half of #3# (depending on which direction you go).

This tells us the lengths of the slanted surface of the triangle which is

#sqrt(6.25) = 2.5 -># for #3#
#sqrt(24.25) -># for #9#

Now we substitute the numbers into the equation for a triangle;

#"area" = 0.5 xx 4.5 xx sqrt(24.25) = 11.079...#
#"area" = 0.5 xx 1.5 xx sqrt(6.25) = 1.875#
Now we add the area of the base #(27)# to the area of the triangles (keep in mind that there are two of both of them, therefore we must add #11.079...# and #1.875# twice).
#27 + (11.079 xx 2) +(1.875 xx 2) = 52.908... ~~ 53#
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Answer 2

The surface area of the pyramid can be calculated using the formula:

[ \text{Surface Area} = \text{Base Area} + \text{Area of Four Triangular Faces} ]

Given the base is a parallelogram, its area can be calculated using the formula for the area of a parallelogram: ( \text{Base Area} = \text{base length} \times \text{height} ).

The area of each triangular face can be calculated using the formula for the area of a triangle: ( \text{Area} = \frac{1}{2} \times \text{base length} \times \text{height} ).

Given the base sides have lengths of 3 and 9, the base area can be calculated as ( 3 \times 2 = 6 ) square units.

To find the height of the triangular faces, we need to use trigonometry. Since one of the base's corners has an angle of ( \frac{3\pi}{8} ), the angle between the base's side and the height is ( \frac{\pi}{8} ). Therefore, using trigonometric ratios, we can find the height of the triangular faces.

Once we find the height of the triangular faces, we can calculate the area of one triangular face and then multiply it by 4 to find the total area of all four triangular faces.

Finally, adding the base area and the total area of the four triangular faces will give us the total surface area of the pyramid.

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Answer from HIX Tutor

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.

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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