The base of a triangular pyramid is a triangle with corners at #(8 ,5 )#, #(6 ,2 )#, and #(5 ,9 )#. If the pyramid has a height of #8 #, what is the pyramid's volume?

Answer 1

The volume is #22 2/3#

The area of the base times the height equals the volume:

#V = 1/3Ah#

It is best to use a determinant to compute the area since we are given three points.

The determinant that will be used to calculate the area given three points is as follows:

#A = +-1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|#

Changing the three points to:

#A = +-1/2 |(8,5,1), (6,2,1), (5,9,1)|#

If find it easier to demonstrate the evaluation of a #3xx3# determinant if the first two rows are repeated:

#A = +-1/2 | (8,5,1,8,5), (6,2,1,6,2), (5,9,1,5,9) |#

Add the results of multiplying each of the major diagonals:

#A = +-1/2 | (color(red)(8),color(green)(5),color(blue)(1),8,5), (6,color(red)(2),color(green)(1),color(blue)(6),2), (5,9,color(red)(1),color(green)(5),color(blue)(9)) | = #

#color(red)((8)(2)(1)) + color(green)((5)(1)(5))+ color(blue)((1)(6)(9)) = 95#

The minor diagonals should be multiplied and then subtracted from the total of the major diagonals.

#A = +-1/2 | (8,5,color(red)(1),color(green)(8),color(blue)(5)), (6,color(red)(2),color(green)(1),color(blue)(6),2), (color(red)(5),color(green)(9),color(blue)(1),5,9) | =#

#95 - color(red)((1)(2)(5)) - color(green)((8)(1)(9)) - color(blue)((5)(6)(1)) = -17#
Because the area cannot be negative, we choose the negative value for the #+-1/2#
#A = -1/2(-17)#
#A = 17/2#
Using the #h = 8# we can compute the volume:
#V = 1/3xx17/2xx8#
#V = 68/3=22 2/3#
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Answer 2

To find the volume of the triangular pyramid, you can use the formula:

( V = \frac{1}{3} \times A_{\text{base}} \times h )

Where ( A_{\text{base}} ) is the area of the base of the pyramid and ( h ) is the height of the pyramid.

First, you need to calculate the area of the base of the pyramid using the coordinates of the triangle's vertices. Then, you can substitute the values into the formula to find the volume.

Using the coordinates provided, you can calculate the area of the base using the formula for the area of a triangle given its vertices.

Then, substitute the calculated area of the base and the given height into the formula for the volume of the pyramid to find the final result.

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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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