The base of a triangular pyramid is a triangle with corners at #(1 ,2 )#, #(5 ,5 )#, and #(2 ,7 )#. If the pyramid has a height of #9 #, what is the pyramid's volume?

Answer 1

25.5

given Triangle ABC's coordinates

C(2,7) A(1,2) B(5,5)

Given the coordinates of the three vertices of any triangle, the area of the triangle is given by: area = #(Ax*( By − Cy ) + Bx*( Cy − Ay ) + Cx *( Ay − By ) )/2#

where Ax and Ay represent the point A's x and y coordinates, respectively.

In our case, area = #( 1*(5-7) + 5*(7-2) + 2*(2-5))/2#
Area = # (1*(-2) + 5*5 + 2*(-3))/2#
=> #(-2+25-6)/2#
=> #17/2# = 8.5

A triangular pyramid's volume is calculated as V = 1/3AH, where A is the triangle base's area and H is the pyramid's height, or the separation between the base and apex.

In our case, H=9. A=8.5 So volume = #1/3*8.5*9# = 25.5
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Answer 2

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

[ V = \frac{1}{3}Bh ]

where (B) is the area of the base triangle and (h) is the height of the pyramid.

First, find the area of the base triangle using the vertices (1, 2), (5, 5), and (2, 7) with the formula for the area of a triangle given by coordinates:

[ A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| ]

Substituting the given points:

[ A = \frac{1}{2} |1(5 - 7) + 5(7 - 2) + 2(2 - 5)| ]

[ A = \frac{1}{2} |-2 + 25 - 6| ]

[ A = \frac{1}{2} |17| ]

[ A = \frac{17}{2} ]

Now that we have the area of the base, we can calculate the volume of the pyramid with a height ((h)) of 9:

[ V = \frac{1}{3}Bh = \frac{1}{3} \times \frac{17}{2} \times 9 ]

[ V = \frac{17}{2} \times 3 ]

[ V = \frac{51}{2} ]

[ V = 25.5 ]

Therefore, the volume of the triangular pyramid is 25.5 cubic units.

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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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