The base of a triangular pyramid is a triangle with corners at #(6 ,2 )#, #(3 ,5 )#, and #(4 ,2 )#. If the pyramid has a height of #9 #, what is the pyramid's volume?

Answer 1

The volume is #9# cubic units.

This response consists of two main sections:

Section 1: The area of the triangular base is equal to half of the height measured from one vertex to the other.

Section 2: The pyramid's volume is equal to one-third the base's area times its height.

Let's get to the math now.

Part 1 -- it helps to draw out the triangle on graph paper. Note that the first and third vertices are both on the horizontal line #y=2# and the third vertex is at #y=5#. So we have a side that's #2# units long (from #(4,2)# to #(6,2)#) and the height to the opposite vertex is three units (from #y=2# along the entire side to #y=5# on the third vertex). So the area of the triangle is #(1/2)×2×3=3# square units.
Part 2 -- The area of the base is #3# square units and the height is #9# units. Pugthose numbers into the formula given above for the volume of a pyramis: #(1/3)×3×9=9# cubic units.
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Answer 2

Volume of pyramid is # 9 # cubic.unit

Vertices of triangular base are #(6,2) , (3,5) , (4,2)#
The area of the triangular base is #A_b=1/2(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2))# or
#A_t=1/2(6(5-2)+3(2-2)+4(2-5)) = 1/2 (18+0-12)=12*6= 3#sq.unit
Volume of pyramid is #V= 1/3* A_b*h ; h= 9 , A_b=3 # or
#V= 1/cancel3*cancel3*9 = 9 # cubic.unit [Ans]
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Answer 3

The volume of the triangular pyramid can be calculated using the formula:

[ \text{Volume} = \frac{1}{3} \times \text{Area of Base} \times \text{Height} ]

First, we need to find the area of the base triangle using the coordinates given. Then, we can calculate the volume using the given height.

Using the coordinates provided, the base triangle has vertices at (6, 2), (3, 5), and (4, 2). We can use the formula for the area of a triangle given its vertices:

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

Substituting the coordinates, we get:

[ \text{Area} = \frac{1}{2} |6(5 - 2) + 3(2 - 2) + 4(2 - 5)| = \frac{1}{2} |12 + 0 - 6| = \frac{1}{2} |6| = 3 ]

Now, we can calculate the volume:

[ \text{Volume} = \frac{1}{3} \times \text{Area of Base} \times \text{Height} = \frac{1}{3} \times 3 \times 9 = 9 ]

Therefore, the volume of the triangular pyramid is 9 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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