The base of a triangular pyramid is a triangle with corners at #(7 ,5 )#, #(6 ,9 )#, and #(3 ,4 )#. If the pyramid has a height of #4 #, what is the pyramid's volume?

Answer 1

#color(crimson)("Volume of Pyramid " V_p = (1/3) * A_b * h = 11.36 " cubic units"#

#color(violet)("Volume of Pyramid " V_p = (1/3) * A_b * h#
#Area of base triangle " A_b = sqrt(s (s-a) (s-b) (s-c)), " using Heron's formula#
#A(7,5), B(6,9), C(3,4), h = 4#
#a = sqrt((6-3)^2 + (9-4)^2) = 5.83#
#b = sqrt((7-3)^2 + (5-4)^2) = 4.12#
#c = sqrt((6-7)^2 + (9-5)^2) = 4.12#

With equal sides b and c, the triangle is isosceles.

#"Semi-perimeter " s = (a + b + c) / 2 #
#s = (5.83 + 4.12 + 4.12) / 2 ~~ 7.04#
#A_b = sqrt(7.04 * (7.04 - 5.83) * (7.04 - 4.12) * (7.04 - 4.12)) = 8.52#
#color(crimson)("Volume of Pyramid " V_p = (1/3) * A_b * h = (1/3) * 8.52 * 4 = 11.36 " cubic units"#
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Answer 2

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

Volume = (1/3) * Base Area * Height

First, calculate the area of the base triangle using the coordinates given. You can use the formula for the area of a triangle given its vertices.

Then, plug in the values into the formula for the volume of a pyramid and calculate.

Here's a step-by-step breakdown:

  1. Calculate the lengths of the three sides of the base triangle using the distance formula.

  2. Use Heron's formula to find the area of the base triangle.

  3. Once you have the area of the base triangle, plug it into the volume formula along with the given height of the pyramid.

  4. Calculate the volume using the formula:

    Volume = (1/3) * Base Area * Height

Using the provided coordinates, the lengths of the sides of the base triangle are approximately:

Side 1: √((7-6)^2 + (5-9)^2) ≈ √(1^2 + (-4)^2) ≈ √(1 + 16) ≈ √17

Side 2: √((7-3)^2 + (5-4)^2) ≈ √(4^2 + 1^2) ≈ √(16 + 1) ≈ √17

Side 3: √((6-3)^2 + (9-4)^2) ≈ √(3^2 + 5^2) ≈ √(9 + 25) ≈ √34

The semiperimeter (s) of the triangle is (17 + 17 + 34)/2 = 34.

Now, using Heron's formula:

Area = √(s(s-a)(s-b)(s-c))

where a, b, and c are the side lengths and s is the semiperimeter.

Area ≈ √(34(34-√17)(34-√17)(34-√34))

Now, calculate the approximate value of the area.

Then, plug the area and the height (given as 4) into the volume formula:

Volume ≈ (1/3) * Area * Height

After plugging in the values, you'll get the approximate volume of the triangular pyramid. Calculate this value.

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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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