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?
With equal sides b and c, the triangle is isosceles.
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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:
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Calculate the lengths of the three sides of the base triangle using the distance formula.
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Use Heron's formula to find the area of the base triangle.
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Once you have the area of the base triangle, plug it into the volume formula along with the given height of the pyramid.
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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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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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