A triangle with vertices of (0,0), (0,6) and (5,0) is plotted on the coordinate plane. the
1 answer:
Answer:
50π ≈ 157.08 cubic units
Step-by-step explanation:
The volume of revolution of a plane figure is the product of the area of the figure and the length of the path of revolution of the centroid of that area. The centroid of a triangle is 1/3 the distance from each side to the opposite vertex. (It is the intersection of medians.)
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<h3>length of centroid path</h3>One side of this triangle is the axis of revolution. Then the radius to the centroid is 1/3 the x-dimension of the triangle, so is 5/3. Then the circumference of the circle along which the centroid is revolved is ...
C = 2πr
C = 2π(5/3) = 10π/3 . . . units
<h3>triangle area</h3>The area of the triangle is found using the formula ...
A = 1/2bh
A = 1/2(5)(6) = 15 . . . square units
<h3>volume</h3>The volume is the product of the area and the path length:
V = AC
V = (15)(10π/3) = 50π . . . cubic units
The volume of the solid is 50π ≈ 157.08 cubic units.
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<em>Additional comment</em>
In the attached figure, the point labeled D is the centroid of the triangle. The label has no significance other than being the next after A, B, C were used to label the vertices.
The volume of revolution can also be found using integration and "shell" or "disc" differential volumes. The result is the same.
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Using the midpoint formula, the coordinates of endpoint H are (4, -6).
<h3>The Midpoint Formula</h3>The midpoint formula is given as:
<em>Where</em>,
= coordinates of the midpoint
= coordinates of the first point
= coordinates of the second point
Given the following:
= M( 6,-4)
= G(8,-2)
= H(?, ?)
Plug in the values into the midpoint formula
Solve for the x-coordinate and y-coordinate separately
- Multiply both sides
Therefore, using the midpoint formula, the coordinates of endpoint H are (4, -6).
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