Use the Theorem of Pappus to find the exact volume of a general cone created by rotating a triangle with vertices ( 0 , 0 ) , ( 1 , 0 ) , and ( 0 , 8 ) around the y -axis.

The exact volume of a general Cone is [tex]\frac{8\pi}{3}[/tex]

To solve this problem, we will make use of the second theorem of Pappus. This theorem is concerned with finding the volume produced when a plane shape is rotated around an axis ( the volume of the solid of revolution ).

The theorem states:

The volume of a solid of revolution obtained by rotating a lamina F (the triangle) about a non-intersecting axis (the y-axis) lying in the same plane is equal to the product of the area A of the lamina F and the distance d traveled by the centroid of F.

That is

[tex]V=A\times d[/tex]

When the centroid travels, it forms a circumference d given by,

[tex]d=2\pi r[/tex]

Therefore

[tex]V=A\times 2\pi r[/tex]

From the diagram below,

The vertices (or, corners) of the triangle are A(0, 0), B(1, 0), and C(0,8).

G is the centroid of the triangle. The centroid of the triangle is the geometric center of the triangle. The centroid would be the balance point if a pin were to be used to balance the triangle.

r is the distance from the centroid to the axis of rotation ( the y-axis in this case). It is like saying the radius of rotation.

The things we need to find are A (the Area) and r.

Calculate A:

[tex]A=\frac{1}{2} \left| \Delta \right|[/tex]

          [tex]\Delta = \left|\begin{array}{cc} x_{B}-x_{A} & y_{B}-y_{A} \\ x_{C}-x_{A} &y_{C}-y_{A}\end{array}\right|= \left|\begin{array}{cc} 1-0 & 0-0 \\ 0-0 & 8-0\end{array}\right|= \left|\begin{array}{cc} 1 & 0 \\ 0 & 8\end{array}\right|[/tex]

          [tex]\Delta =8\times 1 -0\times 0=8[/tex]

[tex]A=\frac{1}{2} \left|8\right|=\frac{1}{2}\times 8=4[/tex]

Calculate r:

[tex]r=\overline{x}=\frac{x_A+x_B+x_C}{3}=\frac{0+1+0}{3}=\frac{1}{3}[/tex]

Finally, Calculate the volume, V:

[tex]V=A\times 2\pi r=4 \times2\pi\times\frac{1}{3}=\frac{8\pi}{3}[/tex]

Learn more about centroids:  https://brainly.com/question/20305516

The figure shows a utility pole with an electrical line and a telephone line.
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133.5 divided by 5 show work!

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A. 4
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Will give brainliest.

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Question 1 options:

3 days


4 days


5 days


6 days.

Show and explain your thought process for the question above. How did you reach your conclusion? What did you do to find that number?

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Calculator
Grant reads a story about a turtle that swam 25 feet in 40 seconds. He assumes the time, t, that it takes the turtle to swim a distance of d feet is a proportional relationship, Grant
starts to create the table shown for other times and distances the turtle swims.
Enter the missing numbers in the table to represent the proportional relationship
Time
(seconds)
Distance
(feet)
1
10
30
40
25

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___N2 + H2 → NH3
Balance the equation. What coefficient goes in from of H2?

N2 + ___ H2 → NH3
Balance the equation. What coefficient goes in from of NH3?

N2 + H2 → ___NH3

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(0,3)(1,2)(1,4)(2,1)

Identify which of the sequences below is a geometric sequence.
A. 2, 4, 6, 8,...
B. 3, 6, 18, 54, ...
C. 2, 6, 18, 54. ...
D. 2, 6, 18, 36, ...

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What volume of fluid does a patient receive in 12 hours with a fluid rate of 24 mL/hr?

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A. 33
B. 38
C. 40
D. 28

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What is Ryan's bank account balance after the deposit is made?

Enter your answer in the box.

$

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How many buses of students are
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Find the sum of 12.01 and 3.8

Evaluate.

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Enter your answer as a simplified fraction in the box.

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(14⋅5)+(5⋅8)
(14⋅5)+(14⋅8)
5+(14⋅8)
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