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# Solids Created by Rotations

## Cross sections of solids and solids created by rotating shapes around lines.

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Connecting Between 2D & 3D, Tri-dimensional Modeling

### Connecting Between 2D & 3D, Tri-dimensional Modeling

General Tip: Try creating "think alouds" to help you better understand your mathematical thinking process.  When you solve a math problem, record yourself and try to speak all your thoughts out loud.  Then, once you have solved the problem, go back and listen to your think-aloud.  What strategies worked best for you to solve the problem? Where did you get stuck and why? What are your mathematical tendencies?

#### Guided Practice

1) Revisit this problem from the read.

The shaded figure below is rotated around the line. What is the volume of the solid that is created?



Ask yourself: What information do you need to know to solve this problem?

a) What is the 3 dimensional shape that is created? __________

b) What is the volume formula for the 3D shape created? __________

c) What information can I plug into the formula? __________

b) Do I need to add/subtract any additional volume?  If so, what? __________

Solution:

a) Sphere

b) V=(43)×π×r3\begin{align*}V=(\frac{4}{3})\times\pi\times r^3\end{align*}

c) V=(43)×π×33\begin{align*}V=(\frac{4}{3})\times\pi\times3^3\end{align*} (volume of large sphere) V=36πin3\begin{align*}V=36\pi in^3\end{align*}

d) Vlarge sphere - Vwhite sphere = Vred shape

Vwhitesphere = (4/3)×pi×23\begin{align*}(4/3)\times pi\times 2^3\end{align*}

Vwhitesphere = 32π3in3\begin{align*}\frac{32\pi}{3}in^3\end{align*}

V=36π32π3=76π3 in3\begin{align*}V=36 \pi-\frac{32 \pi}{3}=\frac{76 \pi}{3} \ in^3\end{align*}

2) Solve the following problem using the guiding questions described above. Identify the solid that is created when the following shape is rotated around the line and find the volume of it.



3) A big storm causes a large tree to fall in your yard. The main portion of the tree trunk measures about 12 feet around and is 50 feet long. You plan to chop up the tree to use and sell as fire wood. Approximately what volume of wood will you get from the tree?

Ask yourself: What information do you need to know to solve this problem?

a) What 3D geometric shape best models the object of the problem? (in this case, the tree) _______

b) What is the formula for this 3D geometric shape? _________

c) What specific information is given in the problem? What essential information can I extract using the given information and my formula?__________

Solution:

a) Cylinder

b)V=π×r2×h\begin{align*}V=\pi\times r^2\times h\end{align*}

c) “The main portion of the tree trunk measures about 12 feet around” ---> circumference = 12 ft.

We can find the radius given the circumference using simple algebra.

2×π×r=12\begin{align*}2\times\pi\times r=12\end{align*}

π×r=6\begin{align*}\pi\times r =6\end{align*}

r1.9ft.\begin{align*}r\approx1.9 ft.\end{align*}

"50 feet long" ----> height = 50 ft.

Now we have all the information to find the volume.

V=π×1.92×50\begin{align*}V=\pi\times1.9^2\times 50\end{align*}

v=180.5π\begin{align*}v=180.5\pi\end{align*}

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