Constructing a Cone with an Optimized Volume

1. Constructing a Cone with an Optimized Volume By Mr. E Calculus (year 2003) b

2. The Materials Needed • Poster Board • Compass • Protractor • Scizzors • Ruler • Tape • Staples • Colored Pencils • Calculator • Computer

3. Procedure • 1. Using a Compass, and ruler, trace a Circle with a radius of 4 inches. • 2. Cut the Circle out of the Poster Board • 3. Repeat steps 1 and 2 two more times • 4. Use the ruler and protractor to cut two of the circles with 45º and 60º slices, respectively. Label them Cone 45º and Cone 60º, respectively. • 5. Form the cones from the slices in the circle by joining the ends and either stapling or taping them together.

4. Inquiry/Question • Which of the two Cones contains largest volume? Take a quess. • To find out, measure the radius of the base circle and the height of the cones using the ruler. • Use the formula Volume=1/3*r²h and calculate the Volume of Cone 45º and Cone 60º, and compare the answers.

5. Hypothesis • Is it possible to construct a cone which will contain the maximum Volume?

6. Data • The Circumference of a circle is determined by the formulu C= d • The height of the cone lives in the relationship: • h² + r²= R² where h is the height of the cone, r is the radius of the cone’s base circle, and R is the radius of the circle which was originally used to construct the cone( see the diagram) • The Volume of a Cone is calculated by using the formula: Volume=1/3*r²h

7. Diagrams X R R h The Original Circle to be cut with arc length X and radius R r The completed cone will have a height of h, a base radius of r, and a face diagonal of R.

8. Diagrams X R The Original Circle to be cut with arc length X and radius R

9. Diagrams Close the ends together R The Original Circle to be cut with arc length X and radius R

10. Diagrams Close the ends together R The Original Circle to be cut with arc length X and radius R

11. Diagrams Pull top up to form cone R .

12. Diagrams Pull top up to form cone R h r The completed cone will have a height of h, a base radius of r, and a face diagonal of R.

13. Diagrams R h r The completed cone will have a height of h, a base radius of r, and a face diagonal of R.

14. Analysis and Exploration • Note that the cone Volume will be dependent on the size X arc length which will be cut away • Note that the arc length X determines the angle that is cut. • Hence the angle cut from the original circle determines the Volume of the cone • The Volume of the cone can be optimized by determining the X that can be cut yielding the greatest Volume by using Calculus or Trial and Error.

15. Trial and Error Cone Volume Optimization • Keep constructing different cones by cutting different angles away from the original circle of posterboard. • This process could take several hours,…or one could…

16. Use the Calculus Approach to maximize the Volume of a Cone • Step 1: Determine the first derivative by differentiating the Volume formula with respect to X. • Volume= V(x) = 1/3*r²h • dV/dx = 1/3*  * [r²*dh/dx + h* 2*r*dr/dx]

17. Use the Calculus Approach to maximize the Volume of a Cone • Step 2: Alter the formula for the Circumference C= d = (2R)= (2*4)=8 The circumference will be reduced by X in order to construct the cone. The new cone will have a base circle circumference of: 2  r = (8  - x). Therefore the base radius r formula will be r = (8  - x)/ (2 ) = 4 – (x/(2 ))

18. Use the Calculus Approach to maximize the Volume of a Cone • Step 3: Determine the first derivative by differentiating the radius r with respect to X. • r = (8  - x)/ (2 ) = 4 – (x/(2 )) • dr/dx = 0 – 1/2  • dr/dx = -1/2 

19. Use the Calculus Approach to maximize the Volume of a Cone • Step 4: Recall that the height of the cone lives in the relationship: h² + r²= R² where h is the height of the cone, r is the radius of the cone’s base circle, and R is the radius of the circle which was originally used to construct the cone( see the diagram) • Performing implicit differentiation on h² + r²= R² yields 2h dh/dt + 2r dr/dt = 0 since R is a constant Solving for dh/dt = -2r/2h* dr/dt = -r/h*(-1/2 )= = r/(2  h)

20. Use the Calculus Approach to maximize the Volume of a Cone • Step 5: Substitute dh/dx into the dV/dx derivative. • Volume= V(x) = 1/3*r²h • dV/dx = 1/3*  * [r²*r/(2 h) + h* 2*r*dr/dx] • dV/dx = 1/3* *[r3/(2 h) + h*2*r*(-1/ 2 )] • dV/dx = 1/3*  * [r3/(2 h) - h*r*/( )] • Set dV/dx = 0 to maximize the Volume • 0 =1/3*  * [r3/(2 h) - h*r*/( )] • Mulitplying both sides by 3/  • 0 = [r3/(2 h) - h*r*/( )] • Multiplying everything by 2 h • 0 = r3 – 2h²r • Adding 2h²r to both sides, • 2h²r = r3 • Dividing both sides by r • 2h²= r2

21. 2h²= r2 • Substituting the value of h and the value of r: • 2(16 – (8  - x) 2/ (2 ) 2) = ((8  - x)/ (2 )) 2 • multiplying both sides by 4  2 • 128  2 - 2 (8  - x) 2= (8  - x) 2 • Adding 2 (8  - x) 2 to both sides • 128  2 = 3(8  - x) 2 • Dividing both sides by 3 • And getting the square root of both sides • (8  - x) = √128/3 • Solving for x we get x = 8  -  √ 42.67= *(8-6.53) • = *1.47= 4.62

22. Conclusion • X= 4.62 will yield the Cone with the largest Volume. • Inguiry: What angle is cut when the arc length x=4.62? • (use the Protractor to find out)

23. Max Volume of Cone

24. Volume of Cone (Excel Chart) The Max Volume = 25.83 at x=4.62

25. Now you may color your Cones!!! • Use the colored pencils to color the cones with your favorite designs