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(1) Bob the Iguana wants...

(1) Bob the Iguana wants to make friendship bracelets for all of his friends, so he is making beads to go on each one.

(a) Each bead is formed by removing a cylinder of radius r from the center of a sphere of radius R. Use calculus to find the volume of the bead with r = 1 and R = 2.

(b) Bob wants to make each bead a specific volume, so he needs to calculate the volume of a bead in general. Use calculus to find the volume V of a bead in terms of r and R for Bob.

(c) The height of the bead, h, is labeled in the figure. Rewrite your formula from (b) to show that V = (π/6)h^3.

(d) While making his beads and measuring each of their volumes, Bob makes a surprising discovery. Since he can express their volumes completely in terms of their height h, any two beads of the same height must have the same volume, regardless of the size of the sphere from which they came! In other words, if he started with a sphere the size of an orange and another sphere the size of a basketball and made each of them

into beads of height 2 inches, the beads would have the same volume! Bob is very befuddled by this; explain to him how this could possibly be true. (While you don’t need to do any calculations to convince Bob, you should still use some math in your response.)

(e) Do beads of the same height, h, all have the same outside surface area (not counting the area of surface the cylindrical center hole)?

(a) Each bead is formed by removing a cylinder of radius r from the center of a sphere of radius R. Use calculus to find the volume of the bead with r = 1 and R = 2.

(b) Bob wants to make each bead a specific volume, so he needs to calculate the volume of a bead in general. Use calculus to find the volume V of a bead in terms of r and R for Bob.

(c) The height of the bead, h, is labeled in the figure. Rewrite your formula from (b) to show that V = (π/6)h^3.

(d) While making his beads and measuring each of their volumes, Bob makes a surprising discovery. Since he can express their volumes completely in terms of their height h, any two beads of the same height must have the same volume, regardless of the size of the sphere from which they came! In other words, if he started with a sphere the size of an orange and another sphere the size of a basketball and made each of them

into beads of height 2 inches, the beads would have the same volume! Bob is very befuddled by this; explain to him how this could possibly be true. (While you don’t need to do any calculations to convince Bob, you should still use some math in your response.)

(e) Do beads of the same height, h, all have the same outside surface area (not counting the area of surface the cylindrical center hole)?

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