Question

An ant starts at a point P on the top edge of a cylinder with radius R and height H. If the ant makes n complete circuits around the cylinder and finishes at a point Q directly below the starting point on the bottom edge, what is the shortest possible path for PQ?

Steps:
a. Draw a cylinder and its net with points P and Q marked.
b. Label all the lengths you can.
c. Solve the problem when R = 4", H = 8', n = 3
d. Write a rule for PQ using R, H, n

Answer 1

The length of the path of the ant from the top of the cylinder down the

point  Q is a helical path.

Correct responses:

• The shortest path for PQ when R = 4'', H = 8' and n = 3 is 40.69 inches

• $The \ shortest \ path \ for \ PQ \ is \ \underline{ \sqrt{\left(2\cdot n \cdot R \right)^2 + H^2 }}$

Steps by which the above values are found

The given description are;

The radius of the cylinder = R

Height of the cylinder = H

Point at which the ant starts = P at the top edge

Point at which the the ant finishes = Q at the bottom edge

a. Please find attached the drawing of the cylinder showing the points P

and Q, the height of the cylinder, H, and the radius R, created with MS

Visio.

b. The length of the radius, R, the height of the cylinder, H, the and a turn

of the ant, n, are labelled.

c. When R = 4 inches; H = 8 feet;  n = 3, we have;

The shortest possible path is a helical line, given by the following equation;

$Length \ of \ helix = \mathbf{ n \cdot \sqrt{C^2 + p^2}}$

Where;

C = Circumference of the circle = 2×π×R

$\displaystyle p = The \ pitch = \mathbf{\frac{H}{n}}$

Therefore;

C = 2 × π × 4 inches  = 8·π inches

$\displaystyle p = \frac{8 \, ft.}{3} = \frac{96 \ inches}{3} = \mathbf{ 32 \ inches}$

Therefore;

Length of the helix = $3 \times \sqrt{(8 \cdot \pi)^2 + 32^2} \approx 40.69$

• The length of the helical path of the ant ≈ 40.69 inches

d. The rule for the cylinder is as follows;

$\displaystyle Length \ of \ ant \ path = n \times \sqrt{\left(2 \cdot R\right)^2 + \left(\frac{H}{n} \right)^2} = \sqrt{\left(2 \cdot R \cdot n\right)^2 + H^2 }$

The rule for the cylinder is therefore;

• $\displaystyle Length \ of \ ant \ path \ from \ P \ to \ \underline{Q = \sqrt{\left(2 \cdot R \cdot n\right)^2 + H^2 }}$

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