Real Numbers – Episode 08, Crossing the River

River Crossing Problem Setup:

• River banks are straight parallel lines 100 meters apart.
• River flow is constant and uniform across all points but the rate is unknown.
• Boat "Wavy" has a motor providing constant speed in the direction it faces, also an unknown rate.

Observed Test Run Data:

• Friend pointed boat directly at the opposite dock (90-degree angle to flow).
• Boat landed 60 meters downstream from the target dock.
• This drift implies the ratio of current speed to boat speed is 0.6 (60m drift / 100m width).

Forward-Looking Challenge:

• Task requires calculating the specific aiming point along the far bank to counteract drift and land exactly at the opposite dock.
• Viewer submissions must include the navigational bearing and supporting diagrams.

Review of Previous Episode (Snow Cone Geometry):

• Objective: Determine the radius of the slush to be scraped from a hemisphere to reshape it into a cone.
• Dimensions: Cone (height 10 cm, radius 5 cm), Hemisphere (radius 5 cm).
• Geometric Modeling:
  • Coordinate system centered at sphere origin.
  • Cone side equation: $y = 2x - 10$ (slope 2, y-intercept -10).
  • Sphere bottom equation: $x^2 + y^2 = 25$.
• Intersection Calculation:
  • Solving the system yielded intersection points at $x = 3$ and $x = 5$.
  • $x = 5$ represents the known rim contact point.
  • $x = 3$ represents the required removal radius, resulting in a 3 cm radius circle.
• Key Finding: The intersection point $(3, -4)$ confirms the formation of a 3-4-5 right triangle.

Calculus Bonus Question Analysis:

• Goal: Calculate the volume of slush moved to reshape the hemisphere.
• Method: Shell method applied to the region where the cone dips below the sphere ($x=0$ to $x=3$).
  • Shell radius $r = x$.
  • Shell height $h = (2x - 10) - (-\sqrt{25 - x^2})$.
  • Result: $\frac{40\pi}{3}$ cm³ (approx. 41.888 cm³).
• Verification:
  • Calculated volume removed from the outer sphere shell ($x=3$ to $x=5$).
  • Confirmed identical result of $\frac{40\pi}{3}$ cm³.

Conclusion:

• Host Sawyer directs viewers to submit the solution to the river crossing problem and any additional questions for the next episode of "Real Numbers."