Calc Final Story Chapter 0 / 4

Survive the Sine Wave

One tropical tale. Four calculus challenges. Answer every question correctly or doom them to a terrible fate.

Alex and Keyshawn are adrift in a small rowboat when they spot an abandoned island on the horizon. The hull is cracked, water is pouring in, and the shore looks impossibly far away. Only YOUR calculus can keep them alive long enough to reach land.

Chapter 1 - Integration

The Sinking Boat

boat taking on water near a tropical island
Hull profile y = 9 − x² · water inflow 0.9 m³/min
Alex and Keyshawn cling to their rowboat as waves strike the sides. They can just make out an island in the distance; however, a large hole in the hull bellows below the waterline. Alex remembers the boat's cross-sectional area is y = 9 − x² m^2. The boat is only in the first quadrant and can hold 27 m³ of water before it floods. Water enters through the hole at a constant 0.9 m³/min.

Combined integral problem

Find (A) the volume of the hull using the given cross-section and (B) how many minutes until the boat is completely submerged.

Select the pair that matches both calculations.

    Chapter 2 - Related Rates

    Race to the Shore

    alex and keyshawn racing toward a tropical beach
    Alex 80 m @ 1.0 m/s · Keyshawn 50 m @ 1.4 m/s
    Oh No! The boat sank! Alex and Keyshawn leap out and try their luck swimming for the island. Alex is 80 m from shore, swimming at a pace of 1.0 m/s. Keyshawn, thrown by a wave, is 50 m from shore while swimming at a pace of 1.4 m/s toward land along the same path.

    Related rates

    Let D(x) be the change in distance between them. How fast is Keyshawn pulling away from Alex at this instant? Find dx/dt.

      Chapter 3 - Derivatives

      Island Survival

      Food supplies crate on a tropical island beach
      They both crawl onto the island, soaked but alive. Upon the shore, they miraculously find a crate of supplies that holds exactly 150 kg of food. The rate of their increase in combined consumption, in kg per day after t full days on the island, is modelled by C(t) = 12t. When F(t) = 0, they will die of starvation. What is the maximum number of days they can both survive for?

      Chapter 4 - Surface Area

      The Final Snack Attack

      Keyshawn's cartoon final form
      Belly profile y = 1/2x^2 · rotated around the y-axis
      In one last zaney twist, Keyshawn gets so hungry that he eats Alex like a cake and instantly balloons into a massive round beast. His stomach is modelled by the parabola y = 1/2x^2 from 0 ≤ x ≤ 2, rotated around the y-axis. Calculate the surface area of Keyshawn's spherical stomach.

      Surface area of revolution

      Use the surface area formula for rotating a curve around the y-axis.

      Surface area S = 2π∫02 x√(1 + x^2) dx
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        Island Conquered!

        Celebration

        Alex and Keyshawn actually made it! Thanks to integrals, related rates, derivatives, and surface area utilised by YOU at exactly the right moments. The island is theirs! For now...