# The Decay of Radioactive Isotopes

Cesium-137 is one of the most problematic radioactive isotopes found in many nuclear weapon tests and nuclear accidents such as the Chernobyl & Fukushima disasters. Exposure to it increases chances of cancer, while exposure to high concentrations can lead to serious burns, and even death.

The isotope has a rate of radioactive decay described by the function:

where

• y = amount of material left
• k = the decay constant

The site of a nuclear accident has an abnormally high concentration of Cesium-137. If the decay constant for Cesium-137 is 0.023, how much time does it take for some initial amount of Cesium-137 to decay to half its original amount?

# RC Car (Electronics)

You've played with RC cars before, but this time you decide to give your favorite childhood toy a little more juice. Instead of the usual 4 AA batteries (6V) you decide to double it (12V) to see it really race!

How much power is your motor now receiving? Also, does your new circuit satisfy the conservation of power?

# (Diff Eq) What's My Particular Solution?

For the ordinary differential equation below:

What is the form of its particular solution?

(Original question contributed by Professor Autar Kaw & modified with permission)

# (Fluids) Pressure! Diver Down!

Dave the Diver's just bought a new underwater watch to wear on his diving expeditions. On the packaging, the watch company claims that the device can withstand absolute pressures of up to 1.0 MPa.

"Oh really..." he says as the gears turn in his head. He calculates that...

From this calculation, Dave reasons that he can safely dive with his watch to a maximum depth of 102 m.

a) What is the flaw in Dave's reasoning?

b) To what maximum depth can Dave actually dive with his watch? Round to the nearest tenth of a meter.

# (Physics) Delivery Across the River

You find yourself on one side of a frozen river in Canada. You have two boxes, block 1 & block 2, with yummy food you want to push along to your friend on the other side of the river. (You're too lazy to drive across the bridge.)

Block 1 is resting on top of a bigger block 2. Mass m1< m2. The surface between block 1 and block 2 has a static coefficient μs, and we idealize the contact surface between block 2 and assume the ice is frictionless. Your friend is hungry and wants to eat the food ASAP.

In which of the above scenarios would you be able to deliver a greater pushing force without the blocks slipping relative to each other?