In a container whose width changes partway up, the “rate” at which the depth increases also changes partway through.
Key Points about Stepped Containers
- Until the lower step is full, calculate using the base area of the lower step
- Once water enters the upper step, calculate using the base area of the upper step (which is larger)
- The larger the base area, the more slowly the depth increases
Example: Water is added at a rate of 40 cm³ per second to a two-step container with a lower step (base area 20 cm², height 10 cm) and an upper step (base area 40 cm², height 15 cm).
① After how many seconds will the lower step be full?
② After how many seconds will the entire container be full?
Solution ①:
Water that can be held in the lower step = 20 × 10 = 200 cm³
Time = 200 ÷ 40 = 5 seconds
Solution ②:
Water that can be held in the upper step = 40 × 15 = 600 cm³
Time until the upper step is full = 600 ÷ 40 = 15 seconds
For the entire container, 5 + 15 = 20 seconds
① After how many seconds will the lower step be full?
② After how many seconds will the entire container be full?
Solution ①:
Water that can be held in the lower step = 20 × 10 = 200 cm³
Time = 200 ÷ 40 = 5 seconds
Solution ②:
Water that can be held in the upper step = 40 × 15 = 600 cm³
Time until the upper step is full = 600 ÷ 40 = 15 seconds
For the entire container, 5 + 15 = 20 seconds
A graph of “time” and “depth” will be a line with a sharp bend partway through. Check the time and depth at the bending point.