Water Wonderland

Learning Focus

Model situations where the rate of change changes using piecewise descriptions and graphs.

Big idea

When the rate changes, the graph is made of different pieces:

• Each piece is linear with its own slope.
• The pieces connect at the input where the rate changes.
• The later piece continues from the previous output: $f(t)=f(t_0)+m(t-t_0)\quad\text{for }t\ge t_0$

A smaller slope means the graph is still rising, but less steeply.

Worked example

A tank fills $4$ gal/min for the first $10$ min, then $2$ gal/min.

• $0\le t\le10$: $V(t)=4t$, so $V(10)=40$
• $t\ge10$: $V(t)=40+2(t-10)$
• $V(15)=40+10=50$
• Reaches $80$: $40+2(t-10)=80\Rightarrow t=30$

Takeaways

• Find the value where the rate changes, then continue from it.
• Use $f(t_0)+m(t-t_0)$ for each piece.
• A slope decreasing in size means the graph flattens, not falls.