# Related Rates

## Related Rates (some examples)

Lecture notes coming soon...

• Q** A candle is placed a distance l1 from a thin block of wood of height H. The block is a distance l2 from a wall as shown in figure 7.1. The candle burns down so that the height of the flame, h1 decreases at the rate of 3 cm/hr. Find the rate at which the length of the shadow y cast by the block on the wall increases. ( note: your answer will be in terms of the constants l1 and l2).
• A** Note that a decrease in the candle height is equivalent to an increase in H. Thus we can use similar triangle to write H/l1 = y/(l1 + l2). Now differentiate the equation with respect to time t, to get H'/l1 = y'/(l1 + l2). Since H' = 3cm/hr (note that it is positive because it's increasing in length), y' = 3(1 + l2/l1) cm/hr.
• Q** Helium is pumped into a spherical balloon at a rate of 4 cubic feet per second. How fast is the radius increasing after 3 minutes? Note: The volume of a sphere is given by :${\displaystyle V=(4/3)\pi r^{3}}$
• A**: Derivative is ${\displaystyle {\frac {dV}{dt}}=4\pi r^{2}{\frac {dr}{dt}}}$. We are given ${\displaystyle dV/dt=4m^{3}/s}$. You can figure out the radius after 3 mins by volume which will be 3*4 or 12 cubic meters. Once we have the radius we can solve for dr/dt.
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