# 4. Related Rates

by M. Bourne

If 2 variables both vary with respect to time and have a relation between them, we can express the rate of change of one in terms of the other.

We need to differentiate both sides w.r.t. (with respect to) time.

That is, we'll be finding `(df)/(dt)` for some function `f(t)`.

### Important!

Recall from implicit differentiation the following for some function `x` of `t`:

`d/(dt)x^2=2x(dx)/(dt)`

`d/(dt)x^3=3x^2(dx)/(dt)`

`d/(dt)x^4=4x^3(dx)/(dt)`

`d/(dt)x^5=5x^4(dx)/(dt)`

We use this concept throughout this section on **related
rates.**

### Example 1

A `20\ "m"` ladder
leans against a wall. The top slides down at a rate of 4 ms^{-1}.
How fast is the bottom of the ladder moving when it is 16 m from
the wall?

**Steps:**

- Make a sketch of the problem
- Identify constant and variable quantities
- Establish relationship between quantities.
- Differentiate w.r.t time.
- Evaluate at point of interest.

### Example 2

A stone is
dropped into a pond, the ripples forming concentric circles which
expand. At what rate is the area of one of these circles
increasing when the radius is `4\ "m"` and increasing at the rate of
0.5 ms^{-1}?

### Example 3

An earth satellite moves in a path that can be described by

`x^2/72.5+y^2/71.5=1`

where *x*
and *y* are in thousands of kilometres.

If `dx/dt = 12900\ "km/h"` for `x = 3200\ "km"` and `y > 0`, find `dy/dt`.

### Example 4

The
tuning frequency *f* of an electronic tuner
is inversely proportional to the square root of the capacitance `C`
in the circuit.

If *f*
= 920 kHz for *C* = 3.5 pF, find how fast *f*
is changing at this frequency if `(dC)/(dt) =0.3\ "pF/s"`.

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