New Zealand
Level 8 - NCEA Level 3

# Related rates (dy/dx=dy/dt x dt/dx)

## Interactive practice questions

Let $y=\left(x+5\right)^5$y=(x+5)5 be defined as a composition of the functions $y=u^5$y=u5 and $u=x+5$u=x+5.

a

Determine $\frac{dy}{du}$dydu.

b

Determine $\frac{du}{dx}$dudx.

c

Hence determine $\frac{dy}{dx}$dydx.

Easy
Approx 3 minutes

A sandbox is being filled at a local playground. The density of sand is given by $\frac{dm}{dV}=4.5$dmdV=4.5 g/cm3, where $m$m is the mass in grams and $V$V is the volume in cubic centimetres.

If sand is being added to the sandbox at a rate of $\frac{dV}{dt}=400$dVdt=400 cm3/min, find $\frac{dm}{dt}$dmdt, the rate at which the sandbox gets heavier over time.

A point moves along the curve $y=5x^3$y=5x3 in such a way that the $x$x-coordinate of the point increases by $\frac{1}{5}$15 units per second.

Let $t$t be the time at which the point reaches $\left(x,y\right)$(x,y).

Find the rate at which the $y$y-coordinate is changing with respect to time when $x=9$x=9.

A spherical hot air balloon, whose volume and radius at time $t$t are $V$V m3 and $r$r m respectively, is filled with air at a rate of $4$4 m3/min.

At what rate is the radius of the balloon increasing when the radius is $2$2 m?

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91578

Apply differentiation methods in solving problems