Question $3[17]$

$3\mathrm{.}1$ The formula $,y=\frac{2F{L}^3}{27EI}$

is used to determine the midpoint deflection of a tubular composite shaft $($shown below$),$ where:

F: applied force

L: support point distance

E: Elasticity of material

I: cross sectional moment of inertia

If the force $(F)$ is increased by $0\mathrm{.}2\%$ and the support point distance $(L)$ is increased by $0\mathrm{.}5\%,$ use the partial differentiation method to approximate the percentage change in the midpoint deflection $($y$).$

$(6)$

$3\mathrm{.}2$ A model for the surface area $($skin$)$ of a human body is given by

$S=72w^{0\mathrm{.}4}{h}^{0\mathrm{.}7}$

where $w$ is the weight in kilogram, $h$ the height in centimetre and $S$ is measured in $c{m}^2.$ If the errors in measurement of $w$ and $h$ are at most $2\%,$ use differentials to estimate the maximum error in the calculated surface area.

$(6)$

$3\mathrm{.}3$ For gas confined in a container the pressure $P$ is given by the equation

$P=M\frac{T}{V}$

pascal where $M$ is a constant equal to $20$ and the temperature $T$ is $80C$ when the volume $V$ is $35m^3.$ Find the rate in change of the volume $V$ if the temperature increases at a rate of $2\frac{C}{?}$ hour and the pressure $P$ is to remain constant.

$(5)$

TOTAL $80$