Span of a bridge is 2.4 km. At 30°C a cable along the span sags by 0.5 km. Taking α =12 x10⁻⁶ per °C , change in length of cable for a change in temperature from 10°C to 42°C is

The correct option is 0.99 m.

To find the change in the length of the cable due to the temperature change, we first need to determine the actual length of the sagging cable.
The length of a sagging cable $S$ in terms of the bridge span $L$ and the sag $d$ is given by the formula:
$S=L+\frac{8d^2}{3L}$

Given:
Span of the bridge, $L=2.4\text{km}$
Sag of the cable, $d=0.5\text{km}$

Substituting these values into the formula to find the length of the cable:
$S\approx 2.58\text{km}=2580\text{m}$

The change in temperature $\mathrm{\Delta T}$ is:
$\mathrm{\Delta T}=42°\text{C}-10°\text{C}=32°\text{C}$

The coefficient of linear expansion is:
$\alpha =12\times {10}^{-6}/°\text{C}$

The change in the length of the cable $\mathrm{\Delta S}$ is given by:
$\mathrm{\Delta S}=S\alpha \mathrm{\Delta T}$
$\mathrm{\Delta S}=2580\text{m}\times 12\times {10}^{-6}/°\text{C}\times 32°\text{C}$
$\mathrm{\Delta S}\approx 0.99\text{m}$