$($a$)$ Following table gives the necessary data about the crops, duty and area under each crop commanded by a canal taking off from a storage reservoir. Assuming a time factor for the canal to be $(\frac{12}{20}),$ calculate the discharge required at the head of the canal. If the capacity factor is $0\mathrm{.}80,$ determine the design discharge.

$11$

Crop

Sugarcane

$(1)$ Sugarcane

$(2)$ Overlap for sugarcane in hot weather

$(3)$ Wheat $($Rabi$)$

$(4)$ Bajra $($Kharif$)$

$(5)$ Vegetables

$($Hot weather$)$

Duty of water at the head of the canal $($ha$/$cumec$)$

$580$

$580$

$1600$

$2000$

$600$