A company knows that it needs to purchase $75,000$ units of a particular asset later this year and decides to hedge against the risk of a price increase with futures contracts on another related asset. Each futures contract is on $5,000$ units. The current spot price of the asset to be purchased is $$83$ and the standard deviation of the change in this price over the life of the hedge is estimated to be $$2\mathrm{.}50.$ The futures price of the related asset is $$50$ and the standard deviation of the change in this over the life of the hedge $$1\mathrm{.}25.$ The coefficient of correlation between the spot price change and futures price change is $0\mathrm{.}85.$

$($a$)$ Should the company take a long or short futures position to hedge this purchase? Explain.

$($b$)$ What is the minimum variance hedge ratio?

$($c$)$ What is the optimal number of futures contracts for the hedge?