A company runs food service concessions for sporting events throughout the country. Their marketing research department chose a particular football stadium to test market a new jumbo hot dog. It was found that the demand for the new hot dog is given approximately by:
p = 7 - ln(x) when 5 ≤ x ≤ 500
, where x is the number of hot dogs (in thousands) that can be sold during one game at a price of p dollars.
If the company pays 1 dollar for each hot dog, how should the hot dogs be priced to maximize the profit per game?

Let T​ : V right arrow→ W be a linear transformation from a vector space V into a vector space W. Prove that the range of T is a subspace of W.​ [Hint: Typical elements of the range have the form ​T(x​) and ​T(w​) for some x and w in​ V.] What would be the most useful method of proving that the range of T is a subspace of​ W? A. Show that the three properties that define a subspace are satisfied by the vectors in the range of T. B. Show that there is a set of vectors that span the range of T. C. Show that the range of T must be all of W. D. Find a matrix such that the range of T is the column space of that matrix. E. Find a matrix such that the range of T is the null space of that matrix. F. Show that all ten axioms for a vector space are satisfied by the vectors in the range of T.

Step-by-step explanation: