Two companies are raising money for an organization that supports patients with breast cancer. Algorithm Corp. purchased 10,000 t-shirts for $27,500. They plan to stamp a cool logo in the front and back of the t-shirt and sell them for $15 each. Pura Vida Social, LLC purchased 20,000 sports water bottles for $15,250. They plan to customize each bottle with a unique message and workout challenge and sell them for $10 each. Let x represent the number of items each company could sell, h(x) represent total potential profit raised by Algorithm Corp. and g(x) represent total potential profit raised by Pura Vida Social, LLC.

Answer:Part A) [tex]h(x)=15x-27,500[/tex]  and  [tex]g(x)=10x-15,250[/tex]Part B) [tex]x=2,450\ items[/tex]Part C) Both companies will have the same profit, selling 2,450 items eachPart D) The domain for h(x) is all whole numbers greater than or equal to 0 t-shirt and less than or equal to 10,000 t-shirtsPart E) The domain for g(x) is all whole numbers greater than or equal to 0 sports water bottle and less than or equal to 20,000 sports water bottlesPart F) The profit of the company Pura Vida Social, LLC could be greater than the profit of the company Algorithm CorpStep-by-step explanation:Part A) write expressions for h(x) and g(x)Letx ----> the number of items each company could sellh(x) ---->  total potential profit raised by Algorithm Corpg(x) ----> total potential profit raised by Pura Vida Social, LLC.we know thatThe profit is equals to revenue minus costsAlgorithm CorpThe cost is $27,500The revenue is the price of sell by the number of items -----> $15(x)soThe total potential profit is[tex]h(x)=15x-27,500[/tex]Pura Vida Social, LLCThe cost is $15,250The revenue is the price of sell by the number of items -----> $10(x)soThe total potential profit is[tex]g(x)=10x-15,250[/tex]Part B) For what value of x does h(x)=g(x) we have [tex]h(x)=15x-27,500[/tex] -----> equation A[tex]g(x)=10x-15,250[/tex] ----> equation BEquate equation A and equation B and solve for x[tex]15x-27,500=10x-15,250[/tex][tex]15x-10x=-15,250+27,500[/tex][tex]5x=12,250[/tex][tex]x=2,450\ items[/tex]Part C) What does it mean for h(x)=g(x)we know thath(x)=g(x)  For x=2,450 itemsFind the value of h(x) [tex]h(2,450))=15(2,450)-27,500=\$9,250[/tex]Find the value of g(x)[tex]g(2,450)=10(2,450)-15,250=\$9,250[/tex] When the company Algorithm Corp sells 2,450 t-shirts its profit is equal to $9,250 When the company Pura Vida Social, LLC sells 2,450 sports water bottles its profit is equal to $9,250thereforeBoth companies will have the same profit, selling 2,450 items each.Part D) What is a reasonable domain for h(x)?we know thatThe variable x represent the number of t-shirts that the company Algorithm Corp, could sellRemember that the company purchased 10,000 t-shirtsthereforeThe domain for h(x) is[tex]0 \leq x \leq 10,000[/tex]All whole numbers greater than or equal to 0 t-shirt and less than or equal to 10,000 t-shirtsPart E) What is a reasonable domain for g(x)?we know thatThe variable x represent the number of sports water bottles that the company Pura Vida Social, LLC, could sellRemember that the company purchased 20,000 sports water bottlesthereforeThe domain for g(x) is[tex]0 \leq x \leq 20,000[/tex]All whole numbers greater than or equal to 0 sports water bottle and less than or equal to 20,000 sports water bottlesPart F) What do the domains tell us about this situation?Algorithm Corp[tex]h(x)=15x-27,500[/tex]Domain -----> the interval [0,10,000]For x=10,000substitute in h(x)[tex]h(10,000)=15(10,000)-27,500=\$122,500[/tex]The maximum profit of this company could be $122,500Pura Vida Social, LLC [tex]g(x)=10x-15,250[/tex]Domain -----> the interval [0,20,000]For x=20,000substitute in g(x)[tex]g(20,000)=10(20,000)-15,250=\$184,750[/tex]The maximum profit of this company could be $184,750thereforeThe profit of the company Pura Vida Social, LLC could be greater than the profit of the company Algorithm Corp