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.

Part A)  and  

Part B)

Part C) Both companies will have the same profit, selling 2,450 items each

Part 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-shirts

Part 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 bottles

Part F) The profit of the company Pura Vida Social, LLC could be greater than the profit of the company Algorithm Corp

Step-by-step explanation:

Part A) write expressions for h(x) and g(x)

Let

x ----> the number of items each company could sell

h(x) ---->  total potential profit raised by Algorithm Corp

g(x) ----> total potential profit raised by Pura Vida Social, LLC.

we know that

The profit is equals to revenue minus costs

Algorithm Corp

The cost is $27,500

The revenue is the price of sell by the number of items -----> $15(x)

so

The total potential profit is

Pura Vida Social, LLC

The cost is $15,250

The revenue is the price of sell by the number of items -----> $10(x)

so

The total potential profit is

Part B) For what value of x does h(x)=g(x)

we have

-----> equation A

----> equation B

Equate equation A and equation B and solve for x

Part C) What does it mean for h(x)=g(x)

we know that

h(x)=g(x)  

For x=2,450 items

Find the value of h(x)

Find the value of g(x)

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 $9,250

therefore

Both companies will have the same profit, selling 2,450 items each.

Part D) What is a reasonable domain for h(x)?

we know that

The variable x represent the number of t-shirts that the company Algorithm Corp, could sell

Remember that the company purchased 10,000 t-shirts

therefore

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-shirts

Part E) What is a reasonable domain for g(x)?

we know that

The variable x represent the number of sports water bottles that the company Pura Vida Social, LLC, could sell

Remember that the company purchased 20,000 sports water bottles

therefore

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 bottles

Part F) What do the domains tell us about this situation?

Algorithm Corp

Domain -----> the interval [0,10,000]

For x=10,000

substitute in h(x)

The maximum profit of this company could be $122,500

Pura Vida Social, LLC

Domain -----> the interval [0,20,000]

For x=20,000

substitute in g(x)

The maximum profit of this company could be $184,750

therefore

The profit of the company Pura Vida Social, LLC could be greater than the profit of the company Algorithm Corp

the dimensions are:

step-by-step explanation:

we want to find the dimensions of a rectangular box with volume of

we factor to obtain;

we rewrite the rightmost factor to obtain;

recall that:

we apply difference of two squares formula to obtain;

hence the dimensions are