Although still a sophomore at college, John O'Hagan's son Billy-Sean has already created several commercial video games and is currently working on his most ambitious project to date: a game called K that purports to be a "simulation of the world." John O'Hagan has decided to set aside some office space for Billy-Sean against the northern wall in the headquarters penthouse. The construction of the partition will cost $8 per foot for the south wall and $12 per foot for the east and west walls.What are the dimensions of the office space with the largest area that can be provided for Billy-Sean with a budget of $432?south wall length ft -east and west wall length -What is its area?

To determine the dimensions of the office space with the largest area within a budget of $432, we need to consider the costs associated with the construction of each wall. The south wall costs $8 per foot, while the east and west walls cost $12 per foot.

Let's denote the south wall length as S and the east and west wall lengths as E. Therefore, the total cost for constructing the south wall would be 8S dollars and for both the east and west walls would be 2 * 12E = 24E dollars. The budget constraint gives us the equation:

8S + 24E = 432

We also know that the area of the office space, A, is the product of the lengths of the south wall and one of the east or west walls:

A = S * E

To maximize the area, let's find the relationship between S and E from the budget equation:

8S = 432 - 24E S = (432 - 24E)/8 S = 54 - 3E

Now let's replace S in the area equation:

A = E * (54 - 3E) A = 54E - 3E^2

To find the maximum value for the area, we need to take the derivative of A with respect to E and equate it to zero:

dA/dE = 54 - 2 * 3E 0 = 54 - 6E 6E = 54 E = 54 / 6 E = 9

Plugging the value of E back into the equation for S:

S = 54 - 3 * 9 S = 54 - 27 S = 27

So the maximum area that can be achieved within the budget is with the south wall being 27 feet and the east and west walls being 9 feet each. The area of the office space is:

A = S * E A = 27 * 9 A = 243 square feet

Therefore, the dimensions of the office with the largest area are: - South wall length: 27 feet - East and West wall length: 9 feet - Area of the office: 243 square feet

Extra: In this scenario, we have applied optimization principles using calculus to determine the maximum area of a rectangular space given a budget constraint. The problem is representative of real-world applications in construction and design, where maximizing utility within financial constraints is essential.

In the budget equation, the south wall has a lower cost per foot than the east and west walls. This means that, economically, it's better to have a longer south wall than longer east or west walls. The area of a rectangle is maximized when the product of its two sides is greatest, which we found by setting the derivative of the area equation with respect to one of the variables (E) to zero, solving for that variable, and then finding the corresponding length of the other side (S).

This type of optimization is used in various fields, from architecture and engineering to economics and business, to make efficient use of resources by finding the most effective distribution of limited resources. The method of taking derivative and setting it to zero to find such optimum points is commonly used in calculus and is very useful for these kinds of problems.