Question

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?

Answer 1

Answer:

27 feet for the south wall and 18 feet for the east/west walls

Maximum area= $486\ ft^2$

Step-by-step explanation:

Optimization

This is a simple case where an objective function must be minimized or maximized, given some restrictions coming in the form of equations.

The first derivative method will be used to find the values of the parameters that control the objective function and the maximum value of that function.

The office space for Billy-Sean will have the form of a rectangle of dimensions x and y, being x the number of feet for the south wall and y the number of feet for the west wall. The total cost of the space is

C=8x+12y

The budget to build the office space is $432, thus

$8x+12y=432$

Solving for y

$\displaystyle y=\frac{432-8x}{12}$

The area of the office space is

$A=xy$

Replacing the value found above

$\displaystyle A=x\cdot \frac{432-8x}{12}$

Operating

$\displaystyle A= \frac{432x-8x^2}{12}$

This is the objective function and must be maximized. Taking its first derivative and equating to 0:

$\displaystyle A'= \frac{432-16x}{12}=0$

Operating

$432-16x=0$

Solving

$x=432/16=27$

$x=27\ feet$

Calculating y

$\displaystyle y=\frac{432-8\cdot 27}{12}$

$y=18\ feet$

Compute the second derivative to ensure it's a maximum

$\displaystyle A'= \frac{-16x}{12}$

Since it's negative for x positive, the values found are a maximum for the area of the office space, which area is

$A=xy=27\ ft\cdot 18\ ft\\\\\boxed{A=486\ ft^2}$