Maximizing Area of Rectangle in x+3y=12 Plane

Click For Summary
SUMMARY

The discussion focuses on maximizing the area of a rectangle with its sides on the x and y axes and a corner on the line defined by the equation x + 3y = 12. The maximum area is determined to be 12 square units, achieved when y equals 2. The area function A is expressed as A = (12 - 3y)y, and the derivative dA/dy is set to zero to find the critical point, leading to the solution. The problem-solving approach emphasizes the use of calculus without the need for Lagrange multipliers.

PREREQUISITES
  • Understanding of basic calculus concepts, particularly derivatives
  • Familiarity with the equation of a line in slope-intercept form
  • Knowledge of area calculations for rectangles
  • Ability to interpret graphical representations of functions
NEXT STEPS
  • Study the application of derivatives in optimization problems
  • Explore the concept of critical points and their significance in finding maxima
  • Learn about the geometric interpretation of functions and their graphs
  • Investigate Lagrange multipliers for constrained optimization scenarios
USEFUL FOR

Students studying calculus, particularly those focusing on optimization problems, as well as educators seeking to clarify concepts related to maximizing areas under constraints.

imsoconfused
Messages
50
Reaction score
0

Homework Statement


A rectangle has sides on the x and y axes and a corner on the plane x+3y=12. Find its maximum area.


Homework Equations



A=xy=(12-3y)y

(A=12, according to the solution manual.)

The Attempt at a Solution



At first I thought the corner it was talking about lay on one of the axes, but now I realize that it is a point around (4,3). I know there is a derivative (partial?) I need to take, but I don't know which one and or what to do it with respect to. I've drawn a graph so I can see what I'm doing, but the professor just skimmed over maximixation and I'm really confused! This chapter is not at a point that Lagrange multipliers have been covered. That's the way I would expect to do it, obviously not.
 
Physics news on Phys.org
Ok, if you want to maximize set dA/dy=0. What's y?
 
dA/dy=12-6y. y=2, and then (12-3(2))*2=12 and that's the area. I knew it couldn't be that hard. thanks!
 

Similar threads

Find the dimensions that will minimize the surface area of a Rectangle
  • · Replies 1 ·
Replies
1
Views
2K
Question on Two Differentiation Techniques
  • · Replies 25 ·
Replies
25
Views
2K
Finding Area of Shaded Segment in Circle Using Calculus
  • · Replies 6 ·
Replies
6
Views
2K
Optimization of Area; Inscribed Rectangles
  • · Replies 1 ·
Replies
1
Views
3K
Finding area of the affine translation of a rectangle
  • · Replies 5 ·
Replies
5
Views
2K
Find the dimensions of the rectangle of greatest area
  • · Replies 11 ·
Replies
11
Views
8K
Can the Strength of Light Through a Semicircle and Rectangle be Maximized?
  • · Replies 11 ·
Replies
11
Views
2K
Find points on a plane at(0,0,0) 2x+3y+yz=12
  • · Replies 12 ·
Replies
12
Views
2K
Maximize Area of Rectangle w/ 2400ft Fence
  • · Replies 1 ·
Replies
1
Views
2K
Evaluating the area under a curve
  • · Replies 24 ·
Replies
24
Views
2K