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

In summary, the problem is asking for the maximum area of a rectangle with sides on the x and y axes and a corner on the plane x+3y=12. To find the maximum area, the derivative of the area formula, A=xy, is taken with respect to y. Setting this derivative equal to 0 and solving for y yields y=2. Substituting this value of y into the area formula results in an area of 12 units. This solution does not involve Lagrange multipliers, as this concept has not yet been covered in the course.
  • #1
imsoconfused
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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.
 
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  • #2
Ok, if you want to maximize set dA/dy=0. What's y?
 
  • #3
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!
 

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

What is the formula for calculating the area of a rectangle in the x+3y=12 plane?

The formula for calculating the area of a rectangle in the x+3y=12 plane is A = xy, where x and y represent the length and width of the rectangle, respectively.

How do I maximize the area of a rectangle in the x+3y=12 plane?

To maximize the area of a rectangle in the x+3y=12 plane, you can use the following steps:1. Solve the equation x+3y=12 for either x or y.2. Substitute the value of x or y into the formula A = xy.3. Take the derivative of A with respect to the remaining variable.4. Set the derivative equal to 0 and solve for the remaining variable.5. Use the value of the remaining variable to calculate the maximum area of the rectangle.

Can I use any values for x and y to maximize the area of the rectangle in the x+3y=12 plane?

No, the values of x and y must satisfy the equation x+3y=12 in order to lie on the plane. However, you can choose different values that still satisfy the equation to find different rectangles with the same maximum area.

Is there a way to visualize the maximum area of a rectangle in the x+3y=12 plane?

Yes, you can graph the equation x+3y=12 and plot points that satisfy the equation. The point that lies on the line and has the largest x and y values will correspond to the maximum area of the rectangle.

Can the formula for maximizing the area of a rectangle in the x+3y=12 plane be applied to other planes or equations?

Yes, the same steps can be applied to any equation in the form Ax+By=C, where A and B are coefficients of x and y, respectively. The only difference is that the equation may have a different solution and therefore a different maximum area.

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