We have just started studying functions of several variables and their derivatives and our professor suggested the following problem as food for thought. Two squares, both with length l=1 intersect in a rectangle that has an area equal with 1/8 . Find the minimum and maximum distance between the centers of the squares.

Austin Rangel 2022-10-06 Answered
We have just started studying functions of several variables and their derivatives and our professor suggested the following problem as food for thought.
Two squares, both with length l=1 intersect in a rectangle that has an area equal with 1/8 . Find the minimum and maximum distance between the centers of the squares.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

omeopata25
Answered 2022-10-07 Author has 5 answers
You need to introduce some coordinates in there to transform that geometry problem into an algebra one.
So let's fix the first square between ( 0 , 0 ) and ( 1 , 1 ), so with center ( 1 / 2 , 1 / 2 ). Next, you have ( x , y ) the coordinates of one edge of the other square, which defines that square up to 4 cases. We will focus on ( x , y ) being the top right corner.
The center of that second square is then ( x 1 / 2 , y 1 / 2 ). The distance between the 2 centers is then d ( x , y ) = ( x 1 ) 2 + ( y 1 ) 2
So now the problem is to find the extremum of d ( x , y ) (or better d 2 ) subject to x y = 1 / 8.
With Lagrange multipliers, that's finding the extremum of ϕ ( x , y , λ ) = ( x 1 ) 2 + ( y 1 ) 2 λ ( x y 1 / 8 ) such that ϕ x = ϕ y = 0 = 2 ( x 1 ) λ y = 2 ( y 1 ) λ x
that implies x = y = 1 2 2 (easy to verify it's a minimum by looking at the 2nd derivative).
As for the maximum, it's obviously not a critical point, so has to be on the border. For instance x = 1 and y = 1 / 8.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-09-08

There are 100 two-bedroom apartments in the apartment building Lynbrook West.. The montly profit (in dollars) realized from renting out x apartments is given by the following function. 
P(x)=12x2+2136x41000 
How many units should be rented out in order to optimize the monthly rental profit?
What is the maximum monthly profit realizable?

asked 2021-02-05
Use polar coordinates to find the limit. [Hint: Let x=rcosandy=rsin , and note that (x, y) (0, 0) implies r 0.] lim(x,y)(0,0)x2y2x2+y2
asked 2021-12-10
INVESTMENT ANALYSIS Paul Hunt is considering two business ventures. The anticipated returns (in thousands of dollars) of each venture are described by the following probability distributions:
Venture AEarningsProbability203504503
Venture BEarningsProbability152305403
a. Compute the mean and variance for each venture.
b. Which investment would provide Paul with the higher expected return (the greater mean)?
c. In which investment would the element of risk be less (that is, which probability distribution has the smaller variance)?
asked 2021-11-19
Graph the sets of points whose polar coordinates satisfy the given polar equation.
The given polar equation is written as follows:
θ=π3,1r3
asked 2022-01-04
First of all, I'm new to multivariable calculus... in a multivariable function, by assuming that its domain is going to be R2 and its image is going to be all real numbers, the graph of that function is defined as a subset of R3 in which the x and y axis are going to receive the inputs, and the output is going to be in z(x,y,f(x,y)) ? Is that correct? Will its graph, in this example, be some kind of surface?
asked 2021-09-11
A value of r close to −1 suggests a strong _______ linear relationship between the variables.
asked 2022-01-22
How do you find critical points of multivariable function f(x,y)=x3+xyy3

New questions