Use lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square. let the sides of the rectangle be x and y and let f and g represent the area (a) and perimeter (p), respectively. find the following. a = f(x, y) = p = g(x, y) = f(x, y) = lambda g = then lambda = 1/2y = implies that x = therefore, the rectangle with maximum area is a square with side length.

the rectangle of maximum area is a square with side length x=y= p/4

Step-by-step explanation:

Since

A = f(x,y) = x*y

p= g(x,y)= 2*x+2*y

therefore using Lagrange multiplier →λ

F(x,y) = f(x,y) - λ*g(x,y)

and

Fx(x,y) = fx (x,y) - λ*gx(x,y) = 0

Fx(x,y) = fy (x,y) - λ*gy(x,y) = 0

g(x,y) = p

where fx and gx are the partial derivatives with respect to x and fy and gy are the ones of y

therefore

fx (x,y) - λ*gx(x,y) = y - λ*2 = 0 → y = 2λ

fy (x,y) - λ*gy(x,y) = x - λ*2 = 0 → x = 2λ

2*x+2*y = p → 2*2λ + 2*2λ = p → 8λ = p →  p/8=λ

therefore

x = y = 2λ = 2*p/8 = p/4

A max = x*y = p/4 * p/4 = p²/16

since the side x = side y → the rectangle of maximum area is a square with side length = p/4

plants store their excess carbohydrate in the form of starch so that it could be used as energy source later on. the excess carbohydrate is converted into starch in the plastid i.e. chloroplast & this process occurs in the presence of sunlight. it is an insoluble polymer of glucose which gets stored in cell organelles named as amyloplasts. in some plants, starch is also stored in roots and embryos.

answer:

total cost for tiles and paints is $924.  

step-by-step explanation:

we have been given that a community hall is in the shape of a cuboid. the hall is 40m long 15m high and 3m wide.

the paint will be required for 4 walls and ceiling.

let us find area of walls and ceiling.

therefore, the area of walls and ceiling is 1410 square meters.

given: cost for 10 litre of paint is $10 and 10 litre paint covers 25 square meter. therefore,

therefore, the total painting cost is $564.

tiles will be required for floor. let us find the area of floor.

given: 1m squared floor tiles costs $3. so,

therefore, the total tiles cost is $360.

now let us find combined total cost of tiles and paint.

therefore, the combined total cost of tiles and paint is $924.