You are asked to build an open cylindrical can (i. e. no top) that will hold 364.5 cubic inches. To do this, you will cut its bottom from a square of metal and form its curved side by bending a rectangular sheet of metal.(a) Express the total amount of material required for the square and the rectangle in terms of r. A(r) = (b) Find the radius and height of the can that will minimize the total amount of material required. r = h =

  (a) A(r) = 4r² +729/r

  (b) r = 4.5 inches; h = 5.73 inches

Step-by-step explanation:

(a) For a given radius, the volume is given by the formula ...

  V = πr²h

so the height is ...

  h = V/(πr²)

Then the area of the rectangle required to form the curved side is ...

  A = 2πrh = (2πr)(V/(πr²)) = 2V/r

The area required for the bottom is a square that is 2r on each side, so is ...

  A = (2r)² = 4r²

The total area of required material is ...

  A(r) = 4r² +2V/r

  A(r) = 4r² +729/r

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(b) The material will be minimized when the derivative of A(r) with respect to r is zero:

  A' = 0 = 8r -2V/r²

  2V/r² = 8r

  V/4 = r³

  r = ∛(V/4) = ∛(364.5/4) ≈ 4.5 . . . inches

  h = (364.5)/(π·4.5²) = 18/π ≈ 5.73 . . . inches

Material will be minimized for a radius of 4.5 inches, and a height of 5.73 inches.

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step-by-step explanation:

geometry: triangles

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answers archive

call the larger triangle abc

then the smaller triangle cde is similar to abc (all angles are equal, aaa).  

ab is the side that is 36, and we wish to find de, the side corresponding to ab

area_cde = area_abc - area_cde --> area_abc / area_cde = 2

let x=length of de

ratio of areas = (ratio of sides)^2

+2+=+%2836%2fx%29%5e2+

+2+=+1296%2fx%5e2+

++x%5e2+=+1296%2f2+=+648+

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