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.

c. (3,2,6) is the answer.

d) by the cross product property, ab^2 = bc multiplied by ad

step-by-step explanation:

we are given

in the given triangle abc

angle a is 90° and segment ad is perpendicular to segment bc

we can see that

triangle(abc) and triangle(abd) are similar because they have common side and same angle (90 degree)

so, triangles(abc) and abd are similar

since, they are similar

so, the ratio of their sides must be equal

we get

now, we can cross multiply

and we get

so, this could be step to prove it