A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through a

Question

grigory [225]

3 years ago

10

A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through a

n angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water?

Mathematics

1 answer:

sukhopar [10] · Answer 1 · 2022-09-04T23:56:55+03:00

Answer:

θ = 60°

Step-by-step explanation:

The cross sectional area of the trapezoid shape will be that of a trapezoid with bases of 10 cm and (10 cm + 2·(10 cm)·cos(θ)) and height (10 cm)·sin(θ).

That area in cm² is ...

A = (1/2)(b1 +b2)h = (1/2)(10 + (10 +20cos(θ))(10sin(θ)

A = 100sin(θ)(1 +cos(θ))

A graphing calculator shows this area to be maximized when ...

θ = π/3 radians = 60°

_____

<em>A</em> will be maximized when its derivative with respect to θ is zero. That derivative can be found to be 2cos(θ)² +cos(θ) -1, so the solution reduces to ...

cos(θ) = 1/2

θ = arccos(1/2) = π/3