Question

A sheet of metal that is 30cm wide and 600cm long is to used to make a rectangular eavestrough by bending the shett an equal amount along the dottedlines.Howmany centimeters should be turned up on each side to obtain the greatest volume for the eavestrough.Show all work

Answer 1

Answer:

The sheet should be turned up 7.5cm on each side to obtain maximum volume.

Step-by-step explanation:

If we make a rectangular eavesdrop, by bending the sheet along dotted line, then.

Height of eaves trough = x cm

Length of eaves trough = 600 cm

Width of eaves trough =  (30 - 2x) cm

We know that Volume is given by:

V = Length · Width · Height

V = (x)(30 - 2x)(600)

V = -1200x² + 18000x

To maximize the volume, we take the derivative and put it equal to zero.

$\frac{dV}{dx} =\frac{d}{dx}( -1200x^2 +18000x)=0\\\frac{d}{dx}( -1200x^2 +18000x) = 0\\-(2)1200x+18000=0\\-2400x+18000=0\\x=\frac{18000}{2400}\\ x=7.5cm$