Question

The Goodsmell perfume producing company has a new line of perfume and is designing a new bottle for it. Because of the expense of the glass required to make the bottle, the surface area must be less than 150 cm2. The company also wants the bottle to hold at least 100mL of perfume. The design under consideration is in the shape of a cylinder. Determine the maximum volume possible for a cylindrical bottle that has a total surface area of less than 150 cm2. Determine the volume to the nearest 10mL. Report the dimensions of the bottle and he corresponding surface are and volume.

Answer 1

Dimensions of the bottle

x (radius of the base ) = 3,99 cm

h (heigh of the bottle ) = 3,99 cm

Surface area = 149,99 cm²

Volume of the bottle = 199,45 cm³

Explanation:

The bottle volume must be V(b) = 100 ml or V(b) = 100 cm³

The shape of the bottle is cylindrical

Surface area of bottle is

S = surface area of the base + lateral area

Area of the base = π*x ² where x is radius of circle

Lateral area is 2*π*x*h where h is the heigh of the bottle

V(b) = π* x²*h (1)

π*x² + 2*π*x*h < 150 cm² we work with the limit 150

π*x² + 2*π*x*h = 150

h = (150 - π*x²) /2*x*π

Plugging that value in equation (1)

V(x) = π*x² * (150 - π*x²) /2*x*π ⇒ V(x) = 150*π*x²/2*x*π - π²*x⁴/2*x*π

V(x) = 75*x - π*x³/2

Taking derivatives on both sides of the equation

V´(x) = 75 - 3*π*x²/2

V´(x) = 0 75 - 3*π*x² /2 = 0

x² = 75*2 /3*π ⇒ x² = 15,92 ⇒ x = 3,99 cm

And h = ( 150 -π*x² )/2*π*x

h = ( 150 - 49,98 )/25,05

h = 3,99 cm

Dimensions of the bottle

x (radius of the base ) = 3,99 cm

h (heigh of the bottle ) = 3,99 cm

Surface area = 149,99 cm²

Volume of the bottle = 199,45 cm³