the pretty perfume company designed a new perfume bottle. The bottle is to be a spherical shape with a diameter of 7cm.

a) determine the volume and surface area of this bottle.

b) The spherical bottle has a conical shaped lid with a diameter of 5cm and a height of 4.5cm. Calculate the volume and surface area of the lid. Before you can use the surface area formula for a cone you will need to find the slant height of the lid.

Please show your work so I can understand how to do it.

a) diameter = 2r

Volume = 4/3πr^3

Surface area = 4πr^2

b) Surface area = πr^2 (base) + πrs (cone), where s is diagonal measure of side

Pythagorean theorem, h^2 + r^2 = s^2, to find value of s.

V = 1/3πr^2h

Insert the appropriate values into these formulas.

a) To determine the volume and surface area of the spherical bottle, we can use the following formulas:

Volume of a sphere: V = 4/3 * π * r^3
Surface area of a sphere: A = 4 * π * r^2

Given that the diameter of the bottle is 7cm, we can find the radius by dividing the diameter by 2: r = 7cm / 2 = 3.5cm.

Now we can substitute this value into the formulas:

Volume: V = 4/3 * π * (3.5cm)^3 = 4/3 * 3.14 * 3.5^3 ≈ 179.59 cm³
Surface area: A = 4 * π * (3.5cm)^2 = 4 * 3.14 * 3.5^2 ≈ 153.94 cm²

Therefore, the volume of the perfume bottle is approximately 179.59 cm³, and the surface area is approximately 153.94 cm².

b) To find the volume and surface area of the conical lid, we first need to find the slant height.

The slant height (l) can be calculated using the Pythagorean theorem:

l = √(r^2 + h^2)

Given that the diameter of the lid is 5cm, the radius is 5cm / 2 = 2.5cm, and the height is 4.5cm. Now we can substitute these values into the formula:

l = √(2.5cm^2 + 4.5cm^2) = √(6.25cm^2 + 20.25cm^2) = √(26.5cm^2) ≈ 5.15cm

Now we can calculate the volume and surface area of the lid using the following formulas:

Volume of a cone: V = 1/3 * π * r^2 * h
Surface area of a cone: A = π * r * (r + l)

Substituting the values into the formulas:

Volume: V = 1/3 * 3.14 * (2.5cm)^2 * 4.5cm ≈ 29.53 cm³
Surface area: A = 3.14 * 2.5cm * (2.5cm + 5.15cm) ≈ 57.99 cm²

Therefore, the volume of the conical lid is approximately 29.53 cm³, and the surface area is approximately 57.99 cm².