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.
So your bottle has a radius of 3.5 cm
volume of sphere = (4/3)π r^3
surface area = 4π r^2
sub in r = 3.5 into both equations.
b) the surface area of a cone, without its base, is
πrs, where s is the slant height, and r is the radius.
we have to first calculate s by
s^2 = 2.5^2 + 4.5^2
s = √26.5
so surface area = π(2.5)(√26.5))
= ....
a) To determine the volume and surface area of a spherical perfume bottle with a diameter of 7cm, we can use the following formulas:
Volume of a sphere = (4/3) * π * r^3
Surface area of a sphere = 4 * π * r^2
Given that the diameter of the bottle is 7cm, we can find the radius (r) by dividing the diameter by 2:
r = 7cm / 2 = 3.5cm
Now we can calculate the volume:
Volume = (4/3) * π * 3.5^3 = (4/3) * 3.14 * 3.5^3 = 179.59 cm^3 (rounded to two decimal places)
And the surface area:
Surface area = 4 * π * 3.5^2 = 4 * 3.14 * 3.5^2 = 153.86 cm^2 (rounded to two decimal places)
Therefore, the volume of the perfume bottle is approximately 179.59 cm^3 and its surface area is approximately 153.86 cm^2.
b) To calculate the volume and surface area of the conical lid, we'll need to find the slant height first.
Given that the lid has a diameter of 5cm and a height of 4.5cm, we can find the slant height (l) using the Pythagorean theorem:
l^2 = r^2 + h^2
Where r is the radius and h is the height (or altitude) of the cone.
First, let's find r:
r = 5cm / 2 = 2.5cm
Now, let's find l:
l^2 = 2.5^2 + 4.5^2 = 6.25 + 20.25 = 26.5
l ≈ √26.5 ≈ 5.148cm (rounded to three decimal places)
Now that we have the slant height, we can calculate the volume and surface area of the lid:
Volume = (1/3) * π * r^2 * h = (1/3) * 3.14 * 2.5^2 * 4.5 = 58.88 cm^3 (rounded to two decimal places)
Surface area = π * r * l = 3.14 * 2.5 * 5.148 = 40.68 cm^2 (rounded to two decimal places)
Therefore, the volume of the lid is approximately 58.88 cm^3 and its surface area is approximately 40.68 cm^2.
Note: These formulas assume that the lid of the perfume bottle is a perfect cone and has a rounded top.
a) To determine the volume of a spherical perfume bottle, we can use the formula:
Volume of a Sphere = (4/3) π r^3
Where r is the radius of the sphere, which is half of the diameter. In this case, the diameter is 7cm, so the radius (r) would be 7cm/2 = 3.5cm.
Volume = (4/3) π (3.5^3) = (4/3) π (42.875) ≈ 180.95 cm^3
Therefore, the volume of the perfume bottle is approximately 180.95 cm^3.
Next, to determine the surface area of the spherical perfume bottle, we can use the formula:
Surface Area of a Sphere = 4π r^2
Surface Area = 4π (3.5^2) = 4π (12.25) ≈ 48.57 cm^2
Therefore, the surface area of the perfume bottle is approximately 48.57 cm^2.
b) To find the volume of the conical lid attached to the spherical perfume bottle, we can use the formula:
Volume of a Cone = (1/3) π r^2 h
Where r is the radius of the lid (diameter/2), and h is the height of the lid. In this case, the diameter of the lid is 5cm, so the radius (r) would be 5cm/2 = 2.5cm. The height (h) of the lid is 4.5cm.
Volume = (1/3) π (2.5^2) (4.5) = (1/3) π (6.25) (4.5) ≈ 27.91 cm^3
Therefore, the volume of the lid is approximately 27.91 cm^3.
To calculate the surface area of the lid, we first need to find the slant height (l) of the cone. We can use the Pythagorean theorem:
l = √(r^2 + h^2)
l = √(2.5^2 + 4.5^2) = √(6.25 + 20.25) = √(26.5) ≈ 5.15 cm
Now, we can use the formula for the surface area of a cone:
Surface Area of a Cone = π r (r + l)
Surface Area = π (2.5) (2.5 + 5.15) = π (2.5) (7.65) ≈ 60.42 cm^2
Therefore, the surface area of the lid is approximately 60.42 cm^2.
I hope this explanation helps you understand how to calculate the volume and surface area of a spherical perfume bottle and its conical lid.