Volume and Surface Area of Pyramids, Cones and Spheres

Question

a. Calculate the cost of painting a spherical monument of radius 8 m if 1 litre of paint which costs $8.50 can cover an area of 8 m²

b. The protective nose cone of a re-entry space capsule can be ejected leaving a crew cabin. The figure shows a model of the space capsule with the crew cabin having a vertical height of 59 cm and a top diameter of 27 cm.

Taking a to be 3.142, find the volume of the crew cabin of the model

c. A hollow beach ball made of plastic has an outer radius of 4 cm and an inner radius of 3.6 cm. Find the volume of plastic required to make it.

d. A hollow spherical steel mould has an external diameter of 16 cm and a uniform thickness of 0.8 cm. Find the volume of steel required to make 100 of such moulds

Answer 1

I can try and do my best to help you understand what each problem is asking you to do, so that you can go ahead and solve them on your own since it would take forever to try and explain through each one with work!

Part A: Painting Spherical Monument

When looking at this problem, there's 3 things we need to find out:

1) What is the Surface Area of the spherical monument?

2) How many cans of paint are needed to cover said surface area if we know each can cover an area of 8 m²

3) What is the total cost of all the cans we need if each one costs $8.50.

In case you weren't sure, the S.A. of a sphere is represented as: S.A._sphere = 4π •r ²

Part B: Finding Volume of Crew Cabin

Without the figure, I don't think it's possible for anybody here to try and solve since we don't know the height of the entire space capsule. However, I think I can guide you in the right direction to solve this problem.

The entire space capsule itself seems to be a cone. The protective nose cone is a cone itself, and that can detach from the space capsule, leaving just the crew cabin.

In order to find the volume of the crew cabin, we need to find:

1. Volume of the entire space capsule
2. Volume of the protective cone

From there: V_{crew cabin} = V_capsule - V_cone

The volume of a cone can be represented as V = (1/3) π r ² h, where instead of π, you're asked to use 3.142

Part C: Hollow Beach Ball

To find the volume of plastic required to make the hollow beach ball, we need to calculate the volume of the space between the outer and inner spheres. The volume of a sphere is given by the formula:

1. V = (4/3) π r ³

Part D: Hollow Steel Mould

This problem has some similarities to the prior one as you need to calculate the outer and inner sphere volumes in order to find the volume of a single mould.

Keep in mind that in the problem, you are given the external diameter, and to find the volume of the outer sphere, you need the radius!

The difference in volumes of the outer and inner spheres is the volume of a single (1) mould. To find the volume of 100, just multiply your result by 100!

Feel free to reach out if you have any questions!