When a concrete structure is built, an engineer often performs a slump test on the concrete to ensure that it is of suitable strength and consistency. The engineer fills a truncated cone with a bottom diameter 200mm, top diameter 100mm and height 300mm with concrete. The engineer then places the cone on the ground and lifts off the cone. After period of time, she measures the amount of that the top of the cone has slumped to decide if the concrete is appropriate.

a) Original cone from which the slump test cone was derived had the height of 600m. What was the volume of the original cone? Express your answer to nearest cubic mm.

Formula to figure out the volume of the cone is -->V= 1/3ð<r2>h

This would have been quite easy if they had only given one diameter, but they have given two and I do not know which one to use. The top or bottom?

Also this is part b)

What was the volume of the slump-test cone?

Is it only the height which changes in this part. Because in the first part it was 600 and now it is 300 right? I am still unsure of which diameter to use, because there are 2.

To calculate the volume of the original cone, you need to use the larger diameter, which is the bottom diameter of 200mm. The height remains the same at 600mm.

a) Volume of the original cone:
V = (1/3)πr^2h
V = (1/3)π(100mm/2)^2(600mm)
V ≈ 1/3 × 3.14 × 50^2 × 600
V ≈ 1/3 × 3.14 × 2500 × 600
V ≈ 1/3 × 3.14 × 1,500,000
V ≈ 1/3 × 4,710,000
V ≈ 1,570,000 cubic mm (approximate)

For part b), the volume of the slump-test cone can be calculated using the smaller diameter of 100mm and the reduced height of 300mm.

b) Volume of the slump-test cone:
V = (1/3)πr^2h
V = (1/3)π(50mm/2)^2(300mm)
V ≈ 1/3 × 3.14 × 25^2 × 300
V ≈ 1/3 × 3.14 × 625 × 300
V ≈ 1/3 × 3,927,500
V ≈ 1,309,166.67 cubic mm (approximate)

To find the volume of the original cone, you would need to use the dimensions provided - the bottom diameter of 200mm, the top diameter of 100mm, and the height of 600mm. Since the original cone is a truncated cone, you'll need to calculate the volumes of both the larger and smaller cones and subtract them to find the volume of the remaining frustum.

First, calculate the radius of the larger cone:

Radius of larger cone = Bottom diameter / 2 = 200mm / 2 = 100mm

Second, calculate the radius of the smaller cone:

Radius of smaller cone = Top diameter / 2 = 100mm / 2 = 50mm

Now, use the formula for the volume of a cone to find the volumes of the larger and smaller cones:

Volume of the larger cone = (1/3) * pi * (Radius of larger cone)^2 * height
= (1/3) * 3.14 * (100mm)^2 * 600mm

Volume of the smaller cone = (1/3) * pi * (Radius of smaller cone)^2 * height
= (1/3) * 3.14 * (50mm)^2 * 600mm

Finally, subtract the volume of the smaller cone from the volume of the larger cone to get the volume of the original cone:

Volume of original cone = Volume of larger cone - Volume of smaller cone

Now that you have the volume of the original cone, you can move on to calculating the volume of the slump-test cone. In this case, you are given the dimensions of the slump-test cone - the bottom diameter of 200mm, the top diameter of 100mm, and the height of 300mm. You can use the same formula for the volume of a cone as before to calculate the volume of the slump-test cone:

Volume of slump-test cone = (1/3) * pi * (Radius of larger cone)^2 * height

Remember, in this case, the height is 300mm.

I hope this helps! Let me know if you have any further questions.