Question

volume of a cone r2h, where r is the radius and h is the height. The shape below is made from a cone and a cylinder. The total volume of the shape is 6150 m³. Work out the radius, r, to 2 d.p.

Answer 1

`r\approx 10.33 \text{ m}`

Explanation:

We can model the total volume of the figure as the sum of the volume of the cone and the volume of the cylinder:

`V_\text{total} = V_\text{cone} + V_\text{cylinder}`

We can substitute in the the given volume formulas:

• 
  `V_\text{cone}=\frac{1}3\pi r^2 \cdot h_\text{cone}`
• 
  `V_\text{cylinder} = \pi r^2 \cdot h_\text{cylinder}` (notice how this is equal to
  `A_\text{base} \cdot h`)

↓↓↓

`V_\text{total} = (1)/(3)\pi r^2 \cdot h_\text{cone} + \pi r^2 \cdot h_\text{cylinder}`

Now, we can plug in the given volume and height values:

• 
  `h_\text{cone} = 16`
• 
  `h_\text{cylinder}=13`
• 
  `V_\text{total} = 6150`

↓↓↓

`6150 = (1)/(3)\pi r^2 \cdot 16 + \pi r^2 \cdot 13`

Finally, we can solve for
`r`:

↓ factoring a
`\pi r^2` out of both terms on the right side

`6150 = \left((16)/(3) + 13\right)\pi r^2`

↓ executing the addition

`6150 = (55)/(3)\pi r^2`

↓ dividing both sides by
`(55\pi)/(3)`

`6150/ (55\pi)/(3) = r^2`

↓ rewriting dividing by a fraction as multiplying by its reciprocal

`6150 \cdot (3)/(55\pi) = r^2`

`(18450)/(55\pi) = r^2`

↓ taking the square root of both sides

`\sqrt{(18450)/(55\pi)} = r`

↓ evaluating using a calculator

`\boxed{r\approx 10.33\text{ m}}`