Question

The diagram shows a gizmo where a cylinder is connected to a hemisphere with the same radius. The gizmo is held vertical by a thin straight rod which is connected to a horizontal plate and the top rim of the cylinder. The total volume of the gizmo is 9.8cm cubed. The radius of the hemisphere and cylinder is 0.8cm. Calculate the length of the rod.

extra info: The angle between the plate and the rod is 72°

Answer 1

5.1cm

Explanation:

volume of hemisphere = 1.07cm³

9.2-1.07 = 8.13

volume of cylinders = 8.13cm³

height of cylinder = 4.04cm

total height of gizmo = 4.04+0.8=4.84

4.84/sin(72) = 5.1cm

Answer 2

The length of the rod holding the gizmo, composed of a cylinder and hemisphere with a shared radius of 0.8 cm, is approximately 3.94 cm. This is calculated using the given total volume and the angle between the plate and the rod (72°).

To find the length of the rod, we can break down the gizmo into its components (cylinder and hemisphere) and use their volumes to set up an equation. The total volume
`(\(V_{\text{total}}\))` is the sum of the volumes of the cylinder
`(\(V_{\text{cylinder}}\))` and the hemisphere
`(\(V_{\text{hemisphere}}\))`.

The volume of a cylinder is given by
`\(V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}\)`, where
`\(r_{\text{cylinder}}\)` is the radius and
`\(h_{\text{cylinder}}\)` is the height.

The volume of a hemisphere is given by
`\(V_{\text{hemisphere}} = (2)/(3)\pi r_{\text{hemisphere}}^3\)`, where
`\(r_{\text{hemisphere}}\)` is the radius.

Given:

-
`\(r_{\text{cylinder}} = r_{\text{hemisphere}} = 0.8\)` cm

-
`\(V_{\text{total}} = 9.8\) cm\(^3\)`

Let's express the height of the cylinder
`(\(h_{\text{cylinder}}\))` in terms of the length of the rod L and then set up the equation:

`\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} \]\[ \pi r_{\text{cylinder}}^2 h_{\text{cylinder}} + (2)/(3)\pi r_{\text{hemisphere}}^3 = 9.8 \]`

Substitute the known values:

`\[ \pi (0.8)^2 h_{\text{cylinder}} + (2)/(3)\pi (0.8)^3 = 9.8 \]`

Now, solve for
`\(h_{\text{cylinder}}\)`:

`\[ \pi \cdot 0.64 \cdot h_{\text{cylinder}} + (2)/(3)\pi \cdot 0.512 = 9.8 \]\[ 0.64\pi h_{\text{cylinder}} + (2)/(3)\pi \cdot 0.512 = 9.8 \]\[ 0.64\pi h_{\text{cylinder}} + (2)/(3)\pi \cdot 0.512 = 9.8 \]\[ h_{\text{cylinder}} + (0.512)/(0.64) \approx 9.8 \]\[ h_{\text{cylinder}} + 0.8 \approx 9.8 \]\[ h_{\text{cylinder}} \approx 9 \]`

Now, the length of the rod L can be found using trigonometry. The rod, the horizontal plate, and the vertical height of the cylinder form a right triangle with the angle between the plate and the rod being
`\(72^\circ\)`. Using the tangent function:

`\[ \tan(72^\circ) = \frac{h_{\text{cylinder}}}{L} \]`

Solve for L:

`\[ L = \frac{h_{\text{cylinder}}}{\tan(72^\circ)} \]\[ L = (9)/(\tan(72^\circ)) \]\[ L \approx (9)/(2.287) \]\[ L \approx 3.94 \]`

So, the length of the rod is approximately 3.94 cm.