Recall the relationship between diameter and radius.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
1.2.1) Calculate the radius of the cylindrical magic cube if the diameter is $67 \text{ mm}$.
Step 1: Recall the relationship between diameter and radius.
The radius is half of the diameter.
$$ \text{Radius} = \frac{\text{Diameter}}{2} $$
Step 2: Substitute the given diameter and calculate the radius.
$$ \text{Radius} = \frac{67 \text{ mm}}{2} $$
$$ \text{Radius} = 33.5 \text{ mm} $$
The radius of the cylindrical magic cube is $\boxed{\text{33.5 mm}}$.
1.2.2) Calculate the total volume of the cylindrical magic cube in $\text{mm}^3$.
The following formula may be used:
Volume of cylinder = $\pi \times \text{radius} \times \text{radius} \times \text{height}$, where $\pi = 3.142$.
Step 1: Identify the given dimensions and $\pi$ value.
From the image, the height of the cylindrical magic cube is $60 \text{ mm}$.
From 1.2.1, the radius is $33.5 \text{ mm}$.
The value of $\pi$ is given as $3.142$.
Step 2: Substitute the values into the volume formula.
$$ \text{Volume} = \pi \times \text{radius}^2 \times \text{height} $$
$$ \text{Volume} = 3.142 \times (33.5 \text{ mm})^2 \times 60 \text{ mm} $$
$$ \text{Volume} = 3.142 \times 1122.25 \text{ mm}^2 \times 60 \text{ mm} $$
$$ \text{Volume} = 3.142 \times 67335 \text{ mm}^3 $$
$$ \text{Volume} = 211516.47 \text{ mm}^3 $$
The total volume of the cylindrical magic cube is $\boxed{\text{211516.47 mm}^3}$.
1.2.3) Calculate the total surface area of the magic cube in $\text{mm}^2$.
The following formula may be used:
Total surface area = side $\times$ side $\times$ 6
Step 1: Identify the side length of the cubic magic cube.
From the image, the side length of the cubic magic cube is $57 \text{ mm}$.
Step 2: Substitute the side length into the surface area formula.
$$ \text{Total surface area} = \text{side}^2 \times 6 $$
$$ \text{Total surface area} = (57 \text{ mm})^2 \times 6 $$
$$ \text{Total surface area} = 3249 \text{ mm}^2 \times 6 $$
$$ \text{Total surface area} = 19494 \text{ mm}^2 $$
The total surface area of the magic cube is $\boxed{\text{19494 mm}^2}$.
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