Step 1: Analyzing the structure of a bathing mug.
- The mug generally has a cylindrical or slightly tapered body.
- A curved handle is attached to one side to facilitate grip.
Step 2: Sketching the hand holding the mug.
- The fingers and thumb should naturally wrap around the handle.
- The grip must be proportionate to the size and shape of the mug.
Step 3: Depicting material characteristics.
- The mug can be made of plastic, metal, or stainless steel.
- Apply texture details and reflective highlights based on the material properties.
Step 4: Enhancing realism.
- Utilize shading techniques to create depth and form.
- Include cast shadows of the mug and hand to indicate light direction.
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
A temperature difference can generate e.m.f. in some materials. Let $ S $ be the e.m.f. produced per unit temperature difference between the ends of a wire, $ \sigma $ the electrical conductivity and $ \kappa $ the thermal conductivity of the material of the wire. Taking $ M, L, T, I $ and $ K $ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $ Z = \frac{S^2 \sigma}{\kappa} $ is: