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 ________.
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____