It appears that you've uploaded an image with a problem related to the equations of the elastic curve. To proceed, I'll help you with the steps and interpretation of the formulas provided.

The problem is asking to determine the equations of the elastic curve for a beam with constant EI (flexural rigidity). The provided formulas for the slope and deflection are related to the displacement of the beam under applied loads.

Here’s a breakdown of the given equations:

1. Deflection at Point 1: $v_1 = \frac{1}{12EI} \left( -x_1^4 + 32x_1^3 - 384x_1^2 \right) \, \text{k} \cdot \text{ft}^3$ This formula calculates the vertical deflection at point 1 of the beam, based on the distance $x_1$ from the left end (point A).

2. Deflection at Point 2: $v_2 = \frac{1}{3EI} \left( -512x_2 + 1024 \right) \, \text{k} \cdot \text{ft}^3$ This gives the deflection at point 2, based on the distance $x_2$.

3. Slope at Point B: $\theta_B = \frac{512k \cdot \ell^2}{3EI}$ This represents the slope at point B, given as a function of the length $\ell$.

4. Deflection at Point B: $v_B = \frac{5120k \cdot \text{ft}^3}{3EI}$ This formula calculates the deflection at point B.

Would you like me to explain the specific steps for solving this problem or elaborate on any particular aspect of the beam's behavior or the derivation of these equations? Let me know how you'd like to proceed!

Here are some related questions that might be useful:

1. How do you interpret and apply the relationship between deflection and the applied load?
2. What is the significance of the constant EI in the equations of the elastic curve?
3. How would the deflection change if the applied load were different?
4. How can the beam’s slope at point B be used to determine further structural behavior?
5. What would happen if the beam material or shape changed, altering EI?

Tip: When dealing with beam deflection problems, always ensure the units for load, length, and deflection are consistent throughout the calculations!