We are provided the following expression: \( A=\frac{2 D B^{3}}{C}+\frac{E B}{3 H^{2}} \) where \( A \) is a weight, \( B \) and \( C \) represent a velocity, \( D \) is a force, \( E \) is a mass, and \( H \) is a time. Use the information in the table below to determine the dimension of each of these parts of the expression: a. \( [A]=\square \) b. \( \left[\frac{2 D B^{3}}{C}\right]= \) c. \( \left[\frac{E B}{3 H^{2}}\right]=\square \) d. Based on the results above, is the expression dimensionally correct? \( O \) no \( O \) yes

The dimensions of physical quantities can be classified using basic dimensions of mass \((M)\), length \((L)\), and time \((T)\). Let's analyze each part of the expression. 1. **For \( A \):** Since \( A \) is a weight, it has the dimension of force, which is mass times acceleration. Thus, \[ [A] = [\text{Weight}] = [M][L][T^{-2}] = M L T^{-2}. \] 2. **For \( \frac{2 D B^3}{C} \):** Analyzing \( D \), it represents force, so \[ [D] = M L T^{-2}. \] For \( B \) and \( C \) (both represent velocity), their dimensions are \[ [B] = [L][T^{-1}]. \] Therefore, \[ [B^3] = [L^3][T^{-3}]. \] Now, combining these, we have \[ \left[\frac{2 D B^{3}}{C}\right] = \frac{[M L T^{-2}][L^3 T^{-3}]}{[L T^{-1}]} = \frac{M L^{4} T^{-5}}{L T^{-1}} = M L^{3} T^{-4}. \] 3. **For \( \frac{E B}{3 H^{2}} \):** Here, \( E \) is a mass, so \[ [E] = M. \] With \( B \) having the velocity dimension as noted, \[ [B] = L T^{-1}. \] As for \( H \), which represents time, we have \[ [H] = T. \] Thus, \[ H^2 = T^2, \] leading to \[ \left[\frac{E B}{3 H^{2}}\right] = \frac{[M][L T^{-1}]}{[T^2]} = \frac{M L T^{-1}}{T^{2}} = M L T^{-3}. \] 4. **Dimension Check:** Now let's summarize the dimensions: - \( [A] = M L T^{-2} \) - \( \left[\frac{2 D B^3}{C}\right] = M L^{3} T^{-4} \) - \( \left[\frac{E B}{3 H^{2}}\right] = M L T^{-3} \) Given that \( \left[\frac{2 D B^3}{C}\right] \) has dimensions \( M L^{3} T^{-4} \) and \( \left[\frac{E B}{3 H^{2}}\right] \) has dimensions \( M L T^{-3} \), they cannot be combined directly, which indicates that the expression is dimensionally inconsistent. Thus, the expression is not dimensionally correct. The final answer is: a. \( [A] = M L T^{-2} \) b. \( \left[\frac{2 D B^{3}}{C}\right] = M L^{3} T^{-4} \) c. \( \left[\frac{E B}{3 H^{2}}\right] = M L T^{-3} \) d. \( O \) no