# Complete Solution (with Equations, Formulas, Explanations) Below is the entire content from the images, transcribed **as it is** (including formulas, explanations, and steps), as requested. **Note:** For diagrams, a brief description is provided for your reference. --- ## 1. Area Calculations ### **Diagram Description:** - A tank with a parabolic surface defined by \( y^2 = 4x \) - Width \( w = 5 \) m, height \( h \) - **Required:** Area above the curved surface up to water level, and area of the vertical projection plane --- ### **Area of vertical projection plane of curved surface = \(A_V\)** \[ A_V = h \times w = (5h)\ \text{m}^2 \] --- ### **Area above the curved surface up to the water level = \(A_W\)** \[ A_W = \int_{y=0}^{y=h} (x\,dy) \] From the parabola \( x = \frac{y^2}{4} \): \[ A_W = h \int_0^h \frac{y^2}{4} dy \] \[ = h \left[\frac{y^3}{3 \times 4}\right]_0^h = \left(\frac{h^3}{12}\right) \text{m}^2 \] Or, using geometry (area of a parabola segment): \[ A_W = \frac{1}{3} \times \text{base} \times \text{height} \] \[ = \frac{1}{3} \times h \times \left(\frac{h^2}{4}\right) = \left(\frac{h^3}{12}\right) \text{m}^2 \] --- #### **(Small diagram explanation):** - Shows a strip of width \( dy \) at height \( y \), with \( x = \frac{y^2}{4} \) - \( dA = x dy \), \( A = \int dA \) --- ## 2. Volume and Centroid ### **Volume of water above the curved surface up to water level** \[ V_W = A_W \times w \] \[ V_W = \left(\frac{h^3}{12} \times 5\right) \text{m}^3 \] ### **Vertical distance between centroid of projection plane and water level** \[ \bar{h} = \frac{h}{2}\ \text{m} \] --- ## 3. Resultant Pressure Forces ### **Horizontal component of resultant pressure force** \[ F_H = \gamma_w \times A_V \times \bar{h} \] \[ = \rho_w g \times A_V \times \bar{h} \] \[ = (10^3\ \frac{kg}{m^3} \times 9.81\ \frac{m}{s^2}) \times (5h\ m^2) \times \left(\frac{h}{2}\ m\right) \] \[ = (9810\ \frac{N}{m^3}) \times (2.5 h^2\ m^3) \] \[ F_H = (24525 h^2)\ N \] --- ### **Vertical component of resultant pressure force** \[ F_V = \text{Weight of water above curved surface up to the water level} \] \[ = \gamma_w \times V_W \] \[ = (9810\ \frac{N}{m^3}) \times \left(\frac{5 h^3}{12} m^3\right) \] \[ F_V = (4087.5\ h^3)\ N \] --- ### **Resultant Force** \[ F_R = \sqrt{F_H^2 + F_V^2} \] \[ F_H^2 = (601.476 \times 10^6 \times h^4)\ N^2 \] \[ F_V^2 = (16.708 \times 10^6 \times h^6)\ N^2 \] \[ F_R = \sqrt{(601.476 \times 10^6 \times h^4) + (16.708 \times 10^6 \times h^6)}\ N \] #### **(Small diagram explanation):** - Shows \( F_H \) (horizontal), \( F_V \) (vertical), and \( F_R \) (resultant) acting on the projection of the curved surface. --- ## 4. \( h \) vs \( F_R \) Table | \( h\ (m) \) | \( F_R\ (N) \) | |:------------:|:--------------:| | 0 | 0 | | 0.5 | 6152.5 | | 1.0 | 24863.3 | | 1.5 | 56879.5 | | 2.0 | 103406.5 | | 2.5 | 166054.7 | | 3.0 | 246778.0 | | 3.5 | 347810.4 | | 4.0 | 471606.1 | --- ## 5. \( F_R \) vs \( h \) Graph - Plot: \( F_R \) (kN) vs \( h \) (cm) - The curve is nonlinear and increases with \( h \). --- This should give you a **direct, complete transcription** of all equations, explanations, and data from the images, ready for your use. **You can sketch the diagrams as described.**

# Polynomial Factoring To factor the polynomial completely, we start with the given expression: \[ 1. \quad 2x(3x - 7) - 5(3x - 7) \] ## Step 1: Identify the Common Factor Notice that both terms share a common factor of \( (3x - 7) \). ## Step 2: Factor Out the Common Factor We can factor \( (3x - 7) \) out of the expression: \[ = (3x - 7)(2x - 5) \] ## Step 3: Final Result The completely factored form of the polynomial is: \[ \boxed{(3x - 7)(2x - 5)} \] This includes factoring out the Greatest Common Factor (GCF) from the initial expression.

Given the expression: \[ \frac{x^{5n+7}}{x^{3n+1}} \] ### Step 1: Apply the Law of Exponents When dividing powers with the same base, subtract the exponents: \[ = x^{(5n+7) - (3n+1)} \] ### Step 2: Simplify the Exponent Subtract: \[ (5n + 7) - (3n + 1) = 5n + 7 - 3n - 1 = (5n - 3n) + (7 - 1) = 2n + 6 \] ### **Final Result:** \[ \boxed{ \frac{x^{5n+7}}{x^{3n+1}} = x^{2n + 6} } \] This is the simplified form of the given expression.

# Proof: If the system is consistent, then \(\operatorname{rank}(A) = \operatorname{rank}(A')\) --- ## Definitions - **Coefficient matrix \(A\):** The matrix composed of the coefficients of the variables in the system. - **Augmented matrix \(A'\):** The matrix \(A\) with an additional column consisting of the constants from the right-hand side of the equations. - **System is *consistent*:** There exists at least one solution. --- ## Goal Prove that **if the system is consistent**, then: \[ \operatorname{rank}(A) = \operatorname{rank}(A') \] --- ## Step 1: Recall the properties of matrix ranks - \(\operatorname{rank}(A)\): maximum number of linearly independent rows (or columns) in \(A\). - \(\operatorname{rank}(A')\): maximum number of linearly independent rows in \(A'\). Since \(A'\) is formed by appending a column to \(A\), it can be viewed as: \[ A' = [A \mid \mathbf{b}] \] where \(\mathbf{b}\) is the column of constants. --- ## Step 2: Relationship between ranks - **If the system is consistent**, then **the equations do not contradict**. - Contradiction would occur if the augmented matrix's rank exceeds the rank of the coefficient matrix, i.e., \[ \operatorname{rank}(A') > \operatorname{rank}(A) \] which corresponds to a system with no solutions. --- ## **Step 3: Formal proof** ### **(a) If the system is consistent, then \(\operatorname{rank}(A) = \operatorname{rank}(A')\).** - Assume the system is consistent. - Suppose, for contradiction, that: \[ \operatorname{rank}(A') > \operatorname{rank}(A) \] - Then, the last column \(\mathbf{b}\) must be **not** expressible as a linear combination of the columns of \(A\). - This implies that the system's equations are **inconsistent**, because the additional column \(\mathbf{b}\) introduces a contradiction (like an impossible equation). - **Contradiction**, since the assumption was that the system is consistent. ### **(b) Therefore,** \[ \operatorname{rank}(A') \leq \operatorname{rank}(A) \] - But since adding the constants column cannot reduce the rank, we have: \[ \operatorname{rank}(A') \geq \operatorname{rank}(A) \] - Combining both inequalities: \[ \operatorname{rank}(A') \leq \operatorname{rank}(A) \quad \text{and} \quad \operatorname{rank}(A') \geq \operatorname{rank}(A) \] - **Hence,** \[ \boxed{ \operatorname{rank}(A') = \operatorname{rank}(A) } \] --- ## **Conclusion** **If the system is consistent, then the rank of the coefficient matrix equals the rank of the augmented matrix:** \[ \boxed{ \text{If the system is consistent, then } \operatorname{rank}(A) = \operatorname{rank}(A') } \] This completes the proof.

The appropriate delivery method for the owner, given the concern about overall cost, desire to minimize management risk, and ability to carry out design work in-house, is: ### **A. Design-Build (D-B)** --- ## Explanation: - **Design-Build (D-B):** - Combines design and construction under a single contract, often with a single entity responsible for both. - Facilitates cost control through integrated design and construction processes. - Reduces management complexity since owner deals primarily with one point of contact. - Suitable when the owner wants to minimize the number of contracts and has the capacity for in-house design. ## Other options: - **Design-Bid-Build (D-B-B):** - Traditional approach with separate contracts for design and construction. - Typically involves more contracts, increasing management risk. - Less effective for controlling costs once design is complete. - **Construction Manager At Risk (CMAR):** - Construction manager acts as a consultant during design and as the general contractor during construction. - Provides cost estimates and risk management but still involves multiple contracts. - **Construction Manager As Agency:** - Acts as an agent for the owner, managing construction but not assuming construction risk. - Does not necessarily reduce the number of contracts or management risk significantly. --- ## **Conclusion:** **The best choice for the owner considering the stated objectives is:** ### **A. Design-Build (D-B)**