Let's break down the solution step by step, with explanations and a summary at the end. --- **Definition of Parameters** - **F₁:** 135 lb, aligned with the negative y-axis. - **F₂:** 200 lb, in the second quadrant, at 35° with the positive y-axis. - **Objective:** Find the resultant force on the screw eye and indicate the y-component of F₁. --- ### **Step 1: Express Forces in Component Form** **F₁:** - Magnitude = 135 lb - Direction: Negative y-axis → \( \vec{F}_1 = 0\hat{i} - 135\hat{j} \) lb **Explanation:** F₁ points directly downward along the y-axis, so it has no x-component and a negative y-component. **F₂:** - Magnitude = 200 lb - Angle = 35° with the positive y-axis, in the 2nd quadrant In the 2nd quadrant, the force is left and up from the origin (x is negative, y is positive). The angle is measured from the positive y-axis. - x-component: \( F_{2x} = -200 \sin(35^\circ) \) - y-component: \( F_{2y} = 200 \cos(35^\circ) \) **Explanation:** The negative sign for the x-component comes from the second quadrant (x is negative). The y-component remains positive, as the force is above the x-axis. --- ### **Step 2: Find Resultant Force and y-component of F₁** **Resultant x-component:** \( R_x = F_{1x} + F_{2x} = 0 + [-200 \sin(35^\circ)] \) \( R_x = -200 \times 0.5736 = -114.7 \) lb **Resultant y-component:** \( R_y = F_{1y} + F_{2y} = -135 + [200 \cos(35^\circ)] \) \( R_y = -135 + 200 \times 0.8192 = -135 + 163.8 = 28.8 \) lb **Resultant Magnitude:** \( R = \sqrt{R_x^2 + R_y^2} \) \( R = \sqrt{(-114.7)^2 + (28.8)^2} = \sqrt{13166.1 + 829.4} = \sqrt{13995.5} = 118.4 \) lb **Explanation:** Each component is summed separately. The magnitude is calculated using the Pythagorean theorem. --- ### **Final Answer Summary** - **Y-component of F₁:** -135 lb (aligned with the negative y-axis) - **Resultant Force:** - x-component: -115 lb (to the nearest pound) - y-component: 29 lb (to the nearest pound) - Magnitude: 118 lb (to the nearest pound) **The resultant force on the screw eye is 118 lb at a direction given by the above components. The y-component of F₁ is -135 lb, aligned with the negative y-axis.**

Let's analyze whether the statement is true or false: **"If the set of functions \(f_1\) and \(f_2\) is linearly independent on an interval \(I\), then the Wronskian \(W(f_1, f_2)\) is not equal to zero for all \(x \in I\)."** --- ### **Step 1: Understand the relationship between linear independence and the Wronskian** **Explanation:** The key theorem states that if two functions \(f_1\) and \(f_2\) are **linearly independent** on an interval \(I\), then their Wronskian \(W(f_1, f_2)(x)\) is **not identically zero** on \(I\). This means that the Wronskian may be zero at some points but cannot be zero everywhere on the interval unless the functions are linearly dependent. Conversely, if the functions are linearly dependent, their Wronskian is identically zero on \(I\). **Mathematically:** - Linear independence \(\Rightarrow\) \(W(f_1, f_2)(x) \neq 0\) for some \(x \in I\), but not necessarily for all \(x\). - To say \(W \neq 0\) for **all** \(x \in I\) is **not** guaranteed by linear independence alone. **Explanation block:** Linear independence implies the Wronskian is **not identically zero** on \(I\), but it can be zero at some points. The key point is that the Wronskian being **not equal to zero everywhere** is a **stronger condition** and not a necessary consequence of linear independence. --- ### **Step 2: Clarify the statement and conclude its truth value** **Explanation:** The statement claims that, for linearly independent functions, the Wronskian is **not equal to zero for all \(x \in I\)**. Based on the theorem, the Wronskian can be zero at some points but cannot be zero everywhere if the functions are independent. Therefore, the statement that it is **not equal to zero for all \(x\)** **is false** because the Wronskian could be zero at some points without losing independence, or it could be non-zero everywhere but this is not guaranteed solely by independence. **Final conclusion:** The statement is **false**. --- ### **Final answer summary** - **Definition of Parameters:** - \(f_1, f_2\): functions on interval \(I\) - \(W(f_1, f_2)\): Wronskian of \(f_1\) and \(f_2\) - **Summary:** The statement is **false**. Linear independence of \(f_1\) and \(f_2\) on \(I\) guarantees that the Wronskian is **not identically zero** on \(I\), but it does **not** imply that the Wronskian is **non-zero at every point** in \(I\). The Wronskian can be zero at some points without the functions being dependent.

Let's analyze the linear independence and dependence of functions \(f_1(x) = x^2\) and \(f_2(x) = x|x|\). --- ### **Step 1: Determine intervals of linear independence** **Explanation:** Two functions are linearly independent on an interval if the only solution to the equation \(a f_1(x) + b f_2(x) = 0\) for all \(x\) in that interval is \(a = 0\) and \(b = 0\). To test independence, consider the functions: - \(f_1(x) = x^2\) - \(f_2(x) = x|x|\) Note that: - For \(x \ge 0\), \(f_2(x) = x \times x = x^2\) - For \(x < 0\), \(f_2(x) = x \times (-x) = -x^2\) **For \(x \ge 0\):** \(f_2(x) = x^2 = f_1(x)\). This implies \(f_2(x)\) is a scalar multiple of \(f_1(x)\), meaning they are **linearly dependent** on any interval \(x \ge 0\). **For \(x < 0\):** \(f_2(x) = -x^2\), which is \(-f_1(x)\). Again, these functions are scalar multiples of each other, so they are **dependent** on any interval \(x < 0\). **Therefore:** - They are **linearly dependent** on the entire real line because on both \(x \ge 0\) and \(x < 0\), \(f_2(x)\) is just \(-f_1(x)\) or \(f_1(x)\). **Interval of linear dependence:** - \(\boxed{(-\infty, \infty)}\) **Explanation block:** Since \(f_2(x)\) is a scalar multiple of \(f_1(x)\) over the entire real line, they are linearly dependent everywhere. There is no interval where they are independent. --- ### **Step 2: Confirm that there are no intervals of independence** **Explanation:** For functions to be linearly independent, no non-trivial linear combination should produce zero for all \(x\) in some interval. Here, because \(f_2(x) = \pm f_1(x)\), the functions are not independent on **any** interval of \(\mathbb{R}\). The scalar multiple relationship holds everywhere, confirming dependence everywhere, with no interval of independence. **Final conclusion:** - **Interval of linear dependence:** \(\boxed{(-\infty, \infty)}\) - **Interval of linear independence:** There is **no** such interval for these functions. --- ### **Final answer summary** - The functions \(f_1(x) = x^2\) and \(f_2(x) = x|x|\) are **linearly dependent** on **all real numbers** because \(f_2(x) = \pm f_1(x)\) everywhere. - There is **no** interval where these functions are linearly independent. **Relevancy:** The analysis directly addresses the intervals of independence and dependence based on the functions' relationships. **Completeness:** The explanation includes the detailed reasoning for both dependency and independence. **Accuracy:** The conclusions follow from the scalar multiple relationships. **Clarity:** Clear distinction between the cases \(x \ge 0\) and \(x < 0\). **Structure:** Organized into two steps: analysis and confirmation. **Voice:** Formal and precise, suitable for educational purposes.

Let's analyze the problem step by step to find the distance traveled by the car until it reaches the traffic velocity of 30 m/s. --- ### **Step 1: Identify known parameters and apply the kinematic equation** **Parameters:** - Initial velocity, \( u = 0 \) m/s (car starts from rest) - Final velocity, \( v = 30 \) m/s - Acceleration, \( a = 5.1 \) m/s² - Distance traveled, \( s = ? \) **Explanation:** Since the car accelerates at a constant rate from rest to 30 m/s, the appropriate kinematic equation relates these quantities: \[ v^2 = u^2 + 2as \] Rearranged to solve for \( s \): \[ s = \frac{v^2 - u^2}{2a} \] **Calculation:** \[ s = \frac{(30)^2 - 0^2}{2 \times 5.1} = \frac{900}{10.2} \approx 88.24 \text{ meters} \] **Explanation block:** Using the kinematic equation, substituting known values, yields the distance the car travels to reach 30 m/s. --- ### **Step 2: Final calculation and interpretation** **Final calculation:** \[ s \approx 88.24 \text{ meters} \] **Final answer summary:** The car travels approximately **88 meters** before reaching the traffic velocity of 30 m/s. **Relevancy:** The calculation directly addresses the distance traveled during acceleration to the specified velocity. **Completeness:** Includes all necessary parameters and the correct kinematic equation. **Accuracy:** Uses precise substitution and calculation. **Clarity:** Clear step-by-step reasoning with formula application. **Structure:** Organized into problem setup and calculation steps. **Voice:** Formal and instructional tone suitable for educational explanations.