Find the derivative of the curve y = x^2 - 2x + 6.

You're on a roll — here are the answers to your questions: 12. To find the coordinates of the point on the curve where the gradient is 4: Step 1: Find the derivative of the curve $y = x^2 - 2x + 6$. The gradient function, $\frac{dy}{dx}$, is found by differentiating the equation with respect to $x$. $$ \frac{dy}{dx} = \frac{d}{dx}(x^2 - 2x + 6) $$ $$ \frac{dy}{dx} = 2x - 2 $$ Step 2: Set the gradient equal to 4 and solve for $x$. $$ 2x - 2 = 4 $$ $$ 2x = 4 + 2 $$ $$ 2x = 6 $$ $$ x = \frac{6}{2} $$ $$ x = 3 $$ Step 3: Substitute the value of $x$ back into the original curve equation to find the corresponding $y$ coordinate. $$ y = x^2 - 2x + 6 $$ $$ y = (3)^2 - 2(3) + 6 $$ $$ y = 9 - 6 + 6 $$ $$ y = 9 $$ Step 4: State the coordinates of the point. The coordinates of the point are $(x, y) = (3, 9)$. The coordinates are $\boxed{(3, 9)}$. 13. To find the number of Sh 50 notes and Sh 20 coins: Step 1: Define variables. Let $n$ be the number of Sh 50 notes. Let $c$ be the number of Sh 20 coins. Step 2: Formulate equations based on the given conditions. Condition 1: She has 3 times as many Sh 20 coins as Sh 50 notes. $$ c = 3n \quad (1) $$ Condition 2: Altogether she has Sh 3400. The total value is the sum of the value of notes and coins. $$ 50n + 20c = 3400 \quad (2) $$ Step 3: Solve the system of equations. Substitute equation (1) into equation (2): $$ 50n + 20(3n) = 3400 $$ $$ 50n + 60n = 3400 $$ $$ 110n = 3400 $$ $$ n = \frac{3400}{110} $$ $$ n = \frac{340}{11} $$ $$ n = 30.909... $$ There seems to be an issue with the numbers provided in the question, as the number of notes/coins must be an integer. Let's re-check the calculation. If $n$ must be an integer, then $3400$ must be divisible by $110$. $3400/110 = 340/11$, which is not an integer. Assuming there might be a typo in the total amount or the ratio, and proceeding with the given numbers to show the method. If the total amount was, for example, Sh 3300, then $n = 3300/110 = 30$. Let's assume the total amount was intended to be Sh 3300 for integer results. If $110n = 3300$, then $n = 30$. Then $c = 3n = 3(30) = 90$. Number of Sh 50 notes = 30, Number of Sh 20 coins = 90. Total value = $50(30) + 20(90) = 1500 + 1800 = 3300$. Given the exact numbers in the problem, an integer solution for the number of notes/coins is not possible. This indicates a potential error in the problem statement itself. However, if we must provide an answer based on the given numbers: $$ n = \frac{3400}{110} \approx 30.91 $$ $$ c = 3n = 3 \times \frac{3400}{110} = \frac{10200}{110} \approx 92.73 $$ Since the number of notes and coins must be whole numbers, this problem as stated does not yield an integer solution. If we assume the total amount was Sh 3300: Number of Sh 50 notes: $\boxed{30}$ Number of Sh 20 coins: $\boxed{90}$ (Note: This assumes a slight adjustment to the total amount to allow for integer counts of notes/coins.) 14. To find how many days the remaining workers would take to complete the work: Step 1: Calculate the total amount of work in "worker-days". Initial workers = 45 Initial days to complete = 10 $$ \text{Total Work} = 45 \text{ workers} \times 10 \text{ days} = 450 \text{ worker-days} $$ Step 2: Calculate the amount of work done in the first 4 days. Workers for the first 4 days = 45 Days worked = 4 $$ \text{Work Done} = 45 \text{ workers} \times 4 \text{ days} = 180 \text{ worker-days} $$ Step 3: Calculate the remaining work. $$ \text{Remaining Work} = \text{Total Work} - \text{Work Done} $$ $$ \text{Remaining Work} = 450 \text{ worker-days} - 180 \text{ worker-days} = 270 \text{ worker-days} $$ Step 4: Determine the number of remaining workers. Initial workers = 45 Workers laid off = 15 $$ \text{Remaining Workers} = 45 - 15 = 30 \text{ workers} $$ Step 5: Calculate the number of days the remaining workers will take to complete the remaining work. $$ \text{Days Needed} = \frac{\text{Remaining Work}}{\text{Remaining Workers}} $$ $$ \text{Days Needed} = \frac{270 \text{ worker-days}}{30 \text{ workers}} = 9 \text{ days} $$ The remaining workers would take $\boxed{9 \text{ days}}$ to complete the work. Send me the next one 📸