Algebra 1 Topic 2: Linear Equations NJGPA REVIEW Part B What equation expresses \( P \), the amount in dollars, of Melissa's mother's donation as a function of \( M \), the number of miles that Melissa walks? A) \( P=M+10 \) B) \( P=2 M+10 \) C) \( P=10 M+2 \) D) \( P=2(M+10) \) Part C If the donation is \( \$ 44 \), how many miles did Melissa walk? A) 15 B) 17 C) 22 D) 98 Part D Melissa wants her mother to donate at least \( \$ 65 \). Which of the listed numbers of miles would make the donation at least \( \$ 65 \) ? Select all that apply. A) 23 miles B) 25 miles C) 27 miles D) 29 miles E) 31 miles F) 33 miles Example 6: Use the given information to answer Part A and Part B. Part A A dump truck weighs 11.25 tons when empty. A conveyor belt pours sand into the truck at a constant rate of \( \frac{1}{4} \) ton per minute until it is full. Let \( t \) represent the elapsed time in minutes. Let \( w \) represent the weight of the truck after \( t \) minutes. Write a linear equation for \( w \) in terms of \( t \). Part B The dump truck from Part A weighs 18 tons when filled. At the same time the dump truck is being filled an identical dump truck filled to capacity is being emptied at a rate of \( \frac{1}{8} \) ton per minute. How much sand is in each dump truck when the trucks are the same weight? 22

Sure, let's work through each part of the Algebra 1 Topic 2: Linear Equations NJGPA REVIEW step by step. --- ### **Part B** **Question:** What equation expresses \( P \), the amount in dollars, of Melissa's mother's donation as a function of \( M \), the number of miles that Melissa walks? **Options:** A) \( P = M + 10 \) B) \( P = 2M + 10 \) C) \( P = 10M + 2 \) D) \( P = 2(M + 10) \) **Answer:** **B) \( P = 2M + 10 \)** **Explanation:** To determine the correct equation, we need to understand the relationship between the miles walked (\( M \)) and the donation amount (\( P \)). - **\( P = 2M + 10 \):** - **\( 2M \)** represents a donation rate of \$2 per mile. - **\( +10 \)** represents a base donation of \$10 regardless of the miles walked. This fits a typical scenario where there's a base donation plus an additional amount per mile. The other options either have incorrect coefficients or structure that doesn't align with this relationship. --- ### **Part C** **Question:** If the donation is \$44, how many miles did Melissa walk? **Options:** A) 15 B) 17 C) 22 D) 98 **Answer:** **B) 17** **Explanation:** Using the equation from Part B: \[ P = 2M + 10 \] Given \( P = 44 \): \[ 44 = 2M + 10 \] Subtract 10 from both sides: \[ 34 = 2M \] Divide both sides by 2: \[ M = 17 \] So, Melissa walked **17 miles**. --- ### **Part D** **Question:** Melissa wants her mother to donate at least \$65. Which of the listed numbers of miles would make the donation at least \$65? **Options (Select all that apply):** A) 23 miles B) 25 miles C) 27 miles D) 29 miles E) 31 miles F) 33 miles **Answer:** **D) 29 miles** **E) 31 miles** **F) 33 miles** **Explanation:** Using the equation: \[ P = 2M + 10 \] We want: \[ 2M + 10 \geq 65 \] Subtract 10 from both sides: \[ 2M \geq 55 \] Divide by 2: \[ M \geq 27.5 \] Since the number of miles walked must be a whole number: - **29 miles, 31 miles,** and **33 miles** are all greater than or equal to 27.5 miles. Therefore, options **D, E,** and **F** make the donation at least \$65. --- ### **Example 6** **Part A** **Question:** A dump truck weighs 11.25 tons when empty. A conveyor belt pours sand into the truck at a constant rate of \( \frac{1}{4} \) ton per minute until it is full. Let \( t \) represent the elapsed time in minutes. Let \( w \) represent the weight of the truck after \( t \) minutes. Write a linear equation for \( w \) in terms of \( t \). **Answer:** \[ w = 11.25 + \frac{1}{4}t \] **Explanation:** - **11.25 tons** is the initial weight of the empty truck. - **\( \frac{1}{4}t \)** represents the additional weight from sand poured in over time \( t \). So, the total weight \( w \) after \( t \) minutes is the sum of the empty truck's weight and the sand added: \[ w = 11.25 + \frac{1}{4}t \] --- **Part B** **Question:** The dump truck from Part A weighs 18 tons when filled. At the same time the dump truck is being filled, an identical dump truck filled to capacity is being emptied at a rate of \( \frac{1}{8} \) ton per minute. How much sand is in each dump truck when the trucks are the same weight? **Answer:** **22** **Explanation:** Let's denote: - **Truck A**: Being filled at \( \frac{1}{4} \) ton per minute. - **Truck B**: Being emptied at \( \frac{1}{8} \) ton per minute. - Both trucks start at 11.25 tons (empty). - Truck A reaches 18 tons when full. First, find the time it takes for Truck A to reach 18 tons: \[ 18 = 11.25 + \frac{1}{4}t \] \[ 6.75 = \frac{1}{4}t \] \[ t = 6.75 \times 4 = 27 \text{ minutes} \] During these 27 minutes, Truck B is being emptied: Initial weight of Truck B: 11.25 + sand = 18 tons (since it's identical and filled to capacity). But it's being emptied at \( \frac{1}{8} \) ton per minute. Amount of sand removed from Truck B in 27 minutes: \[ \frac{1}{8} \times 27 = 3.375 \text{ tons} \] Remaining sand in Truck B: \[ 18 - 3.375 = 14.625 \text{ tons} \] But the question mentions "how much sand is in each dump truck when the trucks are the same weight" with an answer of 22, which may suggest there’s a misunderstanding. If interpreting "same weight" beyond both being filled, re-examining: Alternatively, if both trucks are to have the same weight after the operations: Let \( w \) be the final weight of each truck when they're the same. For Truck A: \[ w = 11.25 + \frac{1}{4}t \] For Truck B: \[ w = 18 - \frac{1}{8}t \] Set them equal: \[ 11.25 + \frac{1}{4}t = 18 - \frac{1}{8}t \] Multiply all terms by 8 to eliminate fractions: \[ 90 + 2t = 144 - t \] \[ 3t = 54 \] \[ t = 18 \text{ minutes} \] Now, find \( w \): \[ w = 11.25 + \frac{1}{4} \times 18 = 11.25 + 4.5 = 15.75 \text{ tons} \] Total sand in each truck: - **Truck A**: 15.75 - 11.25 = 4.5 tons - **Truck B**: 18 - 15.75 = 2.25 tons - **Total sand**: 4.5 + 2.25 = 6.75 tons Given that the user provided "22" possibly as an intended answer, it might relate to a different interpretation or possible typo. Based on standard interpretation and the calculations above, the combined sand is **6.75 tons** when both trucks are the same weight. --- If there are any additional details or clarifications needed for Example 6 Part B, please let me know!