It looks like the text you provided has significant typographical and encoding errors, making it hard to understand. However, I'll do my best to reconstruct the problem and solve it step by step based on what can be inferred: --- ### **Restated Problem (based on context):** A patient's total cholesterol (Tlh) and good cholesterol (Glil) are measured in mg/dL, as a function of weeks after January 1, 2016. - On January 1, the total cholesterol is **116 mg/dL**. - Good cholesterol is **34 mg/dL**. - The doctor prefers that the patient's good cholesterol increases by **1/5 mg/dL per week**. - The doctor prescribes a diet starting January 1, which increases good cholesterol at this rate. - **Question:** After how many weeks will the good cholesterol be at least 15% of the total cholesterol? Round your answer to two decimal places. --- ### **Step-by-Step Solution:** #### **Step 1: Write Equations for Each Quantity** Let \( t \) = weeks after January 1. - **Total cholesterol**: \( T(t) = 116 \) (remains constant, as no change is mentioned) - **Good cholesterol**: \( G(t) = 34 + \frac{1}{5}t \) #### **Step 2: Set Up the Inequality** We want \( G(t) \geq 0.15 \times T(t) \): \[ 34 + \frac{1}{5}t \geq 0.15 \times 116 \] \[ 0.15 \times 116 = 17.4 \] So, \[ 34 + \frac{1}{5}t \geq 17.4 \] But since 34 is already greater than 17.4, the good cholesterol is already at least 15% of the total cholesterol on January 1. #### **Double-check the question:** If the question is when good cholesterol is **at least 41 mg/dL** (the possible intended meaning based on "increase to -41 meld"), let's solve for when \( G(t) \geq 41 \): \[ 34 + \frac{1}{5}t \geq 41 \] \[ \frac{1}{5}t \geq 7 \] \[ t \geq 35 \] #### **Final Step: Box the Answer** **If the question is when good cholesterol is at least 15% of total cholesterol:** - **Answer:** 0 weeks (already satisfied). **If the question is when good cholesterol reaches 41 mg/dL:** - **Answer:** \( t = 35 \) weeks. --- ## **Final Answers:** - **If asking about 15%:** \( \boxed{0.00 \text{ weeks}} \) - **If asking about reaching 41 mg/dL:** \( \boxed{35.00 \text{ weeks}} \) --- If you can clarify the question further, I can update the solution accordingly!