Let's break down and solve the **"finding the explosion"** problem step by step. --- ### **Problem Restatement** **Three microphones** recorded an explosion. - The sound reached the microphone at **site A** **2.5 seconds** later than it reached the microphone at **site B**, and **1.5 seconds** after site C. - Sites A and B are **0.75 mile apart** and sites A and C are **1.3 miles apart**. - **Estimate the location** of the explosion **within 10 feet**, giving **distance and direction from site A**. - **Assume sound moves at 1100 ft/sec**. --- ### **Step 1: Assign Variables and Coordinates** Let: - The location of the explosion be at point \( (x, y) \). - Site **A** at \( (0, 0) \). - Site **B** at \( (d_1, 0) \), with \( d_1 = 0.75 \) miles. - Site **C** at \( (x_C, y_C) \), with \( AC = 1.3 \) miles. - For simplicity, let's place C along the y-axis: \( (0, d_2) \), with \( d_2 = 1.3 \) miles. **Convert miles to feet:** - 1 mile = 5280 feet. - \( d_1 = 0.75 \times 5280 = 3960 \) ft. - \( d_2 = 1.3 \times 5280 = 6864 \) ft. So: - **A**: \( (0, 0) \) - **B**: \( (3960, 0) \) - **C**: \( (0, 6864) \) --- ### **Step 2: Set Up Equations for Sound Arrival Times** Let the explosion occur at \( (x, y) \). Let the time for sound to reach A be \( t_A \), B be \( t_B \), and C be \( t_C \). \[ t_A = \frac{\sqrt{x^2 + y^2}}{1100} \] \[ t_B = \frac{\sqrt{(x - 3960)^2 + y^2}}{1100} \] \[ t_C = \frac{\sqrt{x^2 + (y - 6864)^2}}{1100} \] Given: - \( t_A = t_B + 2.5 \) - \( t_A = t_C + 1.5 \) --- ### **Step 3: Write System of Equations** #### **Equation 1 (A vs B):** \[ \frac{\sqrt{x^2 + y^2}}{1100} = \frac{\sqrt{(x-3960)^2 + y^2}}{1100} + 2.5 \] \[ \sqrt{x^2 + y^2} = \sqrt{(x-3960)^2 + y^2} + 1100 \times 2.5 \] \[ \sqrt{x^2 + y^2} = \sqrt{(x - 3960)^2 + y^2} + 2750 \] #### **Equation 2 (A vs C):** \[ \frac{\sqrt{x^2 + y^2}}{1100} = \frac{\sqrt{x^2 + (y-6864)^2}}{1100} + 1.5 \] \[ \sqrt{x^2 + y^2} = \sqrt{x^2 + (y - 6864)^2} + 1100 \times 1.5 \] \[ \sqrt{x^2 + y^2} = \sqrt{x^2 + (y - 6864)^2} + 1650 \] --- ### **Step 4: Simplify and Solve (Algebraically)** #### **Equation 1** Let \( S = \sqrt{x^2 + y^2} \): \[ S = \sqrt{(x-3960)^2 + y^2} + 2750 \] \[ S - 2750 = \sqrt{(x-3960)^2 + y^2} \] Square both sides: \[ (S - 2750)^2 = (x-3960)^2 + y^2 \] \[ S^2 - 2 \times 2750 \times S + 2750^2 = (x-3960)^2 + y^2 \] But \( S^2 = x^2 + y^2 \), so: \[ x^2 + y^2 - 5500S + 7562500 = x^2 - 2 \times 3960 \times x + 3960^2 + y^2 \] \[ x^2 + y^2 - 5500S + 7562500 = x^2 + y^2 - 7920x + 15681600 \] \[ -5500S + 7562500 = -7920x + 15681600 \] \[ 5500S + 7920x = 15681600 - 7562500 \] \[ 5500S + 7920x = 8129100 \] --- #### **Equation 2** Let \( S = \sqrt{x^2 + y^2} \): \[ S = \sqrt{x^2 + (y-6864)^2} + 1650 \] \[ S - 1650 = \sqrt{x^2 + (y-6864)^2} \] Square both sides: \[ (S - 1650)^2 = x^2 + (y-6864)^2 \] \[ S^2 - 2 \times 1650 \times S + 1650^2 = x^2 + y^2 - 2 \times 6864 \times y + 6864^2 \] \[ x^2 + y^2 - 3300S + 2722500 = x^2 + y^2 - 13728y + 47114304 \] \[ -3300S + 2722500 = -13728y + 47114304 \] \[ 3300S + 13728y = 47114304 - 2722500 \] \[ 3300S + 13728y = 44391804 \] --- ### **Step 5: Substitute and Solve (Numerical Approach)** **We have:** 1. \( 5500S + 7920x = 8129100 \) 2. \( 3300S + 13728y = 44391804 \) But \( S = \sqrt{x^2 + y^2} \). Let’s solve this system numerically. #### **Guess and refine** Let's use an estimation and substitution approach. ##### **Express x and y in terms of S:** From (1): \[ 7920x = 8129100 - 5500S \implies x = \frac{8129100 - 5500S}{7920} \] From (2): \[ 13728y = 44391804 - 3300S \implies y = \frac{44391804 - 3300S}{13728} \] But \( S = \sqrt{x^2 + y^2} \). Let’s try a value for \( S \): #### **Try S = 6000** - \( x = \frac{8129100 - 5500 \times 6000}{7920} = \frac{8129100 - 33000000}{7920} = \frac{-24870900}{7920} \approx -3142 \) ft - \( y = \frac{44391804 - 3300 \times 6000}{13728} = \frac{44391804 - 19800000}{13728} = \frac{24591804}{13728} \approx 1791 \) ft Compute \( S \): \[ S^2 = (-3142)^2 + (1791)^2 = 9871764 + 3209761 = 13081525 \] \[ S = \sqrt{13081525} \approx 3616 \] But we guessed S = 6000, so let's try S = 4000. #### **Try S = 4000** - \( x = \frac{8129100 - 5500 \times 4000}{7920} = \frac{8129100 - 22000000}{7920} = \frac{-13870900}{7920} \approx -1751 \) - \( y = \frac{44391804 - 3300 \times 4000}{13728} = \frac{44391804 - 13200000}{13728} = \frac{31191804}{13728} \approx 2272 \) \[ S^2 = (-1751)^2 + (2272)^2 = 3068001 + 5161984 = 8229985 \] \[ S = \sqrt{8229985} \approx 2868 \] Still too low. Try S = 5000. #### **Try S = 5000** - \( x = \frac{8129100 - 5500 \times 5000}{7920} = \frac{8129100 - 27500000}{7920} = \frac{-19370900}{7920} \approx -2447 \) - \( y = \frac{44391804 - 3300 \times 5000}{13728} = \frac{44391804 - 16500000}{13728} = \frac{27891804}{13728} \approx 2032 \) \[ S^2 = (-2447)^2 + (2032)^2 = 5990409 + 4129024 = 10119433 \] \[ S = \sqrt{10119433} \approx 3182 \] The values for S are much lower than our guess. Try S = 3000. #### **Try S = 3000** - \( x = \frac{8129100 - 5500 \times 3000}{7920} = \frac{8129100 - 16500000}{7920} = \frac{-8370900}{7920} \approx -1057 \) - \( y = \frac{44391804 - 3300 \times 3000}{13728} = \frac{44391804 - 9900000}{13728} = \frac{34491804}{13728} \approx 2512 \) \[ S^2 = (-1057)^2 + (2512)^2 = 1117849 + 6300144 = 7417993 \] \[ S = \sqrt{7417993} \approx 2724 \] Let’s try S = 2000. #### **Try S = 2000** - \( x = \frac{8129100 - 5500 \times 2000}{7920} = \frac{8129100 - 11000000}{7920} = \frac{-2860900}{7920} \approx -361 \) - \( y = \frac{44391804 - 3300 \times 2000}{13728} = \frac{44391804 - 6600000}{13728} = \frac{37791804}{13728} \approx 2753 \) \[ S^2 = (-361)^2 + (2753)^2 = 130321 + 7578709 = 7709030 \] \[ S = \sqrt{7709030} \approx 2776 \] It looks like our guess for S needs to be between 2700 and 3200. Let’s try S = 2800. #### **Try S = 2800** - \( x = \frac{8129100 - 5500 \times 2800}{7920} = \frac{8129100 - 15400000}{7920} = \frac{-7270900}{7920} \approx -918 \) - \( y = \frac{44391804 - 3300 \times 2800}{13728} = \frac{44391804 - 9240000}{13728} = \frac{35151804}{13728} \approx 2561 \) \[ S^2 = (-918)^2 + (2561)^2 = 842724 + 6550721 = 7393445 \] \[ S = \sqrt{7393445} \approx 2718 \] #### **Try S = 3200** - \( x = \frac{8129100 - 5500 \times 3200}{7920} = \frac{8129100 - 17600000}{7920} = \frac{-9460900}{7920} \approx -1195 \) - \( y = \frac{44391804 - 3300 \times 3200}{13728} = \frac{44391804 - 10560000}{13728} = \frac{33831804}{13728} \approx 2464 \) \[ S^2 = (-1195)^2 + (2464)^2 = 1428025 + 6071296 = 7499321 \] \[ S = \sqrt{7499321} \approx 2739 \] #### **Try S = 4000** Already tried; found S ≈ 2868. So, the answer for S is around 2700–2800