Activating Prior Knowledge and Modeling with Mathematics. Mr. Lee wants to park his ice cream cart on Main Street so that he is equidistant from the entrances of the amusement park and the zoo. The entrances are 40 feet apart. Mr. Lee decides to move his cart off the street to the area between Main Street and the entrances to the amusement park and the zoo. How can you find where Mr. Lee should park if he must be 30 feet from both entrances? (**Hint: You must use Pythagorean Theorem at some point)

To determine where Mr. Lee should park his ice cream cart so that it is exactly 30 feet from both entrances of the amusement park and the zoo, we can use the **Pythagorean Theorem**. Here's a step-by-step guide to solving the problem: ### **1. Visualize the Problem** - **Main Street** is the straight road where both entrances are located. - The two entrances (let's call them **Entrance A** and **Entrance B**) are **40 feet apart** along Main Street. - Mr. Lee wants his cart to be **equidistant (30 feet)** from both entrances. - To achieve this, he plans to park **off Main Street**, forming a right triangle with distances from each entrance. ### **2. Set Up a Coordinate System** - **Place Entrance A at the origin (0, 0)** on a coordinate plane. - **Place Entrance B at (40, 0)** since it's 40 feet apart along the street. - Let the **parking spot** be at point **P (x, y)**, where **y** is the perpendicular distance from Main Street. ### **3. Apply the Pythagorean Theorem** Since Mr. Lee's cart needs to be 30 feet from both entrances, the distances PA and PB must both equal 30 feet. - **Distance from Entrance A to P:** \[ \sqrt{(x - 0)^2 + (y - 0)^2} = 30 \] Squaring both sides: \[ x^2 + y^2 = 900 \quad \text{(Equation 1)} \] - **Distance from Entrance B to P:** \[ \sqrt{(x - 40)^2 + (y - 0)^2} = 30 \] Squaring both sides: \[ (x - 40)^2 + y^2 = 900 \quad \text{(Equation 2)} \] ### **4. Solve the System of Equations** Subtract **Equation 1** from **Equation 2** to eliminate \( y^2 \): \[ (x - 40)^2 + y^2 - (x^2 + y^2) = 900 - 900 \] Simplify: \[ x^2 - 80x + 1600 + y^2 - x^2 - y^2 = 0 \] \[ -80x + 1600 = 0 \] \[ 80x = 1600 \] \[ x = 20 \] So, the x-coordinate of the parking spot is **20 feet** from Entrance A along Main Street. ### **5. Find the Perpendicular Distance (y)** Substitute \( x = 20 \) back into **Equation 1** to find \( y \): \[ 20^2 + y^2 = 900 \] \[ 400 + y^2 = 900 \] \[ y^2 = 500 \] \[ y = \sqrt{500} = 10\sqrt{5} \approx 22.36 \text{ feet} \] ### **6. Determine the Parking Location** The coordinates of the parking spot **P** are \( (20, 10\sqrt{5}) \). This means: - **Along Main Street:** 20 feet from Entrance A (and also 20 feet from Entrance B since they're 40 feet apart). - **Off Main Street:** Approximately **22.36 feet** perpendicular from the street. ### **7. Conclusion** Mr. Lee should park his ice cream cart **20 feet along Main Street from either entrance** and **approximately 22.36 feet away from the street** into the area between the entrances. This ensures that the cart is exactly **30 feet** from both the amusement park and zoo entrances. ### **Visual Representation** ``` Entrance A (0,0) ---------------- Main Street ---------------- Entrance B (40,0) \ \ \ P (20, 22.36) ``` - **PA and PB** both represent 30-foot distances from the entrances to the parking spot **P**. - The right triangle **PA-P-Main Street** utilizes the Pythagorean Theorem to solve for the perpendicular distance. By following these steps, you can accurately determine the optimal parking location for Mr. Lee's ice cream cart.