14. There is a rectangular park of length 84 m and foreadth 28 m . Three roads each of width 3 m pass through the field such that two roads are parahlel to the breadth and the third is parallel to the length. Find the area of the field: (a) covered by the roads. (b) not covered by the roads.

To solve the problem, we need to find the area of the park covered by the roads and the area not covered by the roads. Let's break it down step by step. ### Step 1: Calculate the area of the park The area \( A \) of a rectangle is given by the formula: \[ A = \text{length} \times \text{breadth} \] Given: - Length \( l = 84 \, \text{m} \) - Breadth \( b = 28 \, \text{m} \) Calculating the area of the park: \[ A_{\text{park}} = 84 \times 28 \] ### Step 2: Calculate the area covered by the roads There are three roads: - Two roads are parallel to the breadth (width = 3 m each). - One road is parallel to the length (width = 3 m). #### Area of the two roads parallel to the breadth: Each road has a width of 3 m and runs the entire length of the park: \[ A_{\text{roads\_breadth}} = 2 \times (\text{width} \times \text{length}) = 2 \times (3 \times 84) \] #### Area of the one road parallel to the length: This road has a width of 3 m and runs the entire breadth of the park: \[ A_{\text{road\_length}} = 3 \times 28 \] #### Total area covered by the roads: The total area covered by the roads is the sum of the areas of the two roads parallel to the breadth and the one road parallel to the length: \[ A_{\text{roads}} = A_{\text{roads\_breadth}} + A_{\text{road\_length}} \] ### Step 3: Calculate the area not covered by the roads The area not covered by the roads can be found by subtracting the area covered by the roads from the total area of the park: \[ A_{\text{not\_covered}} = A_{\text{park}} - A_{\text{roads}} \] Now, let's perform the calculations. 1. Calculate \( A_{\text{park}} \). 2. Calculate \( A_{\text{roads}} \). 3. Calculate \( A_{\text{not\_covered}} \). Let's compute these values. Calculate the value by following steps: - step0: Calculate: \(84\times 28\) - step1: Multiply the numbers: \(2352\) Calculate or simplify the expression \( 2 * (3 * 84) + (3 * 28) \). Calculate the value by following steps: - step0: Calculate: \(2\left(3\times 84\right)+\left(3\times 28\right)\) - step1: Remove the parentheses: \(2\times 3\times 84+\left(3\times 28\right)\) - step2: Multiply the numbers: \(2\times 3\times 84+84\) - step3: Multiply the terms: \(504+84\) - step4: Add the numbers: \(588\) ### Step 1: Area of the Park The area of the park is calculated as: \[ A_{\text{park}} = 84 \times 28 = 2352 \, \text{m}^2 \] ### Step 2: Area Covered by the Roads The total area covered by the roads is: \[ A_{\text{roads}} = 588 \, \text{m}^2 \] ### Step 3: Area Not Covered by the Roads Now, we can find the area not covered by the roads: \[ A_{\text{not\_covered}} = A_{\text{park}} - A_{\text{roads}} = 2352 - 588 \] Let's calculate this final value. Calculate the value by following steps: - step0: Calculate: \(2352-588\) - step1: Subtract the numbers: \(1764\) ### Final Results 1. **Area covered by the roads**: \[ A_{\text{roads}} = 588 \, \text{m}^2 \] 2. **Area not covered by the roads**: \[ A_{\text{not\_covered}} = 1764 \, \text{m}^2 \] Thus, the area of the field covered by the roads is \( 588 \, \text{m}^2 \) and the area not covered by the roads is \( 1764 \, \text{m}^2 \).