the length and breadth of a rectangular park are in the ratio 8:5. a path 1.5 meter wide running all around the outside of the park has an are of 633m find the length and breadth of the park
Answers
★ Given:
- the park is a rectangle
- ratio of the length to breadth = 8:5
- width of the path = 1.5 m
- area of the park = 633 m²
★ To find:
- the length and the breadth
★ Solution:
Let us take the length and breadth of the rectangular park to be 8x and 5x where x is a common factor:
- length = 8x
- breadth = 5x
Let the rectangular park be ABCD and the outer path MNOP (as in the attachment).
→ Let us find the area of rectangle ABCD.
The area of the rectangle ABCD = l × b
= 8x × 5x
= (40x) m²
→ Let us find the dimensions of the whole rectangle MNOP (including the path).
- length = 8x + (2 × width of the path) = 8x + (2 × 1.5) = 8x + 3
- breadth = 5x + (2 × width of the path) = 5x + (2 × 1.5) = 5x + 3
→ Let us find the area of the whole rectangle MNOP, including the path.
The area of the rectangle MNOP = l × b
= 8x + 3 × 5x + 3
= (40x² + 39x + 9) m²
→ We are given that the path has an area of 633 m². We also know that:
Area of the path = area of whole rectangle MNOP - area of rectangle ABCD
Substituting the values we know into this equation:
⇒ 633 = (40x² + 39x + 9) m² - (40x) m²
- Simplifying,
⇒ 633 = 39x + 9
- Transposing 9,
⇒ 633 - 9 = 39x
⇒ 39x = 624
- Transposing 39,
⇒ x = 624/39
⇒ x = 16
So,
- length = 8x = 8 × 16 = 128 m
- breadth = 5x = 5 × 16 = 80 m
Therefore, the length is 128 m and the breadth is 80 m.
