A stable-owner has four horses. He ties these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length to graze in his farm. Based on this, an... A stable-owner has four horses. He ties these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length to graze in his farm. Based on this, answer the following: (i) Find the area of the square shaped grass field. (ii) Find the area of the total field in which these horses can graze.
Understand the Problem
The question involves finding the area of a square-shaped field and the total area in which the horses can graze based on the information provided about the horses' ropes and field dimensions.
Answer
The area of the square-shaped field is \( 400 \, \text{m}^2 \) and the total area in which the horses can graze is approximately \( 615.76 \, \text{m}^2 \).
Answer for screen readers
The area of the square-shaped field is ( 400 , \text{m}^2 ).
The total area in which the horses can graze is approximately ( 615.76 , \text{m}^2 ).
Steps to Solve
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Find the Area of the Square Field
The area of a square is calculated using the formula: $$ \text{Area} = \text{side}^2 $$ Given that the side length of the square field is 20 m, $$ \text{Area} = 20^2 = 400 , \text{m}^2 $$
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Understanding Grazing Area of Each Horse
Each horse is tied with a 7 m long rope. This means each horse can graze in a circular area. The radius ( r ) of the grazing circle is 7 m.
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Calculate the Area for One Horse's Grazing Circle
The area of a circle is given by the formula: $$ \text{Area} = \pi r^2 $$ Substituting ( r = 7 , \text{m} ): $$ \text{Area} = \pi \times 7^2 \approx \pi \times 49 \approx 153.94 , \text{m}^2 $$
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Total Grazing Area for All Horses
Since there are 4 horses, the total grazing area is: $$ \text{Total Grazing Area} = 4 \times \text{Area of one horse's grazing} $$ $$ \text{Total Grazing Area} \approx 4 \times 153.94 \approx 615.76 , \text{m}^2 $$
The area of the square-shaped field is ( 400 , \text{m}^2 ).
The total area in which the horses can graze is approximately ( 615.76 , \text{m}^2 ).
More Information
- The area of the square field represents the total grazing limits for the horses.
- The circular grazing areas indicate the space each horse requires based on the length of its rope.
- The calculations provide insight into how much space each horse has, and the total area helps in managing the grazing field efficiently.
Tips
- Forgetting to square the side length when calculating the area of a square.
- Confusing the area of a circle with its circumference; ensuring the use of area formulas is crucial.
- Not multiplying the grazing area for all horses can lead to an inaccurate total area.
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