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Solution

UPSC Prelims

Mathematics

Theme:

Speed, Time and Distance / गति, समय और दूरी

Q68 2014, Speed, Time and Distance

Location of B is north of A and location of C is east of A. The distances AB and AC are 5 km and 12 km respectively. The shortest distance (in km) between the locations B and C is

B की स्थिति A के उत्तर में है और C की स्थिति A के पूर्व में है। दूरियाँ AB और AC क्रमशः 5 कि.मी. और 12 कि.मी. हैं। B और C स्थानों के बीच की लघुतम दूरी (कि.मी. में) क्या है?

Explanation

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The correct answer is b: 13.
To find the shortest distance between locations B and C, we can use the Pythagorean theorem. The locations A, B, and C form a right triangle with A as the right angle. Here's how we can solve it step by step:
     1. Identify the Triangle:
            ○ Location A is the reference point.
            ○ Location B is north of A, so AB is the vertical side of the triangle.
            ○ Location C is east of A, so AC is the horizontal side of the triangle.
     2. Given Distances:
            ○ The distance AB = 5 km.
            ○ The distance AC = 12 km.
     3. Apply the Pythagorean Theorem:
            ○ The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
            ○ In this case, BC is the hypotenuse.

B C^{2} = A B^{2} + A C^{2}

$BC^2 = AB^2 + AC^2$

B C^{2} = 5^{2} + 12^{2}

$BC^2 = 5^2 + 12^2$

B C^{2} = 25 + 144

$BC^2 = 25 + 144$

B C^{2} = 169

$BC^2 = 169$
     4. Calculate BC:
            ○ To find BC, take the square root of 169.

B C = \sqrt{169} = 13

$BC = \sqrt{169} = 13$
Therefore, the shortest distance between locations B and C is 13 km.

The correct answer is b: 13. To find the shortest distance between locations B and C, we can use the Pythagorean theorem. The locations A, B, and C form a right triangle with A as the right angle. Here's how we can solve it step by step:      1. Identify the Triangle:             ○ Location A is the reference point.             ○ Location B is north of A, so AB is the vertical side of the triangle.             ○ Location C is east of A, so AC is the horizontal side of the triangle.      2. Given Distances:             ○ The distance AB = 5 km.             ○ The distance AC = 12 km.      3. Apply the Pythagorean Theorem:             ○ The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.             ○ In this case, BC is the hypotenuse.         <mi>B</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>+</mo><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup></math>" role="presentation" style="font-size: 121%; text-align: center; position: relative;">BC2=AB2+AC2<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mi>A</mi><msup><mi>B</mi><mn>2</mn></msup><mo>+</mo><mi>A</mi><msup><mi>C</mi><mn>2</mn></msup></math><script type="math/tex; mode=display" id="MathJax-Element-1">         BC^2 = AB^2 + AC^2         </script>         <mi>B</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><msup><mn>5</mn><mn>2</mn></msup><mo>+</mo><msup><mn>12</mn><mn>2</mn></msup></math>" role="presentation" style="font-size: 121%; text-align: center; position: relative;">BC2=52+122<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><msup><mn>5</mn><mn>2</mn></msup><mo>+</mo><msup><mn>12</mn><mn>2</mn></msup></math><script type="math/tex; mode=display" id="MathJax-Element-2">         BC^2 = 5^2 + 12^2         </script>         <mi>B</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mn>25</mn><mo>+</mo><mn>144</mn></math>" role="presentation" style="font-size: 121%; text-align: center; position: relative;">BC2=25+144<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mn>25</mn><mo>+</mo><mn>144</mn></math><script type="math/tex; mode=display" id="MathJax-Element-3">         BC^2 = 25 + 144         </script>         <mi>B</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mn>169</mn></math>" role="presentation" style="font-size: 121%; text-align: center; position: relative;">BC2=169<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>B</mi><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mn>169</mn></math><script type="math/tex; mode=display" id="MathJax-Element-4">         BC^2 = 169         </script>      4. Calculate BC:             ○ To find BC, take the square root of 169.         <mi>B</mi><mi>C</mi><mo>=</mo><msqrt><mn>169</mn></msqrt><mo>=</mo><mn>13</mn></math>" role="presentation" style="font-size: 121%; text-align: center; position: relative;">BC=√169=13<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>B</mi><mi>C</mi><mo>=</mo><msqrt><mn>169</mn></msqrt><mo>=</mo><mn>13</mn></math><script type="math/tex; mode=display" id="MathJax-Element-5">         BC = \sqrt{169} = 13         </script>      Therefore, the shortest distance between locations B and C is 13 km.

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Q 68. 2014, Speed, Time and Distance

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