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Maths-

General

Easy

Question

Two cars A and B are moving uniformly on two straight roads at right angles to one another at 40 and 20 km/hr respectively. A passes the intersection of the road when B has still to move 50 km to reach it. The shortest distance between the two cars and the time when they are closest are

$20 \sqrt{5} k m, 30$ minutes
20 km, 10 minutes
20 km, 20 minutes
None of these

The correct answer is: $20 square root of 5 k m comma 30$ minutes

Let the two be closest after time t, their initial position are O for A and B such that OB = 50. If they are now at $A to the power of ´ end exponent$ and $B to the power of ´ end exponent$ respectively, then
$O A to the power of ´ end exponent equals 40 t comma O B to the power of ´ end exponent equals O B minus B B to the power of ´ end exponent equals 50 minus 20 t$
$A to the power of ´ end exponent rightwards arrow over short leftwards arrow B to the power of ´ end exponent equals left square bracket left parenthesis 40 t right parenthesis to the power of 2 end exponent plus left parenthesis 50 minus 20 t right parenthesis to the power of 2 end exponent right square bracket to the power of 1 divided by 2 end exponent$
If $A to the power of ´ end exponent B to the power of ´ end exponent$ is least then
$Z equals A to the power of ´ end exponent B to the power of ´ 2 end exponent equals left parenthesis 40 t right parenthesis to the power of 2 end exponent plus left parenthesis 50 minus 20 t right parenthesis to the power of 2 end exponent$ is also least.
\ $fraction numerator d z over denominator d t end fraction equals 2 left parenthesis 40 t right parenthesis 40 plus 2 left parenthesis 50 minus 20 t right parenthesis left parenthesis negative 20 right parenthesis equals 0$
Þ $80 t minus 50 plus 20 t equals 0$
Þ $t equals fraction numerator 1 over denominator 2 end fraction h r equals 30$ minutes
Clearly $fraction numerator d to the power of 2 end exponent z over denominator d t to the power of 2 end exponent end fraction equals 100 equals plus v e$
\ z is least
Putting $t equals fraction numerator 1 over denominator 2 end fraction$ we get z = 20²+ 40²= 20²×5 = $A to the power of ´ end exponent B to the power of ´ 2 end exponent$
\ $A ´ B ´ equals 20 square root of 5 equals 44.8 m$

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