Question 36 - CBSE Class 10 Sample Paper for 2021 Boards - Maths Standard

Transcript

Question 36 A motorboat covers a distance of 16km upstream and 24km downstream in 6 hours. In the same time it covers a distance of 12 km upstream and 36km downstream. Find the speed of the boat in still water and that of the stream.Let the Speed of Boat in still water be x km/hr & let the Speed of Stream be y km/hr Now, Speed Downstream = x + y Speed Upstream = x – y Given that A boat goes 16 km upstream and 24 km downstream in 6 hours Time taken to go 16 km upstream + Time taken to go 24 km downstream (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 16 𝑘𝑚)/(𝑆𝑝𝑒𝑒𝑑 𝑢𝑝𝑠𝑡𝑟𝑒𝑎𝑚) + (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 24 𝑘𝑚)/(𝑆𝑝𝑒𝑒𝑑 𝑑𝑜𝑤𝑛𝑠𝑡𝑟𝑒𝑎𝑚) = 6 𝟏𝟔/(𝒙 − 𝒚) + 𝟐𝟒/(𝒙 + 𝒚) = 6 We know that, Speed = (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 )/𝑇𝑖𝑚𝑒 Time = (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 )/𝑆𝑝𝑒𝑒𝑑 Similarly, A boat goes 12 km upstream and 36 km downstream in 6 hours Time taken to go 12 km upstream + Time taken to go 36 km downstream (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 12 𝑘𝑚)/(𝑆𝑝𝑒𝑒𝑑 𝑢𝑝𝑠𝑡𝑟𝑒𝑎𝑚) + (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 36 𝑘𝑚)/(𝑆𝑝𝑒𝑒𝑑 𝑑𝑜𝑤𝑛𝑠𝑡𝑟𝑒𝑎𝑚) = 6 𝟏𝟐/(𝒙 − 𝒚) + 𝟑𝟔/(𝒙 + 𝒚) = 6 We know that, Speed = (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 )/𝑇𝑖𝑚𝑒 Time = (𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 )/𝑆𝑝𝑒𝑒𝑑 Our equations are 16(1/(𝑥 − 𝑦))+24(1/(𝑥 + 𝑦))=6 …(1) 12(1/(𝑥 − 𝑦))+36(1/(𝑥 + 𝑦))=6 …(2) So, our equations become Solving 16u + 24v = 6 …(3) 12u + 36v = 6 …(4) Let 1/(𝑥 − 𝑦) = u 1/(𝑥 + 𝑦) = v 12u + 36v = 6 From (3) 16u + 24v = 6 16u = 6 – 24v u = (6 − 24𝑣)/16 Putting value of u in (4) 12u + 36v = 6 12((6 − 24𝑣)/16)+36𝑣=6 3((6 − 24𝑣)/4)+36𝑣=6 Multiplying both sides by 4 4 × 3((6 − 24𝑣)/4)+"4 ×" 36𝑣="4 ×" 6 3(6 – 24v) + 144𝑣= 24 18 – 72v + 144v = 24 – 72v + 144v = 24 – 18 72v = 6 v = 𝟔/𝟕𝟐 v = 𝟏/𝟏𝟐 Putting v = 1/12 in equation (3) 12u + 36v = 6 12u + 36(1/12) = 6 12u + 3 = 6 12u = 6 − 3 12u = 3 u = 3/12 u = 𝟏/𝟒 So, u = 1/4 & v = 1/12 But we need to find x & y We know that u = 𝟏/(𝒙 − 𝒚) 1/4 = 1/(𝑥 − 𝑦) x – y = 4 v = 𝟏/(𝒙 + 𝒚) 1/12 = 1/(𝑥 + 𝑦) x + y = 12 So, our equations become x – y = 4 …(6) x + y = 12 …(7) Adding (6) and (7) (x – y) + (x + y) = 4 + 12 2x = 16 x = 16/2 x = 8 Putting x = 8 in (7) x + y = 12 8 + y = 12 y = 12 – 8 y = 4 So, x = 8, y = 4 is the solution of the given equation Hence Speed of boat in still water = x = 8 km/hr Speed of stream = y = 4 km/hr y = 12 – 8 y = 4 So, x = 8, y = 4 is the solution of the given equation Hence Speed of boat in still water = x = 8 km/hr Speed of stream = y = 4 km/hr