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Mathematics

Case Study IV

A motorboat whose speed in still water is 24km/hr, takes 1 hour more to go 32 km upstream than to return downstream to the same spot . Based on this information answer the following questions

1. What is the speed of the motorboat in going upstream, if speed of the stream is x km/hr?

  1. (x − 24) km/hr
  2. (24 − x) km/hr
  3. (x + 24) km/hr
  4. none of these

2. The quadratic equation which represents the given information is:

  1. x2 − 64x − 576 = 0
  2. x2 + 64x + 576 = 0
  3. x2 + 64x − 576 = 0
  4. x2 − 64x + 576 = 0

3. Speed of the motorboat in going downstream is:

  1. 16 km/hr
  2. 8 km/hr
  3. 28 km/hr
  4. 32 km/hr

4. Time taken by the motorboat to go 272 km downstream is:

  1. 8128\dfrac{1}{2} hours

  2. 17 hours

  3. 121212\dfrac{1}{2} hours

  4. 6126\dfrac{1}{2} hours

5. Time taken by the motorboat to go 80 km upstream and then to return back to the same spot is:

  1. 5125\dfrac{1}{2} hours

  2. 6126\dfrac{1}{2} hours

  3. 7127\dfrac{1}{2} hours

  4. 8128\dfrac{1}{2} hours

Quadratic Equations

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Answer

1. Given,

The speed of the motorboat in still water is 24 km/hr

The speed of the stream be x km/hr

The speed of the motorboat in going upstream = boat speed in still water − stream speed = (24 - x) km/hr

Hence, option (2) is the correct option.

2. The speed of the motorboat in going downstream = boat speed in still water + stream speed = (24 + x) km/hr

Distance to be covered by motorboat = 32km

Time = DistanceSpeed\dfrac{\text{Distance}}{\text{Speed}}

In first case:

Time taken by motorboat in going upstream = 3224−x\dfrac{32}{24 - x}

In second case:

Time taken by motorboat in going downstream = 3224+x\dfrac{32}{24 + x}

The difference in time taken between first case and second case = 1 hour more

⇒3224−x−3224+x=1⇒32(24+x)−32(24−x)(24−x)(24+x)=1⇒768+32x−768+32x242−x2=1⇒64x576−x2=1⇒64x=576−x2⇒x2+64x−576=0.\Rightarrow \dfrac{32}{24 - x} - \dfrac{32}{24 + x} = 1 \\[1em] \Rightarrow \dfrac{32(24 + x) - 32(24 - x)}{(24 - x)(24 + x)}= 1 \\[1em] \Rightarrow \dfrac{768 + 32x - 768 + 32x}{24^2 - x^2}= 1 \\[1em] \Rightarrow \dfrac{64x}{576 - x^2}= 1 \\[1em] \Rightarrow 64x = 576 - x^2 \\[1em] \Rightarrow x^2 + 64x - 576 = 0.

Hence, option (3) is the correct option.

3. Solving equation from question 2,

⇒ x2 + 64x - 576 = 0

⇒ x2 - 8x + 72x - 576 = 0

⇒ x(x - 8) + 72(x - 8) = 0

⇒ (x + 72)(x - 8) = 0

⇒ (x + 72) = 0 or (x - 8) = 0     [Using zero-product rule]

⇒ x = -72 or x = 8

⇒ x = 8km/hr [speed of the stream cannot be negative]

Speed of the motorboat downstream = 24 + x = 24 + 8 = 32 km/hr.

Hence, option (4) is the correct option.

4. The downstream speed of motorboat is 32 km/hr (from Question 3).

The time taken by the motorboat to go 272 km downstream = 27232=172\dfrac{272}{32} = \dfrac{17}{2} = 8.5 hours

Hence, option (1) is the correct option.

5. Speed of the motorboat upstream = 16 km/hr

The time taken by the motorboat to go 80 km upstream = 8016\dfrac{80}{16} = 5 hours

The time taken by the motorboat to return 80 km downstreamstream = 8032\dfrac{80}{32} = 2.5 hours

Total time = 5 + 2.5 = 7.5 hours

Hence, option (3) is the correct option.

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