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A man travels 30 km in 30 minutes. Calculate his speed in km/h and m/s. Also solve these simultaneous equations: 2a+b=8 and 3a−b=7

Mathematics

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2 parts
a

60 km/h60 \text{ km/h}

b

16.67 m/s16.67 \text{ m/s}

a=3,b=2a=3, b=2

Step-by-step solution

10 steps

28)

Part a· 3 steps

  1. Identify the given distance and time.

    Distance (DD) = 30 km30 \text{ km}
    Time (TT) = 30 minutes30 \text{ minutes}

  2. Convert the time to hours.

    Since there are 60 minutes60 \text{ minutes} in an hour,

    T=3060 hours=0.5 hoursT = \frac{30}{60} \text{ hours} = 0.5 \text{ hours}
  3. Calculate the speed in kilometers per hour.

    Speed (SS) is calculated as S=DTS = \frac{D}{T}.

    S=30 km0.5 hoursS = \frac{30 \text{ km}}{0.5 \text{ hours}} S=60 km/hS = 60 \text{ km/h}

    His speed in kilometers per hour is 60 km/h\boxed{60 \text{ km/h}}.

Part b

16.67 m/s16.67 \text{ m/s}

  1. Convert the speed from kilometers per hour to meters per second.

    We know that 1 km=1000 m1 \text{ km} = 1000 \text{ m} and 1 hour=3600 seconds1 \text{ hour} = 3600 \text{ seconds}.

    S=60kmh×1000 m1 km×1 h3600 sS = 60 \frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}}
  2. Perform the multiplication and division.

    S=60×10003600msS = \frac{60 \times 1000}{3600} \frac{\text{m}}{\text{s}} S=600003600msS = \frac{60000}{3600} \frac{\text{m}}{\text{s}} S=60036msS = \frac{600}{36} \frac{\text{m}}{\text{s}} S=1006msS = \frac{100}{6} \frac{\text{m}}{\text{s}} S=16.666...msS = 16.666... \frac{\text{m}}{\text{s}}

    Rounding to two decimal places, the speed is 16.67 m/s\boxed{16.67 \text{ m/s}}.

    29)

  3. Write down the given simultaneous equations.

    Equation (1): 2a+b=82a + b = 8
    Equation (2): 3a−b=73a - b = 7

  4. Use the elimination method.

    Add Equation (1) and Equation (2) to eliminate bb.

    (2a+b)+(3a−b)=8+7(2a + b) + (3a - b) = 8 + 7 2a+3a+b−b=152a + 3a + b - b = 15 5a=155a = 15
  5. Solve for aa.

    a=155a = \frac{15}{5} a=3a = 3
  6. Substitute the value of aa into either Equation (1) or Equation (2) to solve for bb.

    Using Equation (1):

    2(3)+b=82(3) + b = 8 6+b=86 + b = 8
  7. Solve for bb.

    b=8−6b = 8 - 6 b=2b = 2

    The solution is a=3,b=2\boxed{a=3, b=2}.

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