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A circle has a radius of 7 cm. Calculate for the circle, the diameter, circumference, and area.

Mathematics

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27. In this question take π\pi to be 227\frac{22}{7}.
A circle has a radius of 7 cm.

a) Calculate for the circle, the diameter.

  1. Use the formula for diameter.

    The diameter (dd) of a circle is twice its radius (rr).

    d=2rd = 2r
  2. Substitute the given radius and calculate.

    d=2×7 cmd = 2 \times 7 \text{ cm} d=14 cmd = 14 \text{ cm}

    The diameter is 14 cm\boxed{14 \text{ cm}}.

    b) Calculate for the circle, the circumference.

  3. Use the formula for circumference.

    The circumference (CC) of a circle is given by C=2πrC = 2\pi r or C=πdC = \pi d.
    Using C=2πrC = 2\pi r:

    C=2×227×7 cmC = 2 \times \frac{22}{7} \times 7 \text{ cm}
  4. Calculate the circumference.

    C=2×22 cmC = 2 \times 22 \text{ cm} C=44 cmC = 44 \text{ cm}

    The circumference is 44 cm\boxed{44 \text{ cm}}.

    c) Calculate for the circle, the area.

  5. Use the formula for area.

    The area (AA) of a circle is given by A=πr2A = \pi r^2.

    A=227×(7 cm)2A = \frac{22}{7} \times (7 \text{ cm})^2
  6. Calculate the area.

    A=227×49 cm2A = \frac{22}{7} \times 49 \text{ cm}^2 A=22×7 cm2A = 22 \times 7 \text{ cm}^2 A=154 cm2A = 154 \text{ cm}^2

    The area is 154 cm2\boxed{154 \text{ cm}^2}.

    28. The table below shows the frequency table of the distribution of time taken by learners in a class to complete a Mathematical question.

    Time (minutes) (xx)345678
    Frequency (ff)47141022

    a) Calculate the mean time taken by pupils to complete the question.

  7. Calculate the sum of (frequency ×\times time) for all categories.

    ∑fx=(3×4)+(4×7)+(5×14)+(6×10)+(7×2)+(8×2)\sum fx = (3 \times 4) + (4 \times 7) + (5 \times 14) + (6 \times 10) + (7 \times 2) + (8 \times 2) ∑fx=12+28+70+60+14+16\sum fx = 12 + 28 + 70 + 60 + 14 + 16 ∑fx=200\sum fx = 200
  8. Calculate the total frequency (total number of pupils).

    ∑f=4+7+14+10+2+2\sum f = 4 + 7 + 14 + 10 + 2 + 2 ∑f=39\sum f = 39
  9. Calculate the mean time.

    Mean=∑fx∑f\text{Mean} = \frac{\sum fx}{\sum f} Mean=20039\text{Mean} = \frac{200}{39} Mean≈5.1282\text{Mean} \approx 5.1282

    The mean time taken is 5.13 minutes\boxed{5.13 \text{ minutes}} (to 3 significant figures).

    b) Pupils who took more than 5 minutes were given another question. A pupil was randomly chosen from the class. Find the probability that the pupil was given another question.

  10. Identify the number of pupils who took more than 5 minutes.

    These are pupils who took 6, 7, or 8 minutes.
    Number of pupils = (Frequency for 6 minutes) + (Frequency for 7 minutes) + (Frequency for 8 minutes)

    Number of pupils=10+2+2=14\text{Number of pupils} = 10 + 2 + 2 = 14
  11. State the total number of pupils.

    From part (a), the total number of pupils is ∑f=39\sum f = 39.

  12. Calculate the probability.

    Probability=Number of pupils who took more than 5 minutesTotal number of pupils\text{Probability} = \frac{\text{Number of pupils who took more than 5 minutes}}{\text{Total number of pupils}} Probability=1439\text{Probability} = \frac{14}{39}

    The probability that the pupil was given another question is 1439\boxed{\frac{14}{39}}.

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