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A running track has a rectangular middle section and two semi-circular ends. The length of the straight sides is 110 m and the diameter of the semi-circles is 70 m. Calculate the perimeter and area of

Mathematics

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Step-by-step solution

3 steps
  1. Identify the dimensions of the running track.

    The running track consists of a rectangular middle section and two semi-circular ends.
    The length of the straight sides (length of the rectangle) is L=110L = 110 m.
    The diameter of each semi-circle is D=70D = 70 m.
    The radius of each semi-circle is r=D2=702=35r = \frac{D}{2} = \frac{70}{2} = 35 m.
    The width of the rectangular section is equal to the diameter of the semi-circles, W=70W = 70 m.
    Use π=227\pi = \frac{22}{7}.

  2. Calculate the distance round the field (perimeter).

    The distance round the field is the sum of the lengths of the two straight sides and the circumference of the two semi-circular ends. The two semi-circular ends together form a full circle.
    Perimeter P=2×L+Circumference of a full circleP = 2 \times L + \text{Circumference of a full circle}
    The circumference of a full circle is πD\pi D.

    P=2L+πDP = 2L + \pi D

    Substitute the given values:

    P=2(110 m)+(227)(70 m)P = 2(110 \text{ m}) + \left(\frac{22}{7}\right)(70 \text{ m}) P=220 m+(22×10) mP = 220 \text{ m} + (22 \times 10) \text{ m} P=220 m+220 mP = 220 \text{ m} + 220 \text{ m} P=440 mP = 440 \text{ m}

    The distance round the field is 440 m\boxed{440 \text{ m}}.

  3. Calculate the area of the field.

    The area of the field is the sum of the area of the rectangular middle section and the area of the two semi-circular ends. The two semi-circular ends together form a full circle.
    Area A=Area of rectangle+Area of a full circleA = \text{Area of rectangle} + \text{Area of a full circle}
    The area of the rectangle is L×WL \times W.
    The area of a full circle is πr2\pi r^2.

    A=(L×W)+πr2A = (L \times W) + \pi r^2

    Substitute the given values:

    A=(110 m×70 m)+(227)(35 m)2A = (110 \text{ m} \times 70 \text{ m}) + \left(\frac{22}{7}\right)(35 \text{ m})^2 A=7700 m2+(227)(1225 m2)A = 7700 \text{ m}^2 + \left(\frac{22}{7}\right)(1225 \text{ m}^2) A=7700 m2+(22×175) m2A = 7700 \text{ m}^2 + (22 \times 175) \text{ m}^2 A=7700 m2+3850 m2A = 7700 \text{ m}^2 + 3850 \text{ m}^2 A=11550 m2A = 11550 \text{ m}^2

    The area of the field is 11550 m2\boxed{11550 \text{ m}^2}.

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