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12 The diagram shows a rectangle and a part from which a quarter-circle was cut. The length of the rectangle = 12 cm and the width = 7 cm. (a) Show that the area of the quarter-circle is 38.5 cm Calculate the area of the shaded part.

Mathematics

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Answers

2 parts
a

38.5 cm2\text{38.5 cm}^2

b

45.5 cm2\text{45.5 cm}^2

Step-by-step solution

5 steps

Part a· 3 steps

  1. Identify the radius of the quarter-circle.

    From the diagram, the radius of the quarter-circle is equal to the width of the rectangle, which is 7 cm7 \, \text{cm}. So, r=7 cmr = 7 \, \text{cm}.

  2. Use the formula for the area of a quarter-circle.

    The area of a full circle is A=πr2A = \pi r^2. For a quarter-circle, the area is 14πr2\frac{1}{4} \pi r^2.
    Given π=227\pi = \frac{22}{7}.

  3. Substitute the values and calculate the area.

    Area of quarter-circle=14×227×(7 cm)2\text{Area of quarter-circle} = \frac{1}{4} \times \frac{22}{7} \times (7 \, \text{cm})^2 Area of quarter-circle=14×227×49 cm2\text{Area of quarter-circle} = \frac{1}{4} \times \frac{22}{7} \times 49 \, \text{cm}^2 Area of quarter-circle=14×22×7 cm2\text{Area of quarter-circle} = \frac{1}{4} \times 22 \times 7 \, \text{cm}^2 Area of quarter-circle=1544 cm2\text{Area of quarter-circle} = \frac{154}{4} \, \text{cm}^2 Area of quarter-circle=38.5 cm2\text{Area of quarter-circle} = 38.5 \, \text{cm}^2

    The area of the quarter-circle is indeed 38.5 cm2\boxed{\text{38.5 cm}^2}.

Part b

45.5 cm2\text{45.5 cm}^2

  1. Calculate the area of the rectangle.

    The length of the rectangle is 12 cm12 \, \text{cm} and the width is 7 cm7 \, \text{cm}.

    Area of rectangle=length×width\text{Area of rectangle} = \text{length} \times \text{width} Area of rectangle=12 cm×7 cm\text{Area of rectangle} = 12 \, \text{cm} \times 7 \, \text{cm} Area of rectangle=84 cm2\text{Area of rectangle} = 84 \, \text{cm}^2

  2. Calculate the area of the shaded part.

    The shaded part is the area of the rectangle minus the area of the quarter-circle.

    Area of shaded part=Area of rectangle−Area of quarter-circle\text{Area of shaded part} = \text{Area of rectangle} - \text{Area of quarter-circle} Area of shaded part=84 cm2−38.5 cm2\text{Area of shaded part} = 84 \, \text{cm}^2 - 38.5 \, \text{cm}^2 Area of shaded part=45.5 cm2\text{Area of shaded part} = 45.5 \, \text{cm}^2

    The area of the shaded part is 45.5 cm2\boxed{\text{45.5 cm}^2}.

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