Question

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Answer 1

36
`√(3)`+ 72

The length of the rectangle = diameter of 2 circles = 4 × 3 = 12 mm

The width of the rectangle = 2 radii + distance d between radii ( lower to upper )

Using the right triangle formed by joining the 3 centres with legs 3 and d and hypotenuse 6

Using Pythagoras' identity, then

d² + 3² = 6²

d² + 9 = 36 ( subtract 9 from both sides )

d² = 27 ( take the square root of both sides )

d = = 3

Thus width of rectangle = 3 + 3 + 3 = 6 + 3

Hence

area = length × width

= 12 × (6 + 3 ) = 36 + 72 mm ²

Answer 2

36
`√(3)` + 72

Explanation:

The length of the rectangle = diameter of 2 circles = 4 × 3 = 12 mm

The width of the rectangle = 2 radii + distance d between radii ( lower to upper )

Using the right triangle formed by joining the 3 centres with legs 3 and d and hypotenuse 6

Using Pythagoras' identity, then

d² + 3² = 6²

d² + 9 = 36 ( subtract 9 from both sides )

d² = 27 ( take the square root of both sides )

d =
`√(27)` = 3
`√(3)`

Thus width of rectangle = 3 + 3
`√(3)` + 3 = 6 + 3
`√(3)`

Hence

area = length × width

= 12 × (6 + 3
`√(3)` ) = 36
`√(3)` + 72 mm ²