To determine the area that the goats will be allowed to roam in, we need to calculate the area of the larger fenced area (for the goats) and subtract the area of the smaller fenced garden.
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Calculate the area of the goat fence: \[ \text{Area of goat fence} = \text{length} \times \text{width} = 12 , \text{yd} \times 15 , \text{yd} = 180 , \text{yd}^2 \]
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Calculate the area of the garden: \[ \text{Area of garden} = \text{length} \times \text{width} = 3 , \text{yd} \times 6 , \text{yd} = 18 , \text{yd}^2 \]
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Calculate the area where the goats can roam: \[ \text{Area for goats} = \text{Area of goat fence} - \text{Area of garden} = 180 , \text{yd}^2 - 18 , \text{yd}^2 = 162 , \text{yd}^2 \]
Thus, the fenced area that the goats will be allowed to roam in is 162 yd.².