To find the third measurement needed for a rectangular prism's net where two of the rectangles have an area of \(4 , \text{ft} \times 3 , \text{ft}\), we first calculate the area of the two rectangles.
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Calculate the area of one rectangle: \[ \text{Area of one rectangle} = 4 , \text{ft} \times 3 , \text{ft} = 12 , \text{ft}^2 \]
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The total area of the two rectangles: \[ \text{Total area of two rectangles} = 2 \times 12 , \text{ft}^2 = 24 , \text{ft}^2 \]
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The net of a rectangular prism consists of 6 rectangles (2 for each dimension). Since we already have the areas for two rectangles, we need to find the area of the remaining 4 rectangles such that the total area of the net adds up to 52 square feet.
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Calculate the remaining area needed: \[ \text{Remaining area} = 52 , \text{ft}^2 - 24 , \text{ft}^2 = 28 , \text{ft}^2 \]
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The remaining 4 rectangles must account for an area of \(28 , \text{ft}^2\). Since these will be the other dimensions of the rectangular prism, we can let the dimensions of the next rectangle (which we need to find) be \(x , \text{ft} \times y , \text{ft}\).
Given that a rectangular prism has pairs of faces with equal dimensions, we assume that the other two unique rectangles are \(y , \text{ft} \times 4 , \text{ft}\) (which would have the same area as the \(3 , \text{ft} \times x , \text{ft}\)) and that both pairs should contribute equally.
Assuming one of those dimensions is the same: If we let one pair of rectangles measure \(4 , \text{ft}\) as one side, and we need the rest (\(y\)), we can distribute: \[ 2 \times (4 \times y) + 2 \times (3 \times y) = 28 \] Solving, \[ 8y + 6y = 28 \] \[ 14y = 28 \] \[ y = 2 \]
So, the third measurement is \(2\) feet.
In conclusion, with a rectangular prism net consisting of two rectangles measuring \(4 , \text{ft} \times 3 , \text{ft}\), the third necessary measurement would be \(2 , \text{ft}\), completing the arrangement that sums the total area up to \(52 , \text{ft}^2\).