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A contractor is to build a new structure at a farm for seed storage. It will have a rectangular base, which is 2020 feet wide by 3535 feet long, with walls of thickness tt feet. The structure is 9090 feet tall. Excluding the volume of the walls, which of the following functions best models the volume, VV, in cubic feet, inside the structure?

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Q. A contractor is to build a new structure at a farm for seed storage. It will have a rectangular base, which is 2020 feet wide by 3535 feet long, with walls of thickness tt feet. The structure is 9090 feet tall. Excluding the volume of the walls, which of the following functions best models the volume, VV, in cubic feet, inside the structure?
  1. Identify Dimensions and Variable: Identify the dimensions of the structure and the variable representing the thickness of the walls.\newlineThe structure has a rectangular base with a width of 2020 feet and a length of 3535 feet. The walls have a thickness of tt feet. The height of the structure is 9090 feet.
  2. Calculate Internal Dimensions: Calculate the internal dimensions of the structure by subtracting the thickness of the walls from the external dimensions.\newlineThe internal width will be the external width minus two wall thicknesses (since there are two walls contributing to the width), and the internal length will be the external length minus two wall thicknesses (since there are two walls contributing to the length).\newlineInternal width = 20−2t20 - 2t\newlineInternal length = 35−2t35 - 2t
  3. Write Volume Function: Write the function for the volume inside the structure.\newlineThe volume, VV, of a rectangular prism is given by the formula V=length×width×heightV = \text{length} \times \text{width} \times \text{height}. Using the internal dimensions, the function for the volume inside the structure is:\newlineV(t)=(20−2t)×(35−2t)×90V(t) = (20 - 2t) \times (35 - 2t) \times 90
  4. Simplify Volume Function: Simplify the function for the volume.\newlineV(t)=(700−40t−70t+4t2)×90V(t) = (700 - 40t - 70t + 4t^2) \times 90\newlineV(t)=(700−110t+4t2)×90V(t) = (700 - 110t + 4t^2) \times 90\newlineV(t)=63000−9900t+360t2V(t) = 63000 - 9900t + 360t^2
  5. Check Function Accuracy: Check the function to ensure it represents the volume inside the structure correctly.\newlineWhen t=0t = 0 (no wall thickness), the volume should be the external dimensions multiplied together:\newlineV(0)=20×35×90=63000V(0) = 20 \times 35 \times 90 = 63000 cubic feet\newlineThis matches the constant term in our function, indicating that the function correctly models the volume inside the structure for t=0t = 0.

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