Sabrina is building a rectangular raised flower bed. The boards on the two shorter sides are 6 inches thick, and the boards on the two longer sides are 4 inches thick. Sabrina wants the outer length of the bed four times it's height and the outer width to be 2 times it's height. She also wants the boards to rise 4 inches above the soil in the bed. What should the outer dimensions of the beds be if she wants it to hold 3136 cubic inches of soil? (How on earth do I start this?)
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To start solving this problem, let's break it down into smaller steps:
Step 1: Determine the dimensions of the inner rectangle
Since Sabrina wants the boards to rise 4 inches above the soil in the bed, the inner rectangle would have dimensions that are 8 inches less in length and width compared to the outer dimensions. This ensures that the soil occupies the desired volume without including the volume occupied by the boards.
Let's denote the length of the inner rectangle as L and the width as W.
Step 2: Find the height of the bed
Since Sabrina wants the outer length of the bed to be four times its height and the outer width to be two times its height, we can express the height as H.
The outer length would then be 4H, and the outer width would be 2H.
Step 3: Calculate the inner dimensions
Using the previously determined information, we can find the dimensions of the inner rectangle.
The length of the inner rectangle would be L = 4H - 2 * 6 inches (for the two boards on the shorter sides).
Similarly, the width of the inner rectangle would be W = 2H - 2 * 4 inches (for the two boards on the longer sides).
Step 4: Calculate the volume of the inner bed
The volume of a rectangular box can be calculated by multiplying its length, width, and height. In this case, the length is L, the width is W, and the height is H.
The volume of the inner bed can be expressed as V_inner = L * W * H.
Step 5: Set up and solve the equation
Now, we can set up an equation to find the desired dimensions. We know that the volume of the inner bed should be 3136 cubic inches.
So, our equation becomes:
V_inner = L * W * H = 3136 cubic inches
Substitute the expressions for L and W from Step 3:
(4H - 2 * 6 inches) * (2H - 2 * 4 inches) * H = 3136 cubic inches
Simplify and solve the equation to find the value of H. Once H is known, you can find L and W using the formulas mentioned earlier.
Note: The thickness of the boards is given in inches, so make sure to use consistent units throughout the calculations.
I hope this explanation helps you get started on finding the outer dimensions of Sabrina's raised flower bed!