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Chapter 3: Problem 68

A solid concrete dam is to be built to hold back a depth \(D\) of water. For ease of construction the walls of the dam must be planar. Your supervisor asks you to consider the following dam cross-sections: a rectangle, a right triangle with the hypotenuse in contact with the water, and a right triangle with the vertical in contact with the water. She wishes you to determine which of these would require the least amount of concrete. What will your report say? You decide to look at one more possibility: a nonright triangle, as shown. Develop and plot an expression for the cross-section area \(A\) as a function of \(a,\) and find the minimum cross-sectional area.

Short Answer

Expert verified
The dam's cross section which would require the least amount of concrete would be a right triangle either with the hypotenuse or the vertical in contact with water.

Step by step solution

01

calculating rectangular area

The rectangle's width will be equal to the depth of the water, \(D\), and its height, \(h\), will be along the water surface. So the cross-sectional area \(A\) of the dam with rectangular shape will be \(A = D \cdot h\). As it's a planar wall, the depth of the dam is equals to \(h\). Thus, \(A = D^2\).
02

calculating right triangle area with hypotenuse in contact with water

In this case, the height of the triangle will be equal to the depth of the water, \(D\), and the base along the water surface will be \(h\). The cross-sectional area \(A\) of the dam with this shape is given by \(A = 0.5 \cdot D \cdot h\). Again, since this is a planar wall dam, \(D = h\). Applying this, we get \(A=0.5 \times D^2\).
03

calculating right triangle with the vertical in contact with water

In this case, both the height and the base of the triangle will be equal to the depth of the water, \(D\). Thus, the area, \(A\) is given by \(A= 0.5 \times D \times D = 0.5 \times D^2\).
04

calculating Non-right triangle area and finding the minimum

Close observation of the diagram will give that the non-right triangle can be divided into two right triangles. Also, since the dam is a planar wall, \(a\) is equal to \(D\), and \(h\) is equal to \(D + D = 2D\). The total area of the dam will then be calculated as \(A =0.5 \times D \times D + 0.5 \times D \times 2D = 1.5 \times D^2\). To find the minimum, take the derivative of \(A\) with respect to \(D\), equate it to zero, and solve for \(D\). As the derivative \(dA/ dD = 3D\), equating this to zero gives no solution. Therefore, the minimum for the non-right triangle does not exist.
05

Comparing the Areas

From Steps 1, 2, 3, and 4, collect all the areas and compare. The rectangular-shaped cross section's area is \(D^2\), right triangle with hypotenuse in contact with the water and right triangle with the vertical in contact with the water both have \(0.5 \times D^2\), and for non-right triangle as there is no minimum, it can't be considered. So, the dam's cross section which would require the least amount of concrete would be right triangle either with hypotenuse or vertical in contact with water.

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