I need help what am I doing wrong
The railroad fill shown in the figure has sloping sides which rise vertically 0.5 feet for each foot horizontally. The top of the fill ABCD is horizontal and the ends are vertical. The depth of the fill at each of the points A, B,C, D is indicated in the figure. Find the cost of making this fill at 75 cents per cubic yard.
This is my solution:
The slope was given which is m=0.5
Using the slope I acquired the horizontal distance of the triangles y=mx
The dimensions of the lower base (front section):
The area of this is:
The dimensions of the upper base (back section):
The area of this is:
The dimensions of the middle section:
The area of the middle section:
The computed perpendicular height between b and B:
Using the formula of a volume of a prismatoid:
The answer in the book is: $858.33
Where did I go wrong? Thanks!
It looks like you have made a mistake by forgetting that the top surface $ABCD$ is horizontal. In all your diagrams, you have assumed that the bottom surface is horizontal. Therefore, you need to redraw the diagrams as described below. By the way, we describe the method for the front section only.
You can start by drawing the segment $AD$ horizontal. Then draw a line joining $A_1$ and $D_1$. Finally, draw two lines with the given slope 0.5 through $A$ and $D$ to meet thee extended line $A_1D_1$ at $P$ and $Q$ respectively. The quadrilateral $APQD$ is the correct front section. Finding its area takes some doing.
Drop three perpendiculars $PU$, $A_1V$, and $QW$ as shown in the diagram. Let $PU=a$, $QW=c$, $A_1U=b$, and $D_1W=d$. Now, we can derive the following equations. $$\dfrac{b}{a}=\dfrac{d}{c}=\dfrac{1}{6}\quad\rightarrow\quad a=6b\quad\text{and}\quad c=6d$$ $$\dfrac{6-b}{a}=\dfrac{8+d}{c}=0.5\quad\rightarrow\quad 12-2b=a \quad\text{and}\quad 16+2d=c$$
Using these equations, we can determine the values of $a$, $b$, $c$, and $d$. $$a=9, \qquad b=1.5,\qquad c=24, \qquad d=4$$
We can calculate the area of the front section ($A_{\text{F}}$) using these values. $$A_{\text{F}}=\dfrac{6\times a}{2}+12\times 8+\dfrac{12\times 2}{2} +\dfrac{8\times c}{2} = \dfrac{6\times 9}{2}+12\times 8+\dfrac{12\times 2}{2} +\dfrac{8\times 24}{2}=207 \text{ ft}^2$$
Once you determine the areas the other two sections, you can calculate the volume of the prismatoid using the equation which you have already used in your text. However, your calculation of the perpendicular distance between the font and the rear sections, i.e., $H$, is wrong. Since $ABCD$ is horizontal, $H=100$.
$\underline{\text{Edit}}$
We give below the areas of the rear section ($A_{\text{R}})$ and the mid-section ($A_{\text{M}})$ of the prismatoid for you to check your calculations. $$A_{\text{R}}=423 \text{ ft}^2$$ $$A_{\text{M}}=306 \text{ ft}^2$$