Question

The cross section of a canal is in the shape of a trapezium. If the canal is 12 m wide at the top and 8 m wide at the bottom and area of its cross section is 84 ${{m}^{2}}$. Determine its depth.

a. 8.4 m
b. 7.4 m
c. 4.8 m
d. 6.4 m

Answer 1

Hint: In order to solve this question, we should know that the area of a trapezium is calculated as the sum of two triangles formed by any one diagonal. So, we will equate the area of the trapezium to the sum of the areas of the triangles ADC and ABC and then we will find the value of h.

Complete step-by-step answer:
In this question, we have been asked to find the depth of the canal, whose cross section is in the shape of a trapezium and its area is 84 ${{m}^{2}}$ and the cross section is 12 m wide at the top and 8 m wide at the bottom. To solve this question, let us first draw the figure.

From the figure, we can see that the area of the trapezium can be calculated as the sum of the areas of the triangles, ADC and ABC. And we know that the area of a triangle is given by, $\dfrac{1}{2}$ (base) (height). So, we can write,
Area of trapezium = Area of $\Delta ADC$ + Area of $\Delta ABC$
$84=\dfrac{1}{2}\times \left( DC \right)\times \left( DE \right)+\dfrac{1}{2}\times \left( AB \right)\times \left( DE \right)$
Now, we will put the values of (AB), (DC) and (DE). So, we get,
$84=\dfrac{1}{2}\times 8\times h+\dfrac{1}{2}\times 12\times h$
We can further write it as,
$\begin{align}
  & 84=4h+6h \\
 & 10h=84 \\
 & h=\dfrac{84}{10}=8.4m \\
\end{align}$
Hence, we can say that the depth of the canal is 8.4 m.
Therefore, option (a) is the correct answer.

Note: While solving this question, we might get confused with the term depth of canal. Depth of canal implies the distance between the top and the bottom of the canal. We can also solve this question by using the formula of trapezium, that is, $\dfrac{1}{2}$ (sum of parallel sides) (height).