Question

The base of a right prism is a Trapezium whose lengths of two parallel sides are 10 cm and 6 cm and the distance between them is 5 cm. If the height of the prism is 8 cm, its volume is:
\[\left( \text{a} \right)\text{ }320\text{ }c{{m}^{3}}\]
\[\left( \text{b} \right)\text{ }300.5\text{ }c{{m}^{3}}\]
\[\left( \text{c} \right)\text{ }310\text{ }c{{m}^{3}}\]
\[\left( \text{d} \right)\text{ }300\text{ }c{{m}^{3}}\]

Answer 1

Hint: To solve the above question, we will first find out what a trapezium is and then we will find the area of the trapezium base with the help of the data given in the question. We will be using the formula for the area of the trapezium as \[\dfrac{1}{2}\times \left( \text{sum of the parallel sides} \right)\times \left( \text{distance between them} \right).\]

Complete step by step solution:
Before we solve the given question, we must know what trapezium is. A trapezium is a quadrilateral having four unequal sides out of which two opposite sides are parallel sides and the other two opposite sides are non-parallel. For a better understanding of the question, a rough sketch of the trapezium is drawn.

Now, we have to calculate the area of this trapezium. For the calculation of the area of a trapezium, we must know the lengths of the parallel sides and the distance between them. The area of the trapezium is calculated by the formula given below,
\[\text{Area of trapezium}=\dfrac{1}{2}\times \left( \text{sum of the parallel sides} \right)\times \left( \text{distance between them} \right)\]
In our case, the parallel sides are AB and CD and the distance between them is represented by the dotted line AE. Thus, we have,
\[\text{Area of trapezium ABCD}=\dfrac{1}{2}\times \left( AB+CD \right)\times AE\]
\[\Rightarrow \text{Area of trapezium ABCD}=\dfrac{1}{2}\times \left( 10+6 \right)\times 5c{{m}^{2}}\]
\[\Rightarrow \text{Area of trapezium ABCD}=\dfrac{1}{2}\times 16\times 5c{{m}^{2}}\]
\[\Rightarrow \text{Area of trapezium ABCD}=40c{{m}^{2}}\]
Now, to find the volume of the prism, we will multiply the area obtained by the height of the prism. Thus, we have,
The volume of prism = Area of trapezium \[\times \] height
In our case, the height of the prism is 8 cm. So, we will get,
\[\Rightarrow \text{Volume of prism}=40c{{m}^{2}}\times 8cm\]
\[\Rightarrow \text{Volume of prism}=320c{{m}^{3}}\]
Hence, option (a) is the correct option.

Note: The area of the trapezium can also be calculated as shown below.

Here, we can see that the trapezium is divided into two triangles and one rectangle. Thus, the area of the trapezium will be given by,
Area of trapezium ABCD = Area of triangle ADE + Area of ABFE + Area of triangle BFC
\[\Rightarrow \text{Required area}=\left( \dfrac{1}{2}\times AE\times DE \right)+\left( AE\times AB \right)+\left( \dfrac{1}{2}\times BF\times FC \right)\]
\[\Rightarrow \text{Required area}=\left( \dfrac{1}{2}\times 5\times y \right)+\left( 5\times 6 \right)+\left( \dfrac{1}{2}\times 5\times x \right)\]
\[\Rightarrow \text{Required area}=\left( \dfrac{5y}{2} \right)+30+\left( \dfrac{5x}{2} \right)\]
\[\Rightarrow \text{Required area}=\left[ \dfrac{5}{2}\left( x+y \right)+30 \right]\text{c}{{\text{m}}^{2}}\]
We know that, x + y + 6 = 10
Therefore, x + y = 4
Therefore, we will get,
\[\Rightarrow \text{Required area}=\left[ \dfrac{5}{2}\left( 4 \right)+30 \right]\text{c}{{\text{m}}^{2}}\]
\[\Rightarrow \text{Required area}=40\text{c}{{\text{m}}^{2}}\]