Question

Calculate the base of the rectangle \[\overline {AD} ?\]

Shaded Area \[ = 200,000c{m^2}\]

Answer 1

Hint: In the question we have given a figure and we have to calculate the base of the rectangle \[\overline {AD} \] which is given as \[xcm\] . Here we see that the shaded area represents the difference between the area of the rectangle and the area of the triangle. So, first of all we will find the area of the rectangle and the area of the triangle. Then we will find the difference between them which will be the area of the shaded region. After that we will equate it to the given shaded area and simplify for \[x\] . Hence, we will get the required answer.

Complete step by step answer:
We have given a rectangle
one of its side, \[AB = 800cm\]
and another side, \[AD = xcm\]
and we have to find the base of the rectangle means we have to find the value of \[x\]
Now we know that
Area of the rectangle is given by the product of the side lengths
i.e., \[Are{a_{\left( {rect} \right)}} = l \times b\]
Here in the figure,
\[l = xcm,{\text{ }}b = 800cm\]
Therefore, we get
\[Are{a_{\left( {rect} \right)}} = x \times 800\]
\[Are{a_{\left( {rect} \right)}} = 800x{\text{ }}c{m^2}{\text{ }} - - - \left( i \right)\]
Now we know that
Area of the triangle is given by half the product of the base and the height
i.e., \[Are{a_{\left( {tri} \right)}} = \dfrac{1}{2} \times b \times h\]
In the figure, the height of the triangle matches the breadth of the rectangle and the base of the triangle is equal to the length of the rectangle because we have two triangles and both have the same height and we need to add the area so we added base here.
Therefore, we get
\[Are{a_{\left( {tri} \right)}} = \dfrac{1}{2} \times p \times 800 + \dfrac{1}{2} \times (x-p) \times 800\]
\[Are{a_{\left( {tri} \right)}} = \dfrac{1}{2} \times x \times 800\]
\[ \Rightarrow Are{a_{\left( {tri} \right)}} = 400x{\text{ }}c{m^2}{\text{ }} - - - \left( {ii} \right)\]
Now the area of shaded region is given by the difference between the area of the rectangle and the area of the triangle
Using equation \[\left( i \right)\] and \[\left( {ii} \right)\] we get
Area of shaded region \[ = 800x - 400x\]
Now it is given that,
Area of shaded region is \[200,000c{m^2}\]
Therefore, we get
\[200,000 = 800x - 400x\]
\[ \Rightarrow 200,000 = 400x\]
On simplifying we get
\[ \Rightarrow x = \dfrac{{200,000}}{{400}} = 500\]
\[ \Rightarrow x = 500{\text{ }}cm\]
Hence, we get the base of the rectangle \[\overline {AD} \] as \[500{\text{ }}cm\]

Note:
In this question students get confused with the base of the triangle. But using the given information we can assume one more variable as the base of one triangle. And we should know that it will be eliminated in calculation.