Question

The sides of a triangular field are 975 m , 1050 m and 1125 m. If the field is sold at the rate of Rs.1000 per hectare . Find its selling price ? $\left[1hectare\text{ = }10000m^2\right]$

Answer 1

Hint: With the given sides we can find the area of the triangular field using heron’s formula , Area of the triangle = $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$ , where $s={\displaystyle \frac{a+b+c}{2}}$ . And in order to convert it into hectares we need to divide it by 10000 and multiplying by 1000 we get the required selling price.

Complete step-by-step answer:
We are given the sides of the triangular field to be 975 m , 1050 m and 1125 m

Here the whole field is sold which means that we need to find the area of the field
Since we are given only the sides we need to use the heron’s formula to find the area of the triangle
Area of the triangle = $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$ , where $s={\displaystyle \frac{a+b+c}{2}}$
Here a = 975 m , b = 1050 m and c = 1125 m
Using this we get
$$\begin{gathered} \Rightarrow s={\displaystyle \frac{975+1050+1125}{2}} \\ \Rightarrow s={\displaystyle \frac{3150}{2}}=1575m \end{gathered}$$
Using this in the heron’s formula we get the area to be
$$\begin{gathered} \Rightarrow Area=\sqrt{1575\left(1575-975\right)\left(1575-1050\right)\left(1575-1125\right)} \\ \Rightarrow Area=\sqrt{1575\left(600\right)\left(525\right)\left(450\right)} \\ \Rightarrow Area=\sqrt{\left(945000\right)\left(236250\right)} \\ \Rightarrow Area=\sqrt{\left(223256250000\right)} \\ \Rightarrow Area=472500m^2 \end{gathered}$$
From this we get the area of the field to be $472500m^2$
Now we know that $1hectare\text{ = }10000m^2$
So now to convert into hectare let's divide by 10000
$\Rightarrow {\displaystyle \frac{472500}{10000}}=47.25hectare$
We are given that the cost per hectare is Rs.1000
Therefore the price of $47.25$ hectare is
$$\begin{gathered} \Rightarrow 47.25\ast 1000 \\ \Rightarrow Rs.47250 \end{gathered}$$
Therefore the selling price of the field is Rs.47250
Note: Here we don’t use the regular area formula as we are not given the height of the triangle
And if we need to convert hectare into $m^2$ it is enough if we multiply by 10000