Question

The side of an equilateral triangle is increasing at the rate of $2cm/\sec$.At what rate is its area increasing when the side of the triangle is $20cm$?

Answer 1

Here, firstly we will use the formula of the area of the equilateral triangle. Then we will differentiate the area with respect to the time. Then we will put the value of rate of increase of the side of the triangle and the value of the side of the triangle in it. Then we will get the rate with which the area increases.

Complete step by step solution:
It is given that the triangle is the equilateral triangle. Equilateral triangle is the triangle which has its all sides equal.
Area of the equilateral triangle is $A = \dfrac{{\sqrt 3 }}{4}{a^2}$ where, $a$ is the side of the triangle.
Now, we will find the rate with which the area is increasing by differentiating the area with respect to the time. Therefore, we get
$\dfrac{{dA}}{{dt}} = \dfrac{{\sqrt 3 }}{4}\left( {2a} \right)\dfrac{{da}}{{dt}}$
It is given that the rate of side increasing is $2cm/\sec$ i.e. $\dfrac{{da}}{{dt}} = 2cm/\sec$ and also it is given that the side of the triangle is $a = 20cm$. Therefore, by putting this value in the equation we get
$\dfrac{{dA}}{{dt}} = \dfrac{{\sqrt 3 }}{4}\left( {2 \times 20} \right)2 = \dfrac{{\sqrt 3 }}{4} \times 40 \times 2 = \sqrt 3 \times 10 \times 2 = 20\sqrt 3 c{m^2}/\sec$

Hence, the area of the triangle is increasing with the rate of $20\sqrt 3 c{m^2}/\sec$.

Note:
Here we should note that the given triangle is an equilateral triangle not the right triangle. In the equilateral triangle all the sides are equal and also all the angles of the triangle are equal. We should also know the basic formula of the area of the triangle.
Area of the triangle $= \dfrac{1}{2} \times Base \times Height$
We should know that whenever the rate of change is asked in the question we have to differentiate with respect to the time. So, we don’t have to differentiate it with respect to the side of the triangle.