Question

The length of the side of a square sheet of metal is increasing at the rate of \[4cm/\sec \] . The rate at which the area of the sheet is increasing when the length of its side is \[8cm\] , is
1) \[16c{{m}^{2}}/\sec \]
2) \[8c{{m}^{2}}/\sec \]
3) \[64c{{m}^{2}}/\sec \]
4) None of these

Answer 1

Hint: In this type of question you need to know that the formula to find the area of square is \[area\_of\_square=sid{{e}^{2}}\] and then we need to differentiate the area we will find to know the rate of it, and with the help of given rate of side of square we will finally get our required rate.

Complete step-by-step solution:
Here we can easily observe that the question is asking to use the concept of application of derivatives therefore first we need to know the use of derivatives and why application of derivatives comes into picture.
There are various applications of derivatives not only in maths and real life but also in other fields like science, engineering, physics, etc. In previous classes, you must have learned to find the derivative of different functions, like, trigonometric functions, implicit functions, logarithm functions, etc. In this section, you will learn the use of derivatives with respect to mathematical concepts and in real-life scenarios.
Derivatives have various important applications in Mathematics such as:
Rate of Change of a Quantity
Increasing and Decreasing Functions
Tangent and Normal to a Curve
Minimum and Maximum Values
Newton’s Method
Linear Approximations
Applications of Derivatives in Math:
The derivative is defined as the rate of change of one quantity with respect to another. In terms of functions, the rate of change of function is defined as \[\dfrac{dy}{dx}=y'\] .
The concept of derivatives has been used in small scale and large scale. The concept of derivatives is used in many ways such as change of temperature or rate of change of shapes and sizes of an object depending on the conditions etc.
So, coming to the question,
Since it is given that the length of a side of the square is \[8cm\] therefore we can write it as,
\[s=8cm\]
Where \[s\] is the side of the square.
Now it is also given that the rate of side of a square is increasing at the rate of \[4cm/\sec \] therefore we can write it as,
\[\dfrac{ds}{dt}=4cm/\sec \]
Since the rate is given positive according to the question therefore we write the rate with a positive sign.
Now we will find the area of square by using the formula,
\[area\_of\_square=sid{{e}^{2}}\]
Let the area of the square be \[A\] .
Therefore we have,
\[\Rightarrow A={{s}^{2}}\]
Now differentiating both sides with respect to time we get,
We will use the differentiation rule which is,
If function is \[f(x)={{x}^{n}}\]
Then the derivative will be,
\[f'(x)=n{{x}^{n-1}}\]
Therefore differentiating area we will get,
\[\Rightarrow \dfrac{dA}{dt}=2s\dfrac{ds}{dt}\]
Now we will put the value of side and the rate with which side is increasing that is value of \[s=8cm\] and \[\dfrac{ds}{dt}=4cm/\sec \]
\[\Rightarrow \dfrac{dA}{dt}=2(8)(4)\]
\[\Rightarrow \dfrac{dA}{dt}=64c{{m}^{2}}/\sec \]
The rate with which area of the square is increasing is \[\dfrac{dA}{dt}=64c{{m}^{2}}/\sec \] .
Therefore our final answer is option \[(3)\].

Note:The modern development of calculus is usually credited to Isaac Newton \[(1643-1727)\] and Gottfried Wilhelm Leibniz \[(1646-1716)\] , who provided independent and unified approaches to differentiation and derivatives which we all know helped in various other discoveries.