# The side of a square sheet is increasing at the rate of 4cm per minute. At what rate is the area increasing when the side is 8cm long?

Last updated date: 21st Mar 2023

•

Total views: 304.8k

•

Views today: 7.82k

Answer

Verified

304.8k+ views

Hint – In this question the rate of increase of the side of a square sheet is given to us. We need to find the rate change of the area. Use the basic formula for area of square in terms of side and differentiate both sides with respect to time. This will help to get the solution.

Complete step-by-step answer:

Let A be the area of the square.

We need to find the rate of change of area w.r.t. time when the side of the square is 8 cm.

Let the side of square be x.

$ \Rightarrow x = 8{\text{ cm}}{\text{.}}$

Therefore we have to find $\dfrac{{dA}}{{dt}}$ at $x = 8{\text{ cm}}{\text{.}}$

As we know that the area (A) of the square is side square.

$ \Rightarrow A = {x^2}$ Square meter.

Differentiate it w.r.t. time

$ \Rightarrow \dfrac{{dA}}{{dt}} = 2x\dfrac{{dx}}{{dt}}$……………….. (1)

Now it is given that the side of the square is increasing at the rate of 4 cm/min.

$ \Rightarrow \dfrac{{dx}}{{dt}} = + 4$ Cm/min. (plus sign indicates it is increasing)

So, put this value in equation (1) we have,

$ \Rightarrow \dfrac{{dA}}{{dt}} = 2\left( 8 \right)\left( 4 \right) = + 64$ Cm/min.

So the rate at which the area is increasing is 64 cm/min.

Note – Whenever we face such types of problems the key concept is to make sure that whether the side is increasing or decreasing with respect to time, as it directly affects the signs of rate change that we need to take into consideration. This concept along with the gist of the basic formula for area will help you get on the right track to get the answer.

Complete step-by-step answer:

Let A be the area of the square.

We need to find the rate of change of area w.r.t. time when the side of the square is 8 cm.

Let the side of square be x.

$ \Rightarrow x = 8{\text{ cm}}{\text{.}}$

Therefore we have to find $\dfrac{{dA}}{{dt}}$ at $x = 8{\text{ cm}}{\text{.}}$

As we know that the area (A) of the square is side square.

$ \Rightarrow A = {x^2}$ Square meter.

Differentiate it w.r.t. time

$ \Rightarrow \dfrac{{dA}}{{dt}} = 2x\dfrac{{dx}}{{dt}}$……………….. (1)

Now it is given that the side of the square is increasing at the rate of 4 cm/min.

$ \Rightarrow \dfrac{{dx}}{{dt}} = + 4$ Cm/min. (plus sign indicates it is increasing)

So, put this value in equation (1) we have,

$ \Rightarrow \dfrac{{dA}}{{dt}} = 2\left( 8 \right)\left( 4 \right) = + 64$ Cm/min.

So the rate at which the area is increasing is 64 cm/min.

Note – Whenever we face such types of problems the key concept is to make sure that whether the side is increasing or decreasing with respect to time, as it directly affects the signs of rate change that we need to take into consideration. This concept along with the gist of the basic formula for area will help you get on the right track to get the answer.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?