# 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?

Answer

Verified

361.8k+ 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.

Last updated date: 28th Sep 2023

•

Total views: 361.8k

•

Views today: 6.61k

Recently Updated Pages

What is the Full Form of DNA and RNA

What are the Difference Between Acute and Chronic Disease

Difference Between Communicable and Non-Communicable

What is Nutrition Explain Diff Type of Nutrition ?

What is the Function of Digestive Enzymes

What is the Full Form of 1.DPT 2.DDT 3.BCG

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

One cusec is equal to how many liters class 8 maths CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE