What is the smallest perimeter possible for a rectangle whose area is 16 in^2, and what are its dimensions?

Smallest possible perimeter 16 inches.

Length : 4 inches,

Width : 4 inches.

Step-by-step explanation:

Let x represent length and y represent width of rectangle.

We have been given that area of a rectangle is 16 square inches.

We know that area of rectangle is product of its length and width. We can represent our given information in an equation as:

We know that perimeter of rectangle is sum of its all sides that is:

From equation (1), we will get:

Upon substituting this value in equation (2), we will get:

Now, we will find the derivative of perimeter equation as:

Now, we will equate our derivative equal to 0 to find critical points as:

Take square root of both sides:

Since length cannot be negative, therefore, .

Now, we will find 2nd derivative as:

We know that where 2nd derivative is positive, the point is a minimum. Let us substitute in 2nd derivative.

Since , therefore   is a minimum.

Let us solve for y using equation as:

Therefore, width and length of 4 inches each will result in smallest perimeter.

Therefore, the smallest possible perimeter would be 16 inches.

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

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step-by-step explanation: