Question

Two identical circles of radius r are inscribed in a rectangle. . a)Find the area of the rectangle as a function of r.. b)Express the perimeter of the rectangle as a function of r..

Answer 1

The area of rectangle can be represented as a function of $r$ as $\boxed{{\mathbf{Area = 8}}{{\mathbf{r}}^{\mathbf{2}}}}$ and the perimeter of rectangle can be represented as a function of $r$ as $\boxed{{\mathbf{Perimeter = 12r}}}$.

Further explanation:

The diameter of the circle is two times of the radius.

It can be mathematically expressed as,

$\text{diameter}=2\times\text{radius}$  

Given:

Two equal circles of radius $r$ are inscribed in a rectangle.

Step by step explanation:

Step 1:

First determine the length and breadth of the rectangle.

The length of the rectangle is two times of the diameter.

Therefore, the length of the rectangle can be found as,

$\text{lenght}=2\times2r$  

Here, $2r,$ is the diameter of the circle.

The breadth of the rectangle is equal to the diameter.

Therefore, the breadth of the rectangle can be found as,

${\text{breadth}} = 2r$  

Step 2:

(a)

The formula of the area of the rectangle can be calculated as,

${\text{area}} = l \times b$  

The area of the rectangle can be found as,

$\begin{aligned}{\text{area}} &= l \times b\\&= 4r \times 2r\\&= 8{r^2}\\\end{aligned}$  

Therefore, the area of rectangle can be represented as a function of $r$ as ${\text{area}} = 8{r^2}$.

Step 3:

(b)

The formula of the perimeter of the rectangle can be calculated as,

${\text{perimeter}} = 2\left( {l + b} \right)$  

The perimeter of the rectangle can be found as,

$\begin{aligned}{\text{perimeter}} &= 2\left( {l + b} \right)\\&= 2\left( {4r + 2r} \right)\\&= 2\left( {6r} \right)\\&= 12r\\\end{aligned}$  

Therefore, the perimeter of rectangle can be represented as a function of $r$ as ${\text{perimeter}} = 12r$.

Learn more:  

1. Learn more about the function is graphed below brainly.com/question/9590016
2. Learn more about the symmetry for a function brainly.com/question/1286775
3. Learn more about midpoint of the segment brainly.com/question/3269852

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Perimeter and area.

Keywords: Perimeter, area, rectangle, inscribed, circle, function of $r$, formula, diameter, radius, length, breadth, two times, mathematically expressed.

Answer 2

The two circles are equally inscribe inside a rectangle which means that its length is equals to the diameter of a circle and the width is two tines the diameter of a circle. 

A. The area of the rectangle using the R. is Area = 2R*4R
B. The perimeter is express as Perimeter = 2(2R*4R)