Mathematics: Apiece of rope there is 28 feet long is cut into two pieces. one is use to form a circle and others used to form a square. write a function g representing the area of the square as a function of the radius of the circle r

61. Just cause 61-25=36. And 61-36=25 but if you put 11-36 it equals -25....

Apiece of rope there is 28 feet long is cut into two

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andrwisawesome0

14.01.2022

Mathematics

The function g representing the area of the square as a function of the radius of the circle r is given as:

$g(r) = 49 - 22r + \frac{121r^2}{49}$

Solution:

Given that,

length of rope = 28 feet

Let "c" be the circumference of circle

Let "p" be the perimeter of square

Therefore,

length of rope = circumference of circle + perimeter of square

c + p = 28 ------- eqn 1

The circumference of circle is given as:

$c = 2 \pi r$

Where, "r" is the radius of circle

Substitute the above circumference in eqn 1

$2 \pi r + p = 28$

$p = 28 - 2 \pi r$ ----------- eqn 2

If "a" is the length of each side of square, then the perimeter of sqaure is given as:

p = 4a

Substitute p = 4a in eqn 2

$4a = 28 - 2 \pi r\\\\a = \frac{28 - 2 \pi r}{4}\\\\a = 7 - \frac{ \pi r}{2}$

The area of square is given as:

$area = (side)^2\\\\area = a^2$

Substitute the value of "a" in above area expression

$area = (7 - \frac{ \pi r}{2})^2$ ------ eqn 3

We know that,

(a - b)^2 = a^2 - 2ab + b^2

Therefore eqn 3 becomes,

$area = 7^2 -2(7)(\frac{\pi r}{2}) + (\frac{ \pi r}{2})^2\\\\area = 49 - 7 \pi r + \frac{ (\pi)^2 r^2 }{4}$

$\text{ substitute } \pi = \frac{22}{7}$

$area = 49 - 7 \times \frac{22}{7} \times r + (\frac{22}{7})^2 \times \frac{r^2 }{4}\\\\area = 49 - 22r + \frac{121}{49} \times r^2\\\\area = 49 - 22r + \frac{121r^2}{49}$

Let g(r) represent the area of the square as a function of the radius of the circle r, then we get

$g(r) = 49 - 22r + \frac{121r^2}{49}$

Thus the function is found

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Apiece of rope there is 28 feet long is cut into two pieces. one is use to form a circle and others used to form a square. write a function g representing the area of the square as a function of the radius of the circle r

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