Respuesta :
Answer:
Ac = (24- (s / 4)) ^ 2 / (4 pi)
Step-by-step explanation:
Because the piece is rope is divided into two pieces we have to:
c + s = 24, where c is the circumference of circle and s is the perimeter of square. Therefore the circumference of circle would be equal to:
c = 24-s
Let As the area of the square be, we know that the perimeter is the sum of all sides and a square has four sides, therefore the value of each side would be the perimeter divided by four. And the area of the square is a side raised to the square, as follows:
As = (s / 4) ^ 2
Now knowing that the circumference of circle is equal to the radius, by the number pi, by two, we have to:
c = 2 * pi * r
rearranging we have:
r = c / (2 * pi)
Let Ac be the area of the circle we have the radius squared by the number pi, like this:
Ac = pi * r ^ 2
But we already know the equivalence of the radius, replacing:
Ac = pi * (c / (2 * pi)) ^ 2
Now replacing the circumference of circle with its equivalence with the perimeter of the square we have to:
Ac = pi * ((24-s) / (2 * pi)) ^ 2
Now knowing that the one side of the square is equal to a quarter of the perimeter of the square we have:
Ac = (24- (s / 4)) ^ 2 / (4 pi)
That would be the area of the circle as a function of one side of the square.
Answer:
function representing the area of a circle as a function of the length of one side of the square is [tex]=\frac{144}{\pi}+\frac{P^{2} }{4\pi} - \frac{P}{4}.\frac{28}{\pi}[/tex]
Step-by-step explanation:
Rope is totally 24 feet longer and is cut into two pieces i.e. circle and square, so we can say that
circumference of the Circle (C)+ perimeter of the square (P) =24
i.e. C+P=24
C=24-P
as we know [tex]C=2\pi\ r[/tex]
[tex]r=\frac{C}{2\pi}[/tex]
Area of a circle is =[tex]\pi\ r^{2}[/tex]
Area of circle [tex]=\pi(\frac{C}{2\pi}) ^{2}[/tex]
putting C=24-P in above equation we get
Area of circle[tex]=\pi(\frac{24-P}{2\pi}) ^{2}[/tex]
simplifying
Area of circle=[tex]=\pi(\frac{24-P}{2\pi}) ^{2}=\pi(\frac{24^{2}+P^{2} -2(P)(24)}{4\pi*\pi})=\frac{24^{2}+P^{2} -2(P)(24)}{4\pi}[/tex]
Area of circle=[tex]=\frac{24^{2} }{4\pi}+\frac{P^{2} }{4}.\frac{1}{\pi}} -\frac{2P(24)}{4\pi}=\frac{144}{\pi}+\frac{P^{2} }{4}.\frac{1}{\pi}} - \frac{P}{4}.\frac{28}{\pi}[/tex]
where P/4 is the length of the side of the square
