Question

Can someone please help me with The Questions #14 and #15 of The Operations On The Polynomials & The Special Products for me, please guys? :)

Answer 1

Answer:

Formulas

$\textsf{Area of a circle}=\pi r^2 \quad \textsf{(where r is the radius)}$

$\textsf{Radius of a circle}=\dfrac{1}{2}d \quad \textsf{(where d is the diameter)}$

$\textsf{Area of a square}=s^2 \quad \textsf{(where s is the side length)}$

$\textsf{Diagonal of a square}=s\sqrt{2} \quad \textsf{(where s is the side length)}$

Question 14

If a circle is inscribed in a square, then the diameter of the circle is equal to the side length of the square.  Therefore, as the radius of a circle is half the diameter, the radius of the circle is half the side length of the square.

Given:

• $s= 12x\:\: \sf cm$

• $r=\dfrac{1}{2}s=6x\:\: \sf cm$

Therefore, the areas of the square and circle are:

$\begin{aligned} \textsf{Area of the circle} & =\pi (6x)^2\\ & = 36 \pi x^2 \:\: \sf cm^2 \end{aligned}$

$\begin{aligned}\textsf{Area of the square}& =(12x)^2\\ & = 144x^2 \:\: \sf cm^2 \end{aligned}$

Therefore, the ratio of the circle to square is:

$\implies \sf circle : square$

$\implies 36 \pi x^2:144x^2$

$\implies 36 \pi :144$

$\implies \pi : 4$

$\implies \dfrac{1}{4} \pi : 1$

So the circle is ¹/₄π the size of the square.

Question 15

If a square is inscribed in a circle, then the diagonal of the square is the diameter of the circle.  Therefore, as the radius of a circle is half the diameter, the radius of the circle is half the diagonal of the square.

Given:

• $r = 5a^2 \:\: \sf cm$

• $d=2r=10a^2 \:\: \sf cm$

$\begin{aligned} \textsf{Area of the circle} & =\pi (5a^2)^2\\ & = 25 \pi a^4 \:\: \sf cm^2 \end{aligned}$

$\begin{aligned}\textsf{Diagonal of a square} & =s\sqrt{2}\\10a^2 & = s \sqrt{2}\\ s & =\dfrac{10a^2}{\sqrt{2}}\\ s & = 5\sqrt{2}a^2\:\: \sf cm^ \end{aligned}$

$\begin{aligned}\textsf{Area of the square}& =(5\sqrt{2}a^2)^2\\ & = 50a^4 \:\: \sf cm^2 \end{aligned}$

Therefore, the ratio of the circle to square is:

$\implies \sf circle : square$

$\implies 25 \pi a^4:50a^4$

$\implies 25 \pi :50$

$\implies \pi : 2$

$\implies \dfrac{1}{2} \pi:1$

So the circle is ¹/₂π  the size of the square.