Question

The perimeter of a semicircle is $4\pi $+ 8. The area of a square which lies inside the semicircle, when rounded off to nearest integer, can be at most

Answer 1

Hint: Here, in this question, first we have to create the diagram according to the information mentioned in the question. Use the Perimeter of the semicircle = $\pi $r + d to find the radius of the semicircle. Now, use the Pythagoras theorem to find the equation which will give us the required result.

Complete step-by-step answer:
Let us first draw the diagram which includes the semicircle and the square lying inside the semicircle.

Here, we can see that the semicircle of diameter D and radius r and a square ABCD is drawn inside the semicircle,
Let us consider the height, ‘h’ of the triangle OBC formed due to the dotted line which represents the radius, and the base of the triangle be OB which is $\dfrac{\text{h}}{2}$.
We know the perimeter is also known as the circumference of the circle. Therefore, the perimeter of the semicircle will be half the circumference of the circle in addition to the base of the semicircle which is the diameter.
Perimeter of the semicircle = $\pi $r + d
We have Perimeter of the given semicircle = $4\pi $+ 8
$4\pi $+ 8 = $\pi $r + d
Comparing both the sides, we get
r = 4 and d = 8 = 2r
In the triangle OBC, it is clearly shown that it is a right-angled triangle with hypotenuse as the radius, OC = 4, height is one of the side of the square whose value is considered to be ‘h’ and it's base as OB which is considered, half of the value of the square, $\dfrac{\text{h}}{2}$.
We know, According to Pythagoras theorem,
$($Hypotenuse${{)}^{2}}$= $($Height${{)}^{2}}$+ $($Base${{)}^{2}}$
$\begin{align}
  & {{\left( 4 \right)}^{2}}={{\left( \text{h} \right)}^{2}}+{{\left( \dfrac{\text{h}}{2} \right)}^{2}} \\
 & 16={{\text{h}}^{2}}+\dfrac{{{\text{h}}^{2}}}{4} \\
 & 16=\dfrac{4{{\text{h}}^{2}}+{{\text{h}}^{2}}}{4} \\
 & 16=\dfrac{5{{\text{h}}^{2}}}{4} \\
 & {{\text{h}}^{2}}=\dfrac{16\times 4}{5} \\
 & =\dfrac{64}{5} \\
 & =12.8
\end{align}$
The value of ${{\text{h}}^{2}}$ is 12.8, we also know that h is the side of the square
The area of the square = ${{\text{h}}^{2}}$ = 12.8 $\approx 13$
Hence, the area of the square when rounded off to the nearest integer is equal to 13.

Note: In this question, we can find the area of the square by dividing the square into three triangles and find their areas and add them together which will give us the area of the square although it would be very tedious.