Chapter 12.4, Problem 3CK

a.

To determine

To find: The perimeter to the nearest centimetre.

Answer to Problem 3CK

Side of the polygon is 45.6 cm and the perimeter is 136.77 cm.

Explanation of Solution

Given information:

The polygon has an area of 900 cm².

Calculation:

The given polygon is an equilateral triangle.

Area of the equilateral triangle is given by

  $A=\frac{\sqrt{3}}{4}{s}^2$ where s is the side of triangle.

Since, the area is 600 mm². Therefore,

  $\begin{array}{l}\frac{\sqrt{3}}{4}{s}^2=900\\ s^2=\frac{900\times 4}{\sqrt{3}}\\ s^2=1200\sqrt{3}\\ s=45.6\text{ cm}\end{array}$

And perimeter of the triangle is given by

  $P=3s$ where s is the measure of each sides.

Therefore,

  $\begin{array}{l}P=3\left(45.6\right)\\ P=136.77\text{ cm}\end{array}$

Hence,

Side of the polygon is 45.6 cm and the perimeter is 136.77 cm.

b.

To determine

To find: The perimeter to the nearest millimetre.

Answer to Problem 3CK

Side of the polygon is 10 cm and perimeter is 120 cm

Explanation of Solution

Given information:

The polygon has an area 900 cm².

Calculation:

The given polygon is a square.

Area of the square is given by

  $A=x^2$ where x is the side of square.

Here, $A=900{\text{ cm}}^2$ . Therefore,

  $\begin{array}{l}{x}^2=900\\ x=\sqrt{900}\\ x=30\text{ cm}\end{array}$

And perimeter of the square is given by

  $P=4x$ where x is the measure of each sides.

Since, $x=30\text{ cm}$ . Therefore,

  $\begin{array}{l}P=4\left(30\right)\\ P=120\text{ cm}\end{array}$

Hence,

Side of the polygon is 10 cm and perimeter is 120 cm

c.

To determine

To find: The perimeter to the nearest millimetre.

Answer to Problem 3CK

Side of the polygon is 18.61 cm and perimeter is 111.6 cm.

Explanation of Solution

Given information:

The polygon has a area of 900 cm²

Calculation:

The given polygon is a regular hexagon.

Area of the regular hexagon is given by

  $A=\frac{3\sqrt{3}}{2}{y}^2$ where y is the side of hexagon.

Here, area $A=900{\text{ cm}}^2$ . Therefore, area is

  $\begin{array}{l}\frac{3\sqrt{3}}{2}{y}^2=900\\ y^2=\frac{900\times 2}{3\sqrt{3}}\\ y^2=200\sqrt{3}\\ y=18.61\text{ cm}\end{array}$

And perimeter of the regular hexagon is given by

  $P=6y$ where y is the measure of each sides.

Here, $y=18.61\text{ cm}$ . Therefore,

  $\begin{array}{l}P=6\left(18.61\right)\\ P=111.6\text{ cm}\end{array}$

Hence,

Side of the polygon is 18.61 cm and perimeter is 111.6 cm.

d.

To determine

To find: The area to the nearest millimetre.

Answer to Problem 3CK

The radius of the circle is 16.92 cm and perimeter is 106.35 cm and the answer suggest that the circle has the minimum perimeter.

Explanation of Solution

Given information:

The area of the circle is 900 cm².+

Calculation:

Area of the circle is given by

  $A=\pi r^2$ where r is the radius of the circle.

Here, $A=900$ . Therefore, area is

  $\begin{array}{l}\pi r^2=900\\ r^2=\frac{900}{\pi }\\ r=16.92\end{array}$

The circumference of the circle is given by

  $C=2\pi r$ where r is the radius of the circle.

Here, $r=16.92$ cm. Therefore,

  $\begin{array}{l}C=2\pi \left(16.92\right)\\ C=106.35\text{ cm}\end{array}$

Hence,

The radius of the circle is 16.92 cm and perimeter is 106.35 cm and the answer suggest that the circle has the minimum perimeter.