Sure, let’s go through each part of the question step-by-step:
### a) The perimeter of a circle is 132 cm
To find:
(i) Radius
(ii) Area
Step 1: Calculate the radius of the circle.
The formula for the perimeter (circumference) of a circle is:
[tex]\[ P = 2 \pi r \][/tex]
where [tex]\( P \)[/tex] is the perimeter and [tex]\( r \)[/tex] is the radius.
Given [tex]\( P = 132 \)[/tex] cm, solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{P}{2 \pi} \][/tex]
[tex]\[ r = \frac{132}{2 \pi} \][/tex]
[tex]\[ r \approx 21.008 \, \text{cm} \][/tex]
Step 2: Calculate the area of the circle.
The formula for the area of a circle is:
[tex]\[ A = \pi r^2 \][/tex]
Using [tex]\( r \approx 21.008 \)[/tex] cm:
[tex]\[ A \approx \pi \times (21.008)^2 \][/tex]
[tex]\[ A \approx 1386.558 \, \text{cm}^2 \][/tex]
Results:
(i) Radius: [tex]\( 21.008 \)[/tex] cm
(ii) Area: [tex]\( 1386.558 \, \text{cm}^2 \)[/tex]
### b) The perimeter of a circular park is 880 m
To find: Area of the park
Step 1: Calculate the radius of the park.
Using the same formula for the perimeter:
[tex]\[ P = 2 \pi r \][/tex]
Given [tex]\( P = 880 \)[/tex] m, solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{P}{2 \pi} \][/tex]
[tex]\[ r = \frac{880}{2 \pi} \][/tex]
[tex]\[ r \approx 140.056 \, \text{m} \][/tex]
Step 2: Calculate the area of the park.
[tex]\[ A = \pi r^2 \][/tex]
Using [tex]\( r \approx 140.056 \)[/tex] m:
[tex]\[ A \approx \pi \times (140.056)^2 \][/tex]
[tex]\[ A \approx 61624.794 \, \text{m}^2 \][/tex]
Result:
Area of the park: [tex]\( 61624.794 \, \text{m}^2 \)[/tex]
### c) The area of the base of a cylindrical tank is 38.5 m²
To find: Perimeter of the base
Step 1: Calculate the radius of the base.
The formula for the area of a circle is:
[tex]\[ A = \pi r^2 \][/tex]
Given [tex]\( A = 38.5 \)[/tex] m², solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{A}{\pi}} \][/tex]
[tex]\[ r \approx \sqrt{\frac{38.5}{\pi}} \][/tex]
[tex]\[ r \approx 3.499 \, \text{m} \][/tex]
Step 2: Calculate the perimeter of the base.
Using the perimeter formula:
[tex]\[ P = 2 \pi r \][/tex]
Using [tex]\( r \approx 3.499 \)[/tex] m:
[tex]\[ P \approx 2 \pi \times 3.499 \][/tex]
[tex]\[ P \approx 21.996 \, \text{m} \][/tex]
Result:
Perimeter of the base: [tex]\( 21.996 \)[/tex] m
### d) The area of a circular pond is 1386 m²
To find: Length of wire required to fence around it with 1 round
Step 1: Calculate the radius of the pond.
Using the area formula:
[tex]\[ A = \pi r^2 \][/tex]
Given [tex]\( A = 1386 \)[/tex] m², solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{A}{\pi}} \][/tex]
[tex]\[ r \approx \sqrt{\frac{1386}{\pi}} \][/tex]
[tex]\[ r \approx 21.008 \, \text{m} \][/tex]
Step 2: Calculate the perimeter of the pond (length of wire needed).
Using the perimeter formula:
[tex]\[ P = 2 \pi r \][/tex]
Using [tex]\( r \approx 21.008 \)[/tex] m:
[tex]\[ P \approx 2 \pi \times 21.008 \][/tex]
[tex]\[ P \approx 131.973 \, \text{m} \][/tex]
Result:
Length of wire required: [tex]\( 131.973 \)[/tex] m
