Here and some coins. The radius of the 5p coin is 9mm The diameter of the 2p coin is 26mm The 2p coin has a greater circumference. Work out how much greater, giving your answer in terms of \pi
The correct answer and explanation is :
To solve this problem, we need to calculate the circumferences of both the 5p and 2p coins and then determine how much greater the circumference of the 2p coin is compared to the 5p coin.
Step 1: Formula for Circumference
The circumference ( C ) of a circle is given by the formula:
[
C = 2\pi r
]
where:
- ( C ) is the circumference,
- ( \pi ) is approximately 3.1416 (but we’ll leave it in terms of ( \pi ) for this question),
- ( r ) is the radius of the coin.
Step 2: Circumference of the 5p Coin
We are given the radius of the 5p coin as 9mm. Using the formula:
[
C_{5p} = 2\pi \times r_{5p}
]
[
C_{5p} = 2\pi \times 9 = 18\pi \, \text{mm}
]
So, the circumference of the 5p coin is ( 18\pi ) mm.
Step 3: Circumference of the 2p Coin
The problem gives us the diameter of the 2p coin as 26mm. The radius is half of the diameter, so:
[
r_{2p} = \frac{26}{2} = 13 \, \text{mm}
]
Now, we can use the same formula for the circumference:
[
C_{2p} = 2\pi \times r_{2p}
]
[
C_{2p} = 2\pi \times 13 = 26\pi \, \text{mm}
]
So, the circumference of the 2p coin is ( 26\pi ) mm.
Step 4: Difference in Circumferences
To find how much greater the circumference of the 2p coin is compared to the 5p coin, we subtract the circumference of the 5p coin from that of the 2p coin:
[
\Delta C = C_{2p} – C_{5p}
]
[
\Delta C = 26\pi – 18\pi = 8\pi \, \text{mm}
]
Final Answer:
The circumference of the 2p coin is ( 8\pi ) mm greater than the circumference of the 5p coin.
Explanation:
The formula for the circumference of a circle is based on its radius. The 2p coin has a larger radius (13mm) compared to the 5p coin (9mm), so it naturally has a larger circumference. By calculating both circumferences and subtracting, we find that the 2p coin’s circumference is ( 8\pi ) mm greater than the 5p coin’s. This shows the relationship between the size of the radius and the circumference of a circle, which is directly proportional.