Answers:
1) Circumference to diameter
2) [tex]\frac{15}{\pi} cm[/tex]
3) [tex]10 m[/tex]
4) [tex]27.004 km[/tex]
Step-by-step explanation:
1st question
Let's begin by explaining that the circumference [tex]C[/tex] of a circle is given by the following formula:
[tex]C=2 \pi r[/tex] (1)
Where [tex]r[/tex] is the radius of the circle
In addition, the diameter [tex]d[/tex] is twice the radius:
[tex]d=2r[/tex] (2)
Hence:
[tex]r=\frac{d}{2}[/tex] (3)
Knowing this, we can try with each option and find the one that results in [tex]\pi[/tex]:
Circumference to radius:
[tex]\frac{C}{r}=\frac{2 \pi r}{r}=2 \pi[/tex]
Radius to circumference:
[tex]\frac{r}{C}=\frac{r}{2 \pi r}=\frac{1}{2 \pi}[/tex]
Diameter to circumference:
[tex]\frac{d}{C}=\frac{2r}{2 \pi r}=\frac{1}{\pi}[/tex]
Circumference to diameter:
[tex]\frac{C}{d}=\frac{2 \pi r}{2r}=\pi[/tex] This is the correct option!
2nd question
If the circumference of a circle is [tex]C=15 cm[/tex], applying (1) we have:
[tex]15 cm=2 \pi r[/tex]
Isolating [tex]r[/tex]:
[tex]r=\frac{15}{2 \pi} cm[/tex]
Multiplying by [tex]2[/tex]:
[tex]d=2r=2\frac{15}{2 \pi} cm[/tex]
Then, the diameter is:
[tex]d=\frac{15}{\pi} cm[/tex]
3rd question
If the circumference of the circle is [tex]C=30 m[/tex], applying (1) we have:
[tex]30 m=2 \pi r[/tex]
Isolating [tex]r[/tex]:
[tex]r=\frac{30}{2 \pi} m[/tex]
Multiplying by [tex]2[/tex]:
[tex]d=2r=2\frac{30}{2 \pi} m[/tex]
Then, the diameter is:
[tex]d=\frac{30}{\pi} cm=9.54 m \approx 10 m[/tex]
4th question
If the diameter of the Large Hadron Collider (LHC) is [tex]d=8.6 km[/tex]
Applying (3) to find the radius:
[tex]r=\frac{8.6 km}{2}[/tex]
[tex]r=4.3 km[/tex]
Calculating the circumference:
[tex]C=2 (3.14)(4.3 km)[/tex]
[tex]C=27.004 km[/tex] This is the circumference of the LHC
