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Math  /  Geometry

QuestionFind the diameter and radius to the nearest tentt C=100\mathrm{C}=100 in C=50C=50 in C=25C=25 in c=314c=314 in

Studdy Solution

STEP 1

What is this asking? We need to find the *diameter* and *radius* of a bunch of circles given their *circumferences*. Watch out! Don't mix up radius and diameter!
Remember, the diameter is *twice* the radius.

STEP 2

1. Remember the formula
2. Calculate for C = 100 in
3. Calculate for C = 50 in
4. Calculate for C = 25 in
5. Calculate for C = 314 in

STEP 3

Alright, let's **kick things off**!
We know the *circumference* of a circle is given by the **magical formula**: C=2⋅π⋅rC = 2 \cdot \pi \cdot r, where CC is the circumference and rr is the radius.
And remember, the *diameter*, dd, is just **double** the radius: d=2⋅rd = 2 \cdot r.

STEP 4

We want to find the radius and diameter, so let's **rearrange** our **magical formula** to solve for rr: r=C2⋅πr = \frac{C}{2 \cdot \pi}.
Once we have rr, finding dd is a **piece of cake**!

STEP 5

Let's **plug in** C=100C = \textbf{100} inches: r=1002⋅π≈1002⋅3.14≈1006.28≈15.92r = \frac{\textbf{100}}{2 \cdot \pi} \approx \frac{100}{2 \cdot 3.14} \approx \frac{100}{6.28} \approx \textbf{15.92} inches.
So, the radius is approximately **15.9** inches to the nearest tenth.

STEP 6

Now for the diameter! d=2⋅r≈2⋅15.92≈31.84d = 2 \cdot r \approx 2 \cdot \textbf{15.92} \approx \textbf{31.84} inches.
That's about **31.8** inches to the nearest tenth!

STEP 7

Now, let's **do it again** with C=50C = \textbf{50} inches: r=502⋅π≈502⋅3.14≈506.28≈7.96r = \frac{\textbf{50}}{2 \cdot \pi} \approx \frac{50}{2 \cdot 3.14} \approx \frac{50}{6.28} \approx \textbf{7.96} inches.
The radius is approximately **8.0** inches to the nearest tenth.

STEP 8

The diameter is d=2⋅r≈2⋅7.96≈15.92d = 2 \cdot r \approx 2 \cdot \textbf{7.96} \approx \textbf{15.92} inches, which is about **15.9** inches to the nearest tenth.
Notice anything **cool**?
It's the same as the radius from the previous calculation!

STEP 9

Time for C=25C = \textbf{25} inches: r=252⋅π≈252⋅3.14≈256.28≈3.98r = \frac{\textbf{25}}{2 \cdot \pi} \approx \frac{25}{2 \cdot 3.14} \approx \frac{25}{6.28} \approx \textbf{3.98} inches.
The radius is approximately **4.0** inches to the nearest tenth.

STEP 10

Diameter time! d=2⋅r≈2⋅3.98≈7.96d = 2 \cdot r \approx 2 \cdot \textbf{3.98} \approx \textbf{7.96} inches, which rounds to **8.0** inches to the nearest tenth.

STEP 11

Last one! C=314C = \textbf{314} inches: r=3142⋅π≈3142⋅3.14≈3146.28≈49.99r = \frac{\textbf{314}}{2 \cdot \pi} \approx \frac{314}{2 \cdot 3.14} \approx \frac{314}{6.28} \approx \textbf{49.99} inches.
That's basically **50.0** inches for the radius!

STEP 12

Finally, the diameter: d=2⋅r≈2⋅50.0=100.0d = 2 \cdot r \approx 2 \cdot \textbf{50.0} = \textbf{100.0} inches.
A nice, even **100** inches!

STEP 13

For C=100C = 100 inches, r≈15.9r \approx 15.9 inches and d≈31.8d \approx 31.8 inches. For C=50C = 50 inches, r≈8.0r \approx 8.0 inches and d≈15.9d \approx 15.9 inches. For C=25C = 25 inches, r≈4.0r \approx 4.0 inches and d≈8.0d \approx 8.0 inches. For C=314C = 314 inches, r≈50.0r \approx 50.0 inches and d≈100.0d \approx 100.0 inches.

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