Math / Geometry

QuestionDetermine whether the following statement is true or false:
If a pie has a circumference of $2 \pi$ and a diameter of 2 dm , then another pie with a circumference of $6 \pi$ has a radius of 3 dm .

Studdy Solution

STEP 1

1. The relationship between the circumference, diameter, and radius of a circle is given by the formulas $C = \pi \times d$ and $C = 2\pi \times r$ .
2. The first pie has a circumference of $2\pi$ and a diameter of $2$ dm.
3. The second pie has a circumference of $6\pi$ .

STEP 2

1. Verify the given information for the first pie.
2. Calculate the radius of the second pie using its circumference.
3. Determine if the calculated radius of the second pie matches the given radius.

STEP 3

Verify the given information for the first pie:
The formula for circumference is $C = \pi \times d$ .
For the first pie: $C = 2\pi$ $d = 2 \text{ dm}$
Check if the formula holds: $2\pi = \pi \times 2$
The given information for the first pie is consistent.

STEP 4

Calculate the radius of the second pie using its circumference:
The formula for circumference is $C = 2\pi \times r$ .
For the second pie: $C = 6\pi$
Substitute into the formula: $6\pi = 2\pi \times r$
Solve for $r$ : $r = \frac{6\pi}{2\pi} = 3 \text{ dm}$

STEP 5

Determine if the calculated radius of the second pie matches the given radius:
The calculated radius is $3 \text{ dm}$ , which matches the given radius.
Therefore, the statement is:
$\boxed{\text{True}}$

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