Math / Geometry

QuestionThe circumference $C$ of a circle is a function of its radius given by $C(r)=2 \pi r$ . a. Express the radius of a circle as a function of its circumference. Call this function $r(C)$ .
Enter the exact answer.
Enclose numerators and denominators in parentheses. For example, $(a-b) /(1+n)$ . $r(C)=$ b. Find $r(34 \pi)$ and interpret its meaning. $r(34 \pi)=$ $\square$ Number

Studdy Solution

STEP 1

1. The circumference $C$ of a circle is related to its radius $r$ by the formula $C = 2 \pi r$ .
2. To find the radius as a function of the circumference, we need to solve the equation $C = 2 \pi r$ for $r$ .
3. The result should be expressed in the form $r(C)$ .
4. In part b, we will substitute $C = 34 \pi$ into the function $r(C)$ to find the corresponding radius.

STEP 2

1. Solve the equation $C = 2 \pi r$ for $r$ to express the radius as a function of the circumference $C$ .
2. Substitute $C = 34 \pi$ into the function $r(C)$ to find the radius.
3. Interpret the meaning of the result in part b.

STEP 3

Start with the equation for the circumference of a circle: $C = 2 \pi r$

STEP 4

Solve for $r$ by dividing both sides of the equation by $2 \pi$ : $r = \frac{C}{2 \pi}$

STEP 5

Express $r$ as a function of $C$ : $r(C) = \frac{C}{2 \pi}$

STEP 6

Substitute $C = 34 \pi$ into the function $r(C)$ : $r(34 \pi) = \frac{34 \pi}{2 \pi}$

STEP 7

Simplify the expression by canceling $\pi$ : $r(34 \pi) = \frac{34}{2}$

STEP 8

Calculate the result: $r(34 \pi) = 17$

STEP 9

Interpret the meaning: When the circumference of a circle is $34 \pi$ units, the radius of the circle is $17$ units.
The solution is: $r(C) = \frac{C}{2 \pi}$ $r(34 \pi) = 17$

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Studdy Solution

STEP 1

STEP 2

1. Solve the equation $C = 2 \pi r$ for $r$ to express the radius as a function of the circumference $C$ .
2. Substitute $C = 34 \pi$ into the function $r(C)$ to find the radius.
3. Interpret the meaning of the result in part b.

STEP 3

Start with the equation for the circumference of a circle: $C = 2 \pi r$

STEP 4

Solve for $r$ by dividing both sides of the equation by $2 \pi$ : $r = \frac{C}{2 \pi}$

STEP 5

Express $r$ as a function of $C$ : $r(C) = \frac{C}{2 \pi}$

STEP 6

Substitute $C = 34 \pi$ into the function $r(C)$ : $r(34 \pi) = \frac{34 \pi}{2 \pi}$

STEP 7

Simplify the expression by canceling $\pi$ : $r(34 \pi) = \frac{34}{2}$

STEP 8

Calculate the result: $r(34 \pi) = 17$

STEP 9

Interpret the meaning: When the circumference of a circle is $34 \pi$ units, the radius of the circle is $17$ units.
The solution is: $r(C) = \frac{C}{2 \pi}$ $r(34 \pi) = 17$

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Studdy Solution

STEP 1

STEP 2

1. Solve the equation C=2πrC = 2 \pi rC=2πr for rrr to express the radius as a function of the circumference CCC.2. Substitute C=34πC = 34 \piC=34π into the function r(C)r(C)r(C) to find the radius.3. Interpret the meaning of the result in part b.

STEP 3

Start with the equation for the circumference of a circle: C=2πr C = 2 \pi r C=2πr

STEP 4

Solve for rrr by dividing both sides of the equation by 2π2 \pi2π: r=C2π r = \frac{C}{2 \pi} r=2πC​

STEP 5

Express rrr as a function of CCC: r(C)=C2π r(C) = \frac{C}{2 \pi} r(C)=2πC​

STEP 6

Substitute C=34πC = 34 \piC=34π into the function r(C)r(C)r(C): r(34π)=34π2π r(34 \pi) = \frac{34 \pi}{2 \pi} r(34π)=2π34π​

STEP 7

Simplify the expression by canceling π\piπ: r(34π)=342 r(34 \pi) = \frac{34}{2} r(34π)=234​

STEP 8

Calculate the result: r(34π)=17 r(34 \pi) = 17 r(34π)=17

STEP 9

Interpret the meaning: When the circumference of a circle is 34π34 \pi34π units, the radius of the circle is 171717 units.The solution is: r(C)=C2πr(C) = \frac{C}{2 \pi}r(C)=2πC​ r(34π)=17r(34 \pi) = 17r(34π)=17

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1. Solve the equation $C = 2 \pi r$ for $r$ to express the radius as a function of the circumference $C$ .
2. Substitute $C = 34 \pi$ into the function $r(C)$ to find the radius.
3. Interpret the meaning of the result in part b.

Start with the equation for the circumference of a circle: $C = 2 \pi r$

Solve for $r$ by dividing both sides of the equation by $2 \pi$ : $r = \frac{C}{2 \pi}$

Express $r$ as a function of $C$ : $r(C) = \frac{C}{2 \pi}$

Substitute $C = 34 \pi$ into the function $r(C)$ : $r(34 \pi) = \frac{34 \pi}{2 \pi}$

Simplify the expression by canceling $\pi$ : $r(34 \pi) = \frac{34}{2}$

Calculate the result: $r(34 \pi) = 17$

Interpret the meaning: When the circumference of a circle is $34 \pi$ units, the radius of the circle is $17$ units.
The solution is: $r(C) = \frac{C}{2 \pi}$ $r(34 \pi) = 17$