Chapter 15: Chapter 15

Problem 285

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Let \(\mathrm{AB}\) be an arc of a circle whose center is \(0 . \mathrm{AB}\) is of length 11 , and \(\mathrm{m} \angle \mathrm{AOB}=10^{\circ}\). (The length of a circular arc is proportional to the central angle which cuts the arc.) a).Compute the circumference of the circle. b). Approximating \(\pi\) by \(22 / 7\), compute the diameter of the circle.

Short Answer

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a) The circumference of the circle is 396 units. b) Approximating π by 22/7, the diameter of the circle is 126 units.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Setup the proportion to find the circumference

Since the ratio of arc length to circumference is equal to the ratio of the central angle to 360 degrees, we have: \[\frac{\text{arc length}}{\text{circumference}} = \frac{\text{central angle}}{360^{\circ}}\] Insert the given values for arc length and central angle: \[\frac{11}{\text{circumference}} = \frac{10^{\circ}}{360^{\circ}}\]

Step 2: Solve for the circumference

Cross-multiply and solve for the circumference: \[11 \times 360^{\circ} = 10^{\circ} \times \text{circumference}\] Now divide by 10°: \[\text{circumference} = \frac{11 \times 360^{\circ}}{10^{\circ}}\] Calculate the value: \[\text{circumference} = 396\] So, the circumference of the circle is 396 units.

Step 3: Calculate the radius from the circumference

We know that the circumference is equal to 2π times the radius. Therefore, we can calculate the radius using the formula: \[r = \frac{\text{circumference}}{2\pi}\] Approximating π by 22/7, we have: \[r = \frac{396}{2 \times \frac{22}{7}}\]

Step 4: Solve for the radius

Calculate the value of the radius: \[r = \frac{396 \times 7}{2 \times 22}\] \[r = \frac{2772}{44}\] \[r = 63\] So, the radius is 63 units.

Step 5: a) Compute the circumference of the circle

As calculated in Step 2, the circumference of the circle is 396 units.

Step 6: b) Approximating π by 22/7, compute the diameter of the circle

With the radius calculated in Step 4, the diameter of the circle is twice the radius: \[d = 2r\] \[d = 2 \times 63\] \[d = 126\] Thus, approximating π by 22/7, the diameter of the circle is 126 units.

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Chapter 15: Chapter 15

Short Answer

Step by step solution

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Step 1: Setup the proportion to find the circumference

Step 2: Solve for the circumference

Step 3: Calculate the radius from the circumference

Step 4: Solve for the radius

Step 5: a) Compute the circumference of the circle

Step 6: b) Approximating π by 22/7, compute the diameter of the circle

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Chapter 7

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