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Problem 285

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.

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

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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