A pendulum swings through an angle of 40^\circ, and the tip of the pendulum travels an arc… — SAT

A pendulum swings through an angle of $40^\circ$ , and the tip of the pendulum travels an arc of length $8\pi$ inches. What is the length of the pendulum, in inches?

Cevap: 36 inches

Cevap

To find the length of the pendulum, which represents the radius

r

of the circular path it sweeps, we can use the arc length formula

s = r\theta

, where

s

is the arc length and

\theta

is the central angle in radians. First, convert the given angle from degrees to radians:

\theta = 40^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{9}

radians. Next, substitute the arc length

s = 8\pi

and the angle

\theta = \frac{2\pi}{9}

into the formula:

8\pi = r \left(\frac{2\pi}{9}\right)

. Solving for

r

by multiplying both sides by

\frac{9}{2\pi}

gives

r = 36

. Alternatively, you can use the ratio of the sector's central angle to the total angle of a circle:

\frac{40^\circ}{360^\circ} = \frac{1}{9}

. This means the arc length is

\frac{1}{9}

of the circumference of the circle:

8\pi = \frac{1}{9}(2\pi r)

. Dividing both sides by

2\pi

yields

4 = \frac{1}{9}r

, so

r = 36

Adım Adım Çözüm

Convert the swing angle of the pendulum from degrees to radians.

\theta = \frac{2\pi}{9}

radians

The arc length formula

s = r\theta

requires the angle

\theta

to be in radians.

Set up the arc length equation using

s = r\theta

, where

s = 8\pi

is the arc length and

r

is the length of the pendulum.

8\pi = r \left(\frac{2\pi}{9}\right)

The tip of the pendulum travels along a circular path whose radius is the length of the pendulum.

Solve the equation for the radius

r

r = 36

Isolating

r

by multiplying both sides by

\frac{9}{2\pi}

yields the length of the pendulum.

Anahtar Kavram

Converting angle measures between degrees and radians and applying the arc length formula.

Soru

Cevap

Adım Adım Çözüm

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