Find the area of the shaded segment to 4 significant figures.

Tuna pembetatu $AOB$ yenye pande $OA = 5$ cm, $OB = 5$ cm (radii), na $AB = 8$ cm.

Badilisha pembe kuwa radiani:

Eneo la sekta $AOB = \frac{1}{2}r^2\theta = \frac{1}{2}(5^2)(1.8546) \approx 23.1825 \text{ cm}^2$.
Eneo la pembetatu $AOB = \frac{1}{2}r^2\sin(\angle AOB) = \frac{1}{2}(5^2)\sin(106.2602^\circ) \approx 12.0000 \text{ cm}^2$.

Eneo la sehemu ndogo = Eneo la sekta $AOB$ - Eneo la pembetatu $AOB = 23.1825 - 12.0000 = 11.1825 \text{ cm}^2$.
Eneo la duara = $\pi r^2 = \pi (5^2) = 25\pi \approx 78.5398 \text{ cm}^2$.

Eneo la sehemu kuu = Eneo la duara - Eneo la sehemu ndogo = $78.5398 - 11.1825 = 67.3573 \text{ cm}^2$.
Kwa tarakimu muhimu 4:
Eneo la sehemu kuu ni $\boxed{\text{67.36 cm}^2}$.

1. (b) Kokotoa pembe kwa radiani ambayo gurudumu huzunguka kwa sekunde moja.

Gurudumu huzunguka mizunguko 300 kwa dakika.
Mizunguko kwa sekunde = $\frac{300 \text{ mizunguko}}{60 \text{ sekunde}} = 5 \text{ mizunguko/sekunde}$.

Mzunguko mmoja ni sawa na $2\pi$ radiani.
Pembe kwa radiani kwa sekunde = $5 \text{ mizunguko/sekunde} \times 2\pi \text{ radiani/mzunguko} = 10\pi \text{ radiani/sekunde}$.

Pembe ambayo gurudumu huzunguka kwa sekunde moja ni $\boxed{\text{31.42 radiani}}$.

2. Tafuta eneo la sehemu iliyotiwa kivuli kwa tarakimu muhimu 4.

Kipenyo = 28 cm, hivyo radius $r = \frac{28}{2} = 14$ cm.
Pembe ya katikati $\theta = 75^\circ$.
Badilisha pembe kuwa radiani:

Eneo la sekta = $\frac{1}{2}r^2\theta = \frac{1}{2}(14^2)\left(\frac{5\pi}{12}\right) = \frac{1}{2}(196)\left(\frac{5\pi}{12}\right) = \frac{245\pi}{6} \approx 128.282 \text{ cm}^2$.
Eneo la pembetatu = $\frac{1}{2}r^2\sin\theta = \frac{1}{2}(14^2)\sin(75^\circ) = 98\sin(75^\circ) \approx 94.6587 \text{ cm}^2$.

Eneo la sehemu iliyotiwa kivuli = Eneo la sekta - Eneo la pembetatu = $128.282 - 94.6587 = 33.6233 \text{ cm}^2$.
Kwa tarakimu muhimu 4:
Eneo la sehemu iliyotiwa kivuli ni $\boxed{\text{33.62 cm}^2}$.

3. Mstatili PQRS ndani ya duara.

3. (a) Kokotoa urefu wa PS wa mstatili.

Radius ya duara $r = 17$ cm. Ulalo wa mstatili $PR$ ni kipenyo cha duara, hivyo $PR = 2r = 2 \times 17 = 34$ cm.

Katika pembetatu ya kulia $PQR$, tuna $PQ = 16$ cm na $PR = 34$ cm.

Kwa kuwa PQRS ni mstatili, $PS = QR$.
Urefu wa PS wa mstatili ni $\boxed{\text{30 cm}}$.

3. (b) Kokotoa pembe POS.

Katika pembetatu $POS$, tuna $OP = OS = 17$ cm (radii) na $PS = 30$ cm.

Pembe POS ni $\boxed{\text{123.90}^\circ}$.

3. (c) Kokotoa eneo la sehemu iliyotiwa kivuli.

Eneo la sekta $POS = \frac{1}{2}r^2\theta = \frac{1}{2}(17^2)(2.1625) \approx 312.061 \text{ cm}^2$.
Eneo la pembetatu $POS = \frac{1}{2}r^2\sin(\angle POS) = \frac{1}{2}(17^2)\sin(123.90^\circ) \approx 119.935 \text{ cm}^2$.

Eneo la sehemu iliyotiwa kivuli = Eneo la sekta $POS$ - Eneo la pembetatu $POS = 312.061 - 119.935 = 192.126 \text{ cm}^2$.
Kwa tarakimu muhimu 4:
Eneo la sehemu iliyotiwa kivuli ni $\boxed{\text{192.13 cm}^2}$.