Mathematics

The given figure shows the cross-section of a water channel consisting of a rectangle and a semi-circle. Assuming that the channel is always full, find the volume of water discharged through it in one minute if water is flowing at the rate of 20 cm per second. Give your answer in cubic meters correct to one place of decimal.

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Let radius of semi-circle be r cm.

From figure,

⇒ 2r = 21 cm

⇒ r = $\dfrac{21}{2}$ cm.

Area of cross-section of water channel = l × b + $\dfrac{1}{2}πr^2$

= 21 × 7 + $\dfrac{1}{2} \times \dfrac{22}{7} \times \dfrac{21}{2} \times \dfrac{21}{2}$

= 147 + $\dfrac{693}{4}$

= $\dfrac{588 + 693}{4} = \dfrac{1281}{4}$

= 320.25 cm².

Length of water column = Water flowing rate × Time

= 20 cm/s × 60 s

= 1200 cm.

Volume of water discharged = Area of cross-section of water channel × Length of water column

= 320.25 × 1200

= 384300 cm³

= $384300 \times \Big(\dfrac{1}{100}\Big)^3$ m³

= 0.3843

≈ 0.4 m³.

Hence, volume of water discharged in one minute = 0.4 m³.

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