A swimming pool is to be drained for cleaning. If L represents the number of liters of water in the pool t seconds after the pool has been plugged off to drain and L = 200(10 – t)², how fast is the water running out at the end of 5 seconds? What is the average rate at which the water flows out during the first 5 seconds?

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Alok Kumar Singh | Contributor-Level 10

8 months ago

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol.

$\begin{array}{l} G i v e n t h a t L = 200 {(10 - t)}^{2} \\ w h e r e L r e p r e s e n t s t h e n u m b e r o f l i t r e s o f w a t e r i n t h e p o o l . \\ D i f f e r e n t i a t i n g b o t h s i d e s w . r . t ., t, w e g e t \\ \frac{d L}{d t} = 200 * 2 (10 - t) (- 1) = - 400 (10 - t) \\ B u t t h e r a t e a t w h i c h t h e w a t e r i s r u n n i n g o u t \\ = - \frac{d L}{d t} = 400 (10 - t) \dots (i) \\ R a t e a t w h i c h t h e w a t e r i s r u n n i n g a f t e r 5 seconds \\ = 400 (10 - 5) = 2000 L / s (f i n a l r a t e) \\ F o r i n i t i a l r a t e p u t t = 0 \\ = 400 (10 - 0) = 4000 L / s \\ T h e a v e r a g e r a t e a t w h i c h t h e w a t e r i s r u n n i n g o u t \\ = \frac{I n i t i a l r a t e + F i n a l r a t e}{2} = \frac{4000 + 2000}{2} = \frac{6000}{2} = 3000 L / s \\ H e n c e, t h e r e q u i r e d r a t e = 3000 L / s . \end{array}$

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