Question Paper of Calculus - Rates of Change

Question 1:

$T h e d i r e c t i o n (s l o p e) f i e l d f o r a c e r t a i n f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n i s s h o w n b e l o w . W h i c h o f t h e f o l l o w i n g b e l o w i s t h e d i f f e r e n t i a l e q u a t i o n ?$

Question 2:

$J a n e i s i n f l a t i n g b a l l o o n s f o r t h e Y e a r 12 F o r m a l . E a c h e m p t y b a l l o o n i s b e i n g i n f l a t e d s o t h a t i t s v o l u m e i n c r e a s e s a t t h e r a t e o f 8 c m^{3} / s . (i) S b o w t h a t t h e r a d i u s a t a n y t i m e t i s r = \sqrt[3]{\frac{6 t}{π}} (i i) F i n d t h e r a t e o f i n c r e a s e o f t h e s u r f a c e a r e a a f t e r 4 s e c o n d s (i i i) T h e b a l l o o n w i l l b u r s t w h e n t h e s u r f a c e a r e a r e a c h e s 3000 c m^{2} . A f t e r h o w m a n y s e c o n d s s h o u l d J a n e c e a s e i n t l a t i o n ?$

Question 3:

$M o l t e n p l a s t i c a t a t e m p e r a t u r e o f 250 ° C, i s p o u r e d i n t o a m o u l d t o f o r m a c a r p a r t . A f t e r 20 m i n u t e s t h e p l a s t i c h a s c o o l e d t o 150 ° C . I f t h e t e m p e r a t u r e a f t e r t m i n u t e s, i s T ° C, a n d t h e s u r r o u n d i n g t e m p e r a t u r e i s 30 ° C, t h e n t h e r a t e o f c o o l i n g i s g i v e n b y : \frac{d T}{d t} = - k (T - 30) w h e r e k i s c o n s \tan t . (i) S h o w t h a t T = 30 + A e^{- k t}, w h e r e A i s a c o n s \tan t, s a t i s f i e s t h i s e q u a t i o n . (i i) S h o w t h a t t h e v a l u e o f A i s 220 ° C . (i i i) F i n d t h e v a l u e o f k t o 2 d e c i m a l p l a c e s . (i v) T h e p l a s t i c c a n b e t a k e n o u t o f t h e m o u l d w h e n t h e t e m p e r a t u r e d r o p s b e l o w 80 ° C . H o w l o n g a f t e r t h e p l a s t i c h a s b e e n p o u r e d w i l l t h i s t e m p e r a t u r e b e r e a c h e d ? G i v e y o u r a n s w e r t o t h e n e a r e s t m i n u t e .$

Question 4:

$A p a r t i c l e m o v e s i n a s t r a i g h t l i n e s o t h a t i t s a c c e l e r a t i o n, a, i s g i v e n b y a = 4 x T h e d i s p l a c e m e n t, x, o f t h e p a r t i c l e i s i n i t i a l l y 1 m e t r e t o t h e r i g h t o f t h e o r i g i n w i t h a v e l o c i t y o f - 2 m s^{- 1} (i) S h o w t h a t v = - 2 x m s^{- 1} (i i) E x p r e s s x a s a f u n c t i o n o f t . (i i i) H e n c e, f i n d t h e d i s p l a c e m e n t w h e n t = 2 s e c o n d s t o 3 d e c i m a l p l a c e s .$

Question 5:

Water is running out of a filled conical funnel at the rate of 3 1 5cm s . The radius of the funnel is 10 cm and the height is 20 cm:

How fast is the water level dropping when the water is 10 cm deep? (answer in exact form)

How long does it take for the water to drop to 10 cm deep? 2 (answer to 2 decimal places)

Question 6:

$During a soccer tournament, Juan is standing 25 m away from the goal line . He kicks a soccer ball off the ground at an angle of 30 ° to the horizontal with an initial velocity of V m per \sec . The ball hits the top bar which is 2.4 m directly above the goal line . Neglecting air resistance and assuming that acceleration due to gravity is 10 m / s^{2}, find : i) The horizontal and vertical components of displacement of the ball in terms of the initial velocity V . ii) The Cartesian equation of the motion for the path of the ball . iii) The initial velocity of the ball, correct to 1 decimal place .$

Question 7:

$O n a c e r t a i n d a y t h e d e p t h o f w a t e r i n a h a r b o u r i s 7 m e t r e s a t l o w t i d e (9 : 25 a m) a n d 10 \frac{2}{3} m e t r e s a t h i g h t i d e (3 : 40 p m) . A s s u m e t h e r i s e a n d f a l l o f t h e s u r f a c e o f t h e w a t e r t o b e i n s i m p l e h a r m o n i c m o t i o n i n t h e f o r m \ddot{x} = - n^{2} (x - b) w h e r e x = b i s t h e c e n t r e o f m o t i o n a n d x = a i s t h e a m p l i t u d e . (i) S h o w t h a t x = b - a \cos n t s a t i s f i e s \ddot{x} = - n^{2} (x - b) (i i) F i n d t h e v a l u e s o f a, b a n d n . (i i i) H e n c e, f i n d t h e e a r l i e s t t i m e b e f o r e 3 : 40 p m o n t h i s d a y, a b o a t m a y s a f e l y e n t e r t h e h a r b o u r i f t h e m i n i m u m d e p t h o f 9 \frac{1}{2} m e t r e s o f w a t e r i s r e q u i r e d .$

Question 8:

$A b u l l e t i s f i r e d h o r i z o n t a l l y w i t h a v e l o c i t y o f 100 m s - 1 f r o m t h e t o p o f a t o w e r 105 m e t r e s h i g h . T h e t o w e r i s a t t h e t o p o f a h i l l, w h i c h s l o p e s d o w n w a r d s a t a n a n g l e o f d e p r e s s i o n o f \frac{π}{4} . T h e b u l l e t l a n d s a t L, o n t h e g r o u n d a s r e p r e s e n t e d o n t h e d i a g r a m b e l o w . (i) C o n s i d e r B, t h e b a s e o f t h e t o w e r, a s t h e o r i g i n, a n d u \sin g a c c e l e r a t i o n d u e t o g r a v i t y a s 10 m s^{- 1} s h o w t h a t t h e e x p r e s s i o n s f o r t h e x a n d y c o o r d i n a t e s o f t h e p o s i t i o n o f t h e b u l l e t a t t i m e t s e c o n d s a r e x = 100 t a n d y = 105 - 5 t^{2} (i i) S h o w t h a t t h e e q u a t i o n o f t h e l i n e B L i s y = - x (i i i) F i n d t h e t i m e t a k e n f o r t h e b u l l e t t o h i t t h e g r o u n d a t L . (i v) F i n d t h e d i s \tan c e B L t o t h e n e a r e s t m e t r e .$

Question 9:

$i) The acceleration in {ms}^{- 2} of an object is given by \ddot{x} = 2 x^{3} + 4 x . If the object is initially 2 m to the right of the origin travelling with velocity 8 {ms}^{- 1}, find an expression for v^{2} (the square of its velocity) in terms of x . ii) What is the minimum speed of the object ? Give a reason for your answer .$

Question 10:

$I n a n O l y m p i c t r i a l, a s h o t p u t t e r r e l e a s e s t h e s h o t f r o m a h e i g h t o f 2.4 m e t r e s a b o v e g r o u n d l e v e l a t a n a n g l e o f 40^{°} t o t h e h o r i z e n t a l, a n d w i t h a s p e e d o f 12 m e t r e s p e r s e c o n d . T a k e t h e o r i g i n O a t a p o i n t o n t h e g r o u n d d i r e c t l y u n d e r t h e p o i n t o f r e l e a s e o f t h e s h o t . (i) U \sin g c a l u l u s, s h o w t h a t t h e p o s i t i o n o f t h e s h o t a t t i m e t s e c o n d s i s g i v e n b y x = 12 \cos 40^{°} t a n d y = 2.4 + 12 \sin 40^{°} t - \frac{1}{2} g t^{2} . (i i) T h e s h o t l a n d s a t a p o i n t A o n t h e g r o u n d . F i n d t h e l e n g t h o f O A i n m e t r e s, c o r r e c t t o o n e d e c i m a l p l a c e . (t a k e g = 9.8) .$