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Katyanochek1 [597]
3 years ago
9

Hungry Harry is a giant ogre with an even bigger appetite. After Harry wakes up from hibernation, his daily hunger H(t)H(t)H, le

ft parenthesis, t, right parenthesis (in \text{kg}kgstart text, k, g, end text of pigs) as a function of time ttt (in hours) can be modeled by a sinusoidal expression of the form a\cdot\cos(b\cdot t)+da⋅cos(b⋅t)+da, dot, cosine, left parenthesis, b, dot, t, right parenthesis, plus, d. When Harry wakes up at t=0t=0t, equals, 0, his hunger is at a maximum, and he desires 30 \text{ kg}30 kg30, start text, space, k, g, end text of pigs. Within 222 hours, his hunger subsides to its minimum, when he only desires 15 \text{ kg}15 kg15, start text, space, k, g, end text of pigs. Find H(t)H(t)H, left parenthesis, t, right parenthesis. \textit{t}tstart text, t, end text should be in radians.

Mathematics
1 answer:
MAXImum [283]3 years ago
3 0

Complete Question

The complete question is shown on the first uploaded image

Answer:

Harry's  hunger is  H(t) =  7.5 cos  (\frac{\pi}{2} t ) + 22.5

Step-by-step explanation:

From the question we are told

 The  hunger of harry is  H(t) =  a \cdot  cos (b \cdot t ) + d    

 The hunger of harry is  maximum at  t =  0  

 The hunger of harry is minimum at  t =  2 hours

 The mass of pig desires at maximum hunger is H(0) =  30 \ kg

 The mass of  pig desires at minimum hunger is  H(2) =  15 \  kg

Generally at maximum hunger we have

        H(0) =  a *  cos (b* 0 )  +d  =  30

=>       a *  cos (0 )  +d  =  30

=>       a + d  =  30 --- [1]

Now  at minimum hunger

          H(2) =  a*  cos  (2 b )  + d  =  15

          a*  cos  (2 b )  + d  =  15

Generally the  minimum value of  cos (\theta )   =  -1

So

  At minimum

                 cos (2b)  =  -1

=>              2b  =  cos^{-1} [-1]      

Generally from trigonometric rules

                cos^{-1} [-1] =  (2n + 1 )\pi    here  n =  0 , 1 , 2 , 3  

Now since we are considering the minimum  =  0  

   So  

           cos^{-1} [-1] =  (2(0)+ 1 )\pi

=>        cos^{-1} [-1] = \pi

So

           2b  = \pi

=>         b =  \frac{\pi }{2 }      

So

       a*  cos  (\pi )  + d  =  15

=>     -a +  d =  15 --- [2]

Adding  equation\  (1 ) \ and  \  (2)

     =>  \   2d  =  45

      =>  d =  22.5  

From equation 1

        a + 22.5  =  30

=>     a =  7.5

So we can represent the harry hunger as  

       H(t) =  7.5 cos  (\frac{\pi}{2} t ) + 22.5

       

   

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