Question

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, left 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.

Answer 1

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  

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