It is estimated that t-weeks into a semester, the average amount of sleep a college math student gets per day s(t) decreases at a rate of −3tet2 hours per day. when the semester begins, math students sleep an average of 8.1 hours per day. what is s(t), 15 week(s) into the semester?

s(15)=

Step-by-step explanation:

Before to getting started ,we have to consider the function which  represents  sleep decline rate S'(t) =

Since  this function is interpreted in this case as a rate of change, we can deduce this funcion is the first derivative of the function of  average amount of sleep (S(t)).

In order to find S(t), we will integrate S'(t)

This case corresponds to intergration by substitution  

Substituting u=

Computting the differential of u

Now, let's go back to our integral and write it in terms of u, using both definition of u and its defferential.

The integral  of a exponential function is itself. So the result is:

Now substitute u back in:

This answer correspond to function of average amount of sleep.

We know that when the semester begins (t=0) , math students sleep an average of 8.1 hours per day so we will use this info in order to find the value of c.

Solve this equation

c=8.1+

c=9.6

Replacing  the value of c on S(t)

S(t)=

Now, let's to find the average of amount of sleep   when 15 weeks has elapsed since the start date of the semester.

s(15)=

answer:  

step-by-step explanation:

odd numbers are only divisible by 1 and itself.

even numbers are divisible by more than 1 and itself.

for example 6 is even because it is divisible by more than 1 and its self. 1 x 6= 6 ; 2 x 3= 6

and 7 is odd because only 7 x 1 equals seveb

substitute,

circle 1 = 5

circle 2 = 20

circle 3 = 44

in this case no matter how you add, you will always get a perfect square..

hence proved

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