The population of a community of foxes is observed to fluctuate on a 10-year cycle due to variations in the availability of prey. When population measurements began (t=0, the population was 35 foxes. The growth rate in units of foxes>>year was observed to be P′(t)=5+10sinπt/5  a. What is the population 15 years later? 35 years later?
b. Find the population P(t) at any time t≥0.

Population 15 years later  P(15)  =  100 + 50/π

Population 35 years later P(35)  =  200 + 50/π

Population any t ≥ 0    P(t) = 35 + 50 /π + 5*t + 10*cos(π*t/5)  

Step-by-step explanation:

P´(t)  =  5  + 10*sinπt/5       ⇒    dP/dt =  5  + 10*sinπt/5    

dP = ( 5  + 10*sinπt/5  ) *dt

P(t) = ∫  ( 5  + 10*sinπt/5 ) dt

P(t)  =  5*t  +  10 * ∫ sinπ*t/5* dt

P(t)  =  5*t  -  10*5/π *cos πt/5 + k

To determine k    t = 0   P(t) = 35

P(0) = 5*0  - 50/π (1) + k

35 = - 50/π  + k

k  =  35  + 50/π     and

P(t)  =  5*t  + 10*cos(π*t/5) + 35 + 50/π

b)P(t) = 35 + 50 /π + 5*t + 10*cos(π*t/5)    (1)

a) Population 15 years later

P(15)  =  35 + 50/π  + 5*15 - 10

P(15)  =  100 + 50/π

Again from equation  (1)

P(35)  =  35 + 50 /π + 5*35 + 10*cos(35*π/5)

P(35)  =  35 + 50/π  + 175 + 10*cos (7*π )

P(35)  =  210 + 50/π  - 10

P(35)  =  200 + 50/π

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answer set (b) 0.5, 3/4 80% is in order from least to greatest.  

step-by-step explanation:

in order to compare rational numbers to each other, it is much easier if they are in the same form.   typically, the best form in which to evaluate rational numbers is in decimals.   in order to convert a percent to a decimal, you would divide the given percentage by 100.   you can also accomplish the conversion by moving the decimal point over two places to the left.   for example, the decimal value of 80% is 80/100, or 0.8.   in order to convert a fraction to a decimal, you need to divide the numerator by the denominator.   for example, 3/4 is 0.75.   so, for choice (b) 0.5 3/4 (0.75), 80% (0.8) is the only choice in which the numbers are in order from least to greatest.