In many population growth problems, there is an upper limit beyond which the population cannot grow. many scientists agree that the earth will not support a population of more than 16 billion. there were 2 billion people on earth in 1925 and 4 billion in 1975. if is the population years after 1925, an appropriate model is the differential equationdy/dt=ky(16-y)note that the growth rate approaches zero as the population approaches its maximum size. when the population is zero then we have the ordinary exponential growth described by y'=16ky. as the population grows it transits from exponential growth to stability.(a) solve this differential equation.(b) the population in 2015 will be(c) the population will be 9 billion some time in the year

a) (y-16)/y = -7*e∧(-0.016946*t)

b) y = 6.34

c) t = 129.66 years in 2055

Step-by-step explanation:

a) dy/dt = ky*(16-y)

Solving the differential equation we have

dy / (y*(y-16)) = -k dt

∫ dy / (y*(y-16)) = ∫ -k dt

(-1/16)*Ln (y) + (1/16)*Ln (y-16) = -k*t + C

(1/16) Ln ((y-16)/y) = -k*t + C

Ln ((y-16)/y) = -16*k*t + C  

(y-16)/y = C*e∧(-16*k*t)

If  t = 0  and y = 2

(2-16)/2 = C*e∧(0)  

C = -7   then we have

(y-16)/y = -7*e∧(-16*k*t)

In 1975 we have t = 1975 - 1925= 50 years   and  y = 4

(4-16)/4 = -7*e∧(-16*k*50)

k= - Ln (3/7) / 800 = 0.001059

Finally, the differential equation will be

(y-16)/y = -7*e∧(-16*0.001059*t)

(y-16)/y = -7*e∧(-0.016946*t)

b)  In 2015 we have t = 2015 – 1925 = 90 years

(y-16)/y = -7*e∧(-0.016946*90)

Solving the equation we get

y = 6.34

c) If  y = 9

(9-16)/9 = -7*e∧(-0.016946*t)

t = 129.66 years  in 2055

were is the equation

< xcy ≅ < xby

step-by-step explanation:

in δcax and δbax,

so, δcax ≅ δbax (side-angle-side or sas)

since δcax ≅ δbax, we can conclude < acx ≅ < abx

in δacy and δaby,

so, δacy ≅ δaby (side-angle-side or sas)

since δacy ≅ δaby, we can conclude < acy ≅ < aby

now,

< acy ≅ < aby

=> < acx + < xcy ≅ < abx + < xby

=> < acx + < xcy ≅ < acx + < xby   (since < acx ≅ < abx already proved above)

subtracting < acx from both sides, we get

< acx + < xcy -< acx ≅ < acx + < xby -< acx

cancelling out < acx and -< acx from both sides, we get

< xcy ≅ < xby (proved)