Let w(t) be the number of words that a parrot knows t days after you start teach

Question

Let w(t) be the number of words that a parrot knows t days after you start teaching it. Then, suppose that the number of words grows at a rate of dw/dt = k(-M - w), where M is the number of words the parrot knows initially, and k is some constant. (a) Find a general equation for w(t). (b) Suppose you have a parrot named Diana. Diana knows how to say 'hello', 'goodbye', 'food', and 'shenanigans' before you start to teach her. What is the particular solution w(t) for Diana? (c) Suppose that after one week, Diana knows 24 words. What must the value of k be?

Explanation / Answer

From the given question,

dw/dt= k(-M-w)

dw/(-M-w)=k dt

integrating both sides,

-ln(M+w)=kt +C

when t=0, w=M

-ln(M+M)=0+C

C=-ln(2M)

-ln(M+w)=kt +-ln(2M)

ln[(2M)/(M+w)]=kt

[(2M)/(M+w)]=ekt

(M+w)/2M=e-kt

(1/2 + w/2M)=e-kt

e-kt - 1/2= w/2M

w= 2M(e-kt - 1/2)

(b) replacing M=4

w= 8(e-kt - 1/2)

c) 24=8(e-7k - 1/2)

3=e-7k - 1/2

3.5=e-7k

ln(3.5)=-7k

k=(-1/7)ln(3.5)

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.