1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Zolol [24]
3 years ago
8

An advertising company designs a campaign to introduce a new product to a metropolitan area of population 3 Million people. Let

P(t) denote the number of people (in millions) who become aware of the product by time t. Suppose that P increases at a rate proportional to the number of people still unaware of the product. The company determines that no one was aware of the product at the beginning of the campaign, and that 50% of the people were aware of the product after 50 days of advertising. The number of people who become aware of the product at time t is:
Mathematics
1 answer:
Advocard [28]3 years ago
5 0

Answer:

P(t)=3,000,000-3,000,000e^{0.0138t}

Step-by-step explanation:

Since P(t) increases at a rate proportional to the number of people still unaware of the product, we have

P'(t)=K(3,000,000-P(t))

Since no one was aware of the product at the beginning of the campaign and 50% of the people were aware of the product after 50 days of advertising

<em>P(0) = 0 and P(50) = 1,500,000 </em>

We have and ordinary differential equation of first order that we can write

P'(t)+KP(t)= 3,000,000K

The <em>integrating factor </em>is

e^{Kt}

Multiplying both sides of the equation by the integrating factor

e^{Kt}P'(t)+e^{Kt}KP(t)= e^{Kt}3,000,000*K

Hence

(e^{Kt}P(t))'=3,000,000Ke^{Kt}

Integrating both sides

e^{Kt}P(t)=3,000,000K \int e^{Kt}dt +C

e^{Kt}P(t)=3,000,000K(\frac{e^{Kt}}{K})+C

P(t)=3,000,000+Ce^{-Kt}

But P(0) = 0, so C = -3,000,000

and P(50) = 1,500,000

so

e^{-50K}=\frac{1}{2}\Rightarrow K=-\frac{log(0.5)}{50}=0.0138

And the equation that models the number of people (in millions) who become aware of the product by time t is

P(t)=3,000,000-3,000,000e^{0.0138t}

You might be interested in
Which angles are linear pairs? Check all that apply.
faust18 [17]

Answer:

A,C,D

Step-by-step explanation:

8 0
3 years ago
Find the slope of each line.
zheka24 [161]

Answer:

D

Step-by-step explanation:

6 0
3 years ago
What are the vertex, focus, and directrix of the parabola with the equation x2+8x+4y+4=0
Novay_Z [31]
<span>x2 +8x +4y +4 = 0
</span>4y=<span> -x2 -8x -4</span>y = -.25*x^2 -2x -1
a = -.25b = -2c = -1
x position of vertex:
h = -b / 2a
h = 2 / 2*-.25h = 2 / -.5h = -4
y position of vertex:
k = ah^2 + bh + ck = -.25*-4^2 + -2*-4 + -1k = -4 +8 -1k = 3
VERTEX = (-4, 3)**************************************************************************
x value of focus =x value of vertex = -4
y value of focus =(1 (-b^2 -4ac)) / 4a

a = -.25 b = -2 c =-1
y value = (1 (-4 -4*-.25*-1)) / 4*-.25
y value = (1 (-4 -4*-.25*-1)) / -1
y value = (1 -4 +1) / -1y value = (-2 / -1)y value = 2
focus value = (-4, 2)
Answer is the last one.

7 0
3 years ago
Choose the correct simplification of the expression (-11x^3y^12z^3)^0
Maurinko [17]

Answer:

Step-by-step explanation:

any thing raised to 0 will equal 1

6 0
4 years ago
What is 15.09 ÷ 0.01 = _______<br><br> This is a 5th grade question
Paladinen [302]

Answer:

1509

Step-by-step explanation:

3 0
3 years ago
Read 2 more answers
Other questions:
  • Which line is steeper, y=7/4x+5 or y=5/3x-2? ​
    7·2 answers
  • Solve for x. 3x−8/2=x−6 Enter your answer in the box.
    6·1 answer
  • Can someone please help me??? :) Please?
    14·2 answers
  • A theater sold 980 tickets for a ballet. The orchestra seats cost $50 each and the balcony seats cost $35 each. The theater coll
    11·2 answers
  • PLEASE HELP ME PLEASE HURRY
    7·1 answer
  • A ladder 15 metres long leans against a vertical wall so that the top of the ladder is 12 metres above the ground. What angle do
    13·1 answer
  • Which graph represents the function f(x) = x– 4?
    11·2 answers
  • The length of a rectangle is 4yd longer than its width. If the perimeter of the rectangle is 72yd, find its length and width.
    7·1 answer
  • Rotate shape A 90° anti-clockwise with centre of rotation (0,0).
    5·1 answer
  • Someone pls help me ASAP! Express the following as the sum of its partial fraction: 2x+4/x(x+2)(x-4)​
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!