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
A door delivery florist wishes to estimate the proportion of people in his city that will purchase his flowers. Suppose the true
kvv77 [185]

Answer:

99.74% probability that the sample proportion will be less than 0.1

Step-by-step explanation:

I am going to use the binomial approximation to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

\sqrt{V(X)} = \sqrt{np(1-p)}

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that \mu = E(X), \sigma = \sqrt{V(X)}.

In this problem, we have that:

n = 276, p = 0.06

So

\mu = E(X) = np = 276*0.06 = 16.56

\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{276*0.06*0.94} = 3.9454

What is the probability that the sample proportion will be less than 0.1

This is the pvalue of Z when X = 0.1*276 = 27.6. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{27.6 - 16.56}{3.9454}

Z = 2.8

Z = 2.8 has a pvalue of 0.9974

99.74% probability that the sample proportion will be less than 0.1

5 0
3 years ago
I need your help if the airplane was at altitude of 800 feet above sea level. It landed at airport 45 feet below sea level. Whic
kipiarov [429]

Answer:

800 ft above sea level

Step-by-step explanation:

6 0
3 years ago
The product of x and 6 is less than or equal to 24.
iren2701 [21]

Answer:

x= 4 or <

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
How much time will it take for a bug to travel 5m across the floor if it is traveling at 0.5m/s?
goldenfox [79]

Answer:

10 seconds

Step-by-step explanation:

Every second the bug travels 0.5m/s or 0.5 meters per second.

Given that, every 2 seconds the bug will travel a full 1 meter.

To solve this one can simply divide 5m by 0.5 and get 10 seconds.

5 0
3 years ago
Round 6,567 to the nearest hundred.
riadik2000 [5.3K]

Answer:

6,600

\sf Rounding~rules

  • 4 or below? Round down or keep the same.
  • 5 or above? Round up.
4 0
1 year ago
Other questions:
  • Please help me with this
    12·1 answer
  • tennis balls with a diameter of 2.5in are sold in cans of three. the can is a cylender. what is the volume of the area not occup
    11·1 answer
  • How do you solve 144= -12(x+5)
    9·2 answers
  • I'LL GIVE BRAINLIEST !!! FASTER<br><br>please explain how do you get the answer​
    6·2 answers
  • Now, if you solve for x, what is the result?
    13·1 answer
  • Suppose a city with population 900,000 has been growing at a rate of ​8% per year. If this rate​ continues, find the population
    10·1 answer
  • Enter the correct letter to match each summation expression with the property or formula. N Σ i=1 cai is equal to . A n(n + 1)(2
    7·1 answer
  • A farmer was counting his chickens and dogs. He counted a total of 60 legs. If the amount of chickens he has is four times the a
    15·1 answer
  • I need help with this question for math. I would love for some help thank you.
    14·2 answers
  • write two equations that when graphed look like this *see image*. please help its due soon and i will give ya 75 points for answ
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!