please show steps. A small town has 4000 inhabitants. At 8 AM, 320 people have heard a rumor. By noon half the town has heard it. At what time will 90% of the population have heard the rumor? (Round the answer to one decimal place.
Use this equation to solve, y=yo/((1-yo)e^-kt +yo)
I don't see any differential equation. This is just logistical growth. And, you left out a scale factor. The general equation is
y = (Ly0)/(y0+(L-y0)e^(-kt))
Now just plug in your y0 and you get
y = 4000L/(4000+(L-4000)e^(-kt))
Now just plug in your given points:
4000L/(4000+(L-4000)e^(-8k)) = 320
4000L/(4000+(L-4000)e^(-12k)) = 2000
a quick google search will provide you with the actual differential equation.
To solve this problem, we can use the given equation:
y = yo / ((1-yo)e^(-kt) + yo)
where:
- y represents the proportion of the population that has heard the rumor at a specific time t,
- yo represents the initial proportion of the population that has heard the rumor,
- k is the growth constant.
We are asked to find the time when 90% of the population has heard the rumor, so our goal is to find t when y = 0.90.
To begin, let's plug in the given values into the equation:
yo = 320/4000 = 0.08
y = 0.90
Now, we need to find k. We can do this by using the information that by noon (at 12 PM), half the town has heard the rumor. Let's calculate the time difference between 8 AM and 12 PM:
12 PM - 8 AM = 4 hours
Since half the town (2000 people) has heard the rumor after 4 hours, we can substitute these values into the equation:
y = yo / ((1-yo)e^(-4k) + yo)
0.50 = 0.08 / ((1-0.08)e^(-4k) + 0.08)
Now, we can solve this equation to find the value of k.
To simplify the equation, let's divide both sides by 0.08:
0.50/0.08 = 1 / ((1-0.08)e^(-4k) + 0.08)
6.25 = 1 / ((0.92)e^(-4k) + 0.08)
Now, let's take the reciprocal of both sides of the equation:
1 / 6.25 = (0.92)e^(-4k) + 0.08
Let's subtract 0.08 from both sides to isolate (0.92)e^(-4k):
1 / 6.25 - 0.08 = (0.92)e^(-4k)
0.16 = (0.92)e^(-4k)
Now, let's divide both sides by 0.92:
0.16 / 0.92 = e^(-4k)
Let's take the natural logarithm (ln) of both sides to remove the exponential:
ln(0.16 / 0.92) = -4k
Now, we can solve for k by dividing both sides by -4:
k = ln(0.16 / 0.92) / -4
Using a calculator, we find that k is approximately 0.0693.
Now, we can substitute the value of k into the equation to find the time when 90% of the population has heard the rumor:
0.90 = 0.08 / ((1-0.08)e^(-0.0693t) + 0.08)
To solve for t, we can cross-multiply:
0.90 * ((1-0.08)e^(-0.0693t) + 0.08) = 0.08
Expand the equation:
0.90 * (0.92e^(-0.0693t) + 0.08) = 0.08
0.90 * 0.92e^(-0.0693t) + 0.90 * 0.08 = 0.08
0.828e^(-0.0693t) + 0.072 = 0.08
Subtract 0.072 from both sides:
0.828e^(-0.0693t) = 0.008
Now, divide both sides by 0.828:
e^(-0.0693t) = 0.008 / 0.828
Take the natural logarithm (ln) of both sides to remove the exponential:
ln(e^(-0.0693t)) = ln(0.008 / 0.828)
Simplify:
-0.0693t = ln(0.008 / 0.828)
Now, solve for t by dividing both sides by -0.0693:
t = ln(0.008 / 0.828) / -0.0693
Using a calculator, we find that t is approximately 4.05.
Therefore, 90% of the population will have heard the rumor at around 4.05 hours after 8 AM, which is approximately 12:05 PM (rounded to one decimal place).
To solve this problem, we can use the provided equation:
y = yo / ((1 - yo) * e^(-kt) + yo)
where:
- y is the proportion of the population that has heard the rumor,
- yo is the initial proportion of the population that has heard the rumor,
- k is a constant,
- t is the time in hours.
Let's break down the problem step by step:
Step 1: Find yo, the initial proportion that has heard the rumor.
We are told that at 8 AM, 320 people have heard the rumor out of a total population of 4000.
So the initial proportion yo = 320 / 4000 = 0.08.
Step 2: Find the constant k.
Given that at noon half the town has heard the rumor, and the initial proportion was 0.08, we can set up the equation:
0.5 = 0.08 / ((1 - 0.08) * e^(-k * 4) + 0.08)
Solving this equation will give us the value of k.
Step 3: Solve for t when y = 0.9.
Now, we need to determine at what time 90% of the population (y = 0.9) will have heard the rumor. We can use the equation:
0.9 = 0.08 / ((1 - 0.08) * e^(-k * t) + 0.08)
Rearrange this equation to solve for t:
(take the reciprocal of both sides)
1/0.9 = ((1 - 0.08) * e^(-k * t) + 0.08) / 0.08
Simplify further:
1.1111 = (0.92 * e^(-k * t) + 0.08) / 0.08
Multiply both sides by 0.08:
0.0889 = 0.92 * e^(-k * t) + 0.08
Subtract 0.08:
0.0089 = 0.92 * e^(-k * t)
Divide by 0.92:
0.009663 = e^(-k * t)
Take the natural logarithm of both sides:
ln(0.009663) = -k * t
Finally, solve for t:
t = ln(0.009663) / -k
Using this equation, you can calculate the value of t (time in hours) when 90% of the population will have heard the rumor.