Domain of ln(2x+3/×-1)
1 answer:
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Answer:
305 °F
Step-by-step explanation:
The core temperature of the object after 4 hours can be found using an exponential decay formula to model the decay of the difference between core temperature and ambient.
<h3>Cooling Model</h3>The solution to the differential equation described by Newton's law of cooling is the exponential equation ...
y = ab^t +c
where 'a' is the initial core temperature difference from ambient, 'b' is the decay factor of that difference in 1 unit of time period t. 'c' is the ambient temperature.
For this problem, the ambient temperature is c=80, and the differences of interest are ...
a = 1200 -80 = 1120
b = (830 -80)/1120 = 75/112
Using these values in the model gives ...
y = 1120(75/112)^t +80 . . . . . . where y(t) is the core temperature at time t
Note that units of time are hours.
<h3>Application</h3>We want y when t=4.
y = 1120(75/112)^4 +80 ≈ 1120(0.20108) +80 ≈ 305.212
The core temperature after 4 hours is about 305 °F.
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<em>Additional comment</em>
The differential equation will have a solution of the form ...
where k = ln(75/112) ≈ -0.40101
In the above, we defined b = e^k = 75/112. Accuracy with this fraction can be better than using a truncated value of k.
To solve our questions, we are going to use the kinematic equation for distance: 
where
is distance
is speed
is time
1. Let
be the speed of the wind, 
be time of the westward trip, and 
the time of the eastward trip. We know from our problem that the distance between the cities is 2,400 miles, so 
. We also know that the speed of the plane is 450 mi/hr, so 
. Now we can use our equation the relate the unknown quantities with the quantities that we know:
<span>Going westward:
The plane is flying against the wind, so we need to subtract the speed of the wind form the speed of the plane:
</span>
Going eastward:
The plane is flying with the wind, so we need to add the speed of the wind to the speed of the plane:
We can conclude that you should complete the chart as follows:
Going westward -Distance: 2400 Rate:
Time:
Going eastward -Distance: 2400 Rate:
Time:
2. Notice that we already have to equations:
Going westward: equation(1)
Going eastward: equation (2)
Let
be the time of the round trip. We know from our problem that the round trip takes 11 hours, so 
, but we also know that the time round trip is the time of the westward trip plus the time of the eastward trip, so 
. Using this equation we can express in terms of :
equation
equation (3)
Now, we can replace equation (3) in equation (1) to create a system of equations with two unknowns:
We can conclude that the system of equations that represent the situation if the round trip takes 11 hours is:
equation (1)
equation (2)
3. Lets solve our system of equations to find the speed of the wind:
equation (1)
equation (2)
Step 1. Solve for in equation (2)
equation (3)
Step 2. Replace equation (3) in equation (1) and solve for
:
We can conclude that the speed of the wind is 78 mi/hr.
where
1. Let
<span>Going westward:
The plane is flying against the wind, so we need to subtract the speed of the wind form the speed of the plane:
</span>
Going eastward:
The plane is flying with the wind, so we need to add the speed of the wind to the speed of the plane:
We can conclude that you should complete the chart as follows:
Going westward -Distance: 2400 Rate:
Going eastward -Distance: 2400 Rate:
2. Notice that we already have to equations:
Going westward:
Going eastward:
Let
Now, we can replace equation (3) in equation (1) to create a system of equations with two unknowns:
We can conclude that the system of equations that represent the situation if the round trip takes 11 hours is:
3. Lets solve our system of equations to find the speed of the wind:
Step 1. Solve for
Step 2. Replace equation (3) in equation (1) and solve for
We can conclude that the speed of the wind is 78 mi/hr.