#$&*
Mth 174
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Work to Lift Weight With Chain
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Problem Number 6
Use Riemann Sums to obtain the integral required to solve the following, and evaluate the integral: How much work is required to lift a weight of 300 pounds a vertical distance of 23 feet using a chain that weighs 3 lb/ft?
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F(x) = sum( weight of chain + weight of the weight)
= 3(23 - x) + 300
= 69 - 3x + 300
= 369 - 3x
W = integral(369 - 3x dx, x, 0, 23)
= 369x - 3 x^2 from 0 to 23
= 6900 ft/lb
Is this solution correct?
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You have the right integral but no Riemann Sum.
The typical interval has sample point x_i* and length `dx.
When the weight is at position x_i*, the length of chain below the 23 foot mark is 23 ft - x_i*, so the weight of that section is 3 lb / ft * (23 ft - x_i*) and the work to raise it through the increment `dx is
`dW_i = (300 + 3 ( 23 - x_i*) ) * `dx
where the units have not been included.
The Riemann sum is therefore
sum( (300 + 3(23 - x_i*)) `dx)
which approaches the integral you have given, except that you need an additional sign of grouping:
W = integral( (369 - 3x) dx, x, 0, 23)
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#$&*
Mth 174
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Temperature Change Question
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Problem Number 7
If an object's temperature moves 35% closer to the 24 degree temperature of a room in any 6-minute time period, then if the object's initial temperature is 42.8 degrees, what function models the temperature as a function of time? What will be the temperature after 34 minutes?
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I'm not sure what the differential equation is for this problem?
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The rate at which and object's temperature approaches room temperature is proportional to the difference between the object's temperature and that of the room.
That rate is dT/dt.
The temperature difference is (T - T_room).
So the equation is
dT/dt = k (T - T_room).
If you solve the equation, which you can easily do using separation of variables, you get
T = T_room + C e^(kt)
The given conditions allow you to solve for C and k.
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