4875567755×44576456667
2 answers:
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Answer:
- The smaller one is x = 3
- and the larger one is x = 6
Step-by-step explanation:
A "critical number" is a function argument where the function has zero or undefined slope. A polynomial function never has undefined slope, so we're looking for the x-values where f'(x) = 0.
A graphing calculator easily shows the points that have zero slope. They are found at x = 3 and x = 6.
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Setting the derivative to zero is another way to find the points with zero slope:
f'(x) = 6x^2 -54x +108 = 6(x^2 -9x +18) = 6(x-3)(x-6)
The derivative will be zero where the factors are zero, at x=3 and x=6.
The critical numbers are x=3 and x=6.
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.
