Two cylindrical containers, C and D, with different capacities are shown below. What is the total volume of liquid held in these
containers if half of container C is full and one third of container D is full?
1 answer:
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Answer:
The point estimate for population mean is 8.129 Mpa.
Step-by-step explanation:
We are given the following in the question:
Data on flexural strength(MPa) for concrete beams of a certain type:
11.8, 7.7, 6.5, 6.8, 9.7, 6.8, 7.3, 7.9, 9.7, 8.7, 8.1, 8.5, 6.3, 7.0, 7.3, 7.4, 5.3, 9.0, 8.1, 11.3, 6.3, 7.2, 7.7, 7.8, 11.6, 10.7, 7.0
a) Point estimate of the mean value of strength for the conceptual population of all beams manufactured
We use the sample mean, as the point estimate for population mean.
Formula:
Thus, the point estimate for population mean is 8.129 Mpa.
Derivative of the denominator:
Hmm our numerator is 2x+7. Ok this let's us know that a simple u-substitution is NOT going to work. But let's apply some clever Algebra to the numerator splitting it up into two separate fractions. Split the +7 into +2 and +5.
and then split the fraction,
Based on our previous test, we know that a simple substitution will work for the first integral:
So the first integral changes,
integrating to a log,
Other one is a little tricky. We'll need to complete the square on the denominator. After that it will look very similar to our arctangent integral so perhaps we can just match it up to the identity.
So we have this going on,
Let's factor the 5 out of the intergral,
and the 4 from the denominator,
Bringing all that stuff together as a single square,
Making the substitution:
giving us,
simplying a lil bit,
and hopefully from this point you recognize your arctangent integral,
undo your substitution as a final step,
and include a constant of integration,
Hope that helps!
Lemme know if any steps were too confusing.