The volume as a function of the location of that vertex is
... v(x, y, z) = x·y·z = x·y·(100-x²-y²)
This function is symmetrical in x and y, so will be a maximum when x=y. That is, you wish to maximize the function
... v(x) = x²(100 -2x²) = 2x²(50-x²)
This is a quadratic in x² that has zeros at x²=0 and x²=50. It will have a maximum halfway between those zeros, at x²=25. That maximum volume is
... v(5) = 2·25·(50-25) = 1250
The maximum volume of the box is 1250 cubic units.
Answer:
The interval is constructed at 93% confidence.
Step-by-step explanation:
Confidence interval concepts:
A confidence interval has two bounds, a lower bound and an upper bound.
A confidence interval is symmetric, which means that the point estimate used is the mid point between these two bounds, that is, the mean of the two bounds.
The margin of error is the difference between these two bounds, divided by 2.
Confidence interval of proportions concepts:
In a sample with a number n of people surveyed with a probability of a success of , and a confidence level of
, we have the following confidence interval of proportions.
In which
z is the zscore that has a pvalue of .
The margin of error is:
In this problem, we have that:
2050 people, so n = 2050.
Lower bound: 0.878
Upper bound: 0.903
Confidence level:
We have to find z.
has a pvalue of 0.965.
That is:
Finally
The interval is constructed at 93% confidence.