Point) a norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the wi
dth of the rectangle. find the dimensions w×hw×h of the norman window whose perimeter is 1000in.1000in. that has maximal area.
1 answer:
Let d represent the diameter of the semicircle (and the width of the window). Then the length of the semicircular arch is π/2*d. The remaining perimeter available for the height of the rectangular shape is then
.. 1000 -(d +π/2*d)
and that amount will be divided between the two sides of the window.
Then the height of the rectangular section is
.. h = (1/2)*(1000 -d(1 +π/2))
.. = 500 -d(1/2 +π/4)
The total area of the window is ...
.. A = dh +(1/2)(π/4)d^2
.. = d(500 -d(1/2 +π/4)) +π/8d^2
.. = 500d +d^2*(π/8 -π/4 -1/2)
.. = d(500 -d(1/2 +π/8))
This downward-opening quadratic function has its vertex (maximum) at
.. d = 500/(1 +π/4) = 2000/(4 +π)
The corresponding rectangle height is
.. h = 500 -500/(1 +π/4)*(1/2 +π/4)
.. = 500(1 - (2 +π)/(4 +π)) = 500(4 +π -2 -π)/(4 +π) = 1000/(4 +π)
That is, the height of the rectangular section is 1/2 the diameter of the circular section, so the overall window has the same overall height and width.
The height and width of the window with maximum area are ...
2000/(4+π) in ≈ 280.0 in
_____
The graph shows both the area function and the (d, h) dimensions of the rectangle.
.. 1000 -(d +π/2*d)
and that amount will be divided between the two sides of the window.
Then the height of the rectangular section is
.. h = (1/2)*(1000 -d(1 +π/2))
.. = 500 -d(1/2 +π/4)
The total area of the window is ...
.. A = dh +(1/2)(π/4)d^2
.. = d(500 -d(1/2 +π/4)) +π/8d^2
.. = 500d +d^2*(π/8 -π/4 -1/2)
.. = d(500 -d(1/2 +π/8))
This downward-opening quadratic function has its vertex (maximum) at
.. d = 500/(1 +π/4) = 2000/(4 +π)
The corresponding rectangle height is
.. h = 500 -500/(1 +π/4)*(1/2 +π/4)
.. = 500(1 - (2 +π)/(4 +π)) = 500(4 +π -2 -π)/(4 +π) = 1000/(4 +π)
That is, the height of the rectangular section is 1/2 the diameter of the circular section, so the overall window has the same overall height and width.
The height and width of the window with maximum area are ...
2000/(4+π) in ≈ 280.0 in
_____
The graph shows both the area function and the (d, h) dimensions of the rectangle.
You might be interested in
Answer:
Between 0.01 and 0.20
Step-by-step explanation:
I am going to use the normal approximation to the binomial to solve this question.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
The standard deviation of the binomial distribution is:
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that ,
.
In this problem, we have that:
So
. So this is 1 subtracted by the pvalue of Z when X = 29.5. Then
has a pvalue of 0.8212
1 - 0.8212 = 0.1788
So the correct option is:
Between 0.01 and 0.20