Upper A 4-ft-tall fence runs parallel to a wall of a house at a distance of 24 ft. (a) Find the length of the shortest ladder t
hat extends from the ground to the house without touching the fence. Assume the vertical wall of the house and the horizontal ground have infinite extent. (a) Let L be the length of the ladder, x be the distance from the base of the ladder to the fence, d be the distance from the fence to the house and b be the distance from the ground to the point the ladder touches the house. What is the objective function, in terms of x?
1 answer:
Answer:
(a) shortest ladder length ≈ 35.7 ft (rounded to tenth)
(b) L = (d/x +1)√(16+x²) . . . . where 16 is fence height squared
Step-by-step explanation:
It works well to solve the second part of the problem first, then put in the specific numbers.
We have not been asked anything about "b", so we can basically ignore it. Using the Pythagorean theorem, we find the length GH in the attached drawing to be ...
GH = √(4²+x²) = √(16+x²)
Then using similar triangles, we can find the ladder length L to be that which satisfies ...
L/(d+x) = GH/x
L = (d +x)/x·√(16 +x²)
The derivative with respect to x, L', is ...
L' = (d+x)/√(16+x²) +√(16+x²)/x - (d+x)√(16+x²)/x²
Simplifying gives ...
L' = (x³ -16d)/(x²√(16+x²))
Our objective is to minimize L by making L' zero. (Of course, only the numerator needs to be considered.)
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(a) For d=24, we want ...
0 = x³ -24·16
x = 4·cuberoot(6) ≈ 7.268 . . . . . feet
Then L is
L = (24 +7.268)/7.268·√(16 +7.268²) ≈ 35.691 . . . feet
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(b) The objective function is the length of the ladder, L. We want to minimize it.
L = (d/x +1)√(16+x²)
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Answer:
a) 30.85% of people earn less than $40,000
b) 37.21% of people earn between $45,000 and $65,000.
c) 15.87% of people earn more than $70,000
Step-by-step explanation:
Problems of normally distributed samples are 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.
In this problem, we have that:
a.What percent of people earn less than $40,000?
This is the pvalue of Z when X = 40000. So
has a pvalue of 0.3085.
30.85% of people earn less than $40,000
b.What percent of people earn between $45,000 and $65,000?
This is the pvalue of Z when X = 65000 subtracted by the pvalue of Z when X = 45000. So
X = 65000
has a pvalue of 0.7734.
X = 45000
has a pvalue of 0.4013.
0.7734 - 0.4013 = 0.3721
37.21% of people earn between $45,000 and $65,000.
c.What percent of people earn more than $70,000?
This is 1 subtracted by the pvalue of Z when X = 70000. So
has a pvalue of 0.8413.
1 - 0.8413 = 0.1587
15.87% of people earn more than $70,000