SOLUTION: At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer. If a person who is 36 feet above the water can see 7.4 miles, find

Question 503964: At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer. If a person who is 36 feet above the water can see 7.4 miles, find how far a person 64 feet above the water can see. Round to the nearest tenth of a mile.
The amount P of pollution varies directly with the population N of people, Kansas City has a population of 442,000 and produces 260,00 tons of pollutants. Find how many tons of pollution we should expect St. Louis to produce, if we know that its population is 348,000. Round to the nearest whole ton
Charles's law states the if the pressure P stays the same, the volume V of a gas is directly proportional to its temperature T. If a balloon is filled with 20 cubic meters of a gas at a temperature of 300 K, find the new volume if the temperature rises 360K while the pressure stays the same.
Here are my three problems. Please help!! Please include the formula used to solve.
Thank you, Lori
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(1) At sea level...
d = h*k, where k = some constant
7.4 miles = 36 ft * k
k = 7.4 miles / 36 ft
k = .2055 miles/ft of elevation
.
h = 64 ft
h*k = 64*.2055 = 13.152 miles
.
Always check your answer.
In this case we can use proportions.
Is the ratio of 64 to 36 the same as 13.152 to 7.4?
64/36 = 16/9
16/9 = 1.777
13.152/7.4 = 1.777
OK.
.
Re-read the question to see what the answer should be.
Rounded to the nearest tenth.
.
Answer: At a height of 64 ft you can see 13.2 miles.
(I applied the rounding rule that with a '.5' you round to the next higher even number.)
.
.
(2) Pollution...
.
Pollution varies directly with population.
p*k = amount of pollution produced, where p = population and k = a constant
442000*k = 260000 tons
k = 260000/442000
k = 10/17
.
St. Louise has 348000 people, so we expect it to produce:
348000 * 10/17 = 204705.88 or about 204706 tons of pollution
.
Always check your work.
Again we can use proportions.
442000/348000 = 221/74 = 1.270
260000/204706 = 1.270
OK
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Answer: We expect St. Louis to produce 204,706 tons.
.
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(3) Charles' Law
.
At constant pressure, V = T*k, where V = volume, T = temperature, and k = constant.
V = 20 cubic meters
T = 300 Kelvin
k = V/T = 20/300 = 1/15
T changes to 360
V = 360* 1/15
V = 24 cubic meters
.
Use can use proportions to check this one, too.
V₁ is to V₂ as T₁ is to T₂.
20/24 = .8333
300/360 = .8333
OK.
.
Answer: At constant pressure, the volume increases to 24 cubic meters.
Done.
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