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Answer Key
Common Core Math Standards:
Grade 6 – CCSS.Math.Content.6.RP.A.3b Solve unit rate problems including those involving unit pricing and
constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed
in 35 hours? At what rate were lawns being mowed?
Grade 7 – CCSS.Math.Content.7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of
lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in
each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
Problem 1 – A Marathon runner finishes the race and gets into her car. The outdoor
o
temperature is 75
F and the humidity is 80%. Explain why her actions may be dangerous if
she immediately drives away, and a simple remedy.
o
Answer: The dew point temperature is T = 75 – (100-80)/5 = 71
F. As she expels moist air
from her lungs, the humidity inside the car steadily increases and the dew point temperature
o
o
rises from 71
F to 75
F. Then the inside of her windows begin to fog up as water condenses
on their surfaces. This makes it harder for her to see outside and creates a dangerous
situation.
A remedy is to first turn on the car air conditioner, which will condense the remaining
moisture in the air onto the cooling pipes. Then turn on the heater which will warm the windows
and evaporate the moisture droplets on the windows. With the AC still on, this moisture will
also condense on the cooling pipes, resolving the situation. Typically this takes only a minute
or two for an average-sized car.
o
Problem 2 – It’s a warm sunny day and the air is rather humid with a dew point of 75
F .The
o
o
ground temperature is 85
F. If the air temperature decreases at a rate of 3.5
F/1000 feet, at
what altitude will a cloud begin to appear? Answer: The rising air near the ground has to drop
o
o
o
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in temperature by 85
F – 75
F = 10
F. It is decreasing by 3.5
F every 1000 feet, so the dew
o
o
o
point temperature of 75
F will be reached at an elevation of 10
F/3.5
F = 2857 feet.
Problem 3 – On a comfortable summer day, the humidity is only 20% and the outside
o
temperature is 80
F. At what altitude might clouds start to form overhead if the air temperature
o
is decreasing at a ‘dry’ lapse rate of 5.5
F/1000 feet?
o
Answer: First use the dew point formula to calculate T
. For P=20 and T
=80
F we get
dewpoint
air
o
T
=80 – (100-20)/5 so T
=64
F.
dewpoint
dewpoint
Next, calculate the altitude from the lapse rate. The difference between the ground
o
o
o
temperature and the dew point temperature is 80
F – 64
F = 16
F. The altitude will be A =
o
o
16
F/(5.5
F/1000feet) = 2909 feet.
Space Math
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