12 multiple-choice questions, progressively harder.
North covers 3 square kilometers at an average of 640 residents per square kilometer. South covers 12 square kilometers at an average of 165 residents per square kilometer. What is the average population density across the two nonoverlapping regions together?
Solution
Correct answer: D
Weight each rate by the area it describes. Combine counts and areas before dividing.
The result is 260 residents per square kilometer.
A reserve averages 36 nesting sites per square kilometer across 5 square kilometers. A 1.25-square-kilometer strip is known to contain 75 of those sites. What is the average nesting-site density in the remainder?
Correct answer: A
Subtract the strip’s actual count and area before dividing.
The remaining average is 28 nesting sites per square kilometer.
A survey team has a boundary that covers 45 square centimeters on an old map. The old scale is 1 cm for 80 m. On a new map the same boundary covers 20 square centimeters. What actual length does 1 cm represent on the new map?
Correct answer: C
Both maps represent the same ground area. Find the ground area per square centimeter on the new map, then take the positive square root.
One new map centimeter represents 120 actual meters.
A wetland restoration map uses 1 cm for 0.15 km. The planted part of the wetland covers 24 square centimeters on that map. A target of 350 plants per square kilometer applies to the planted part. If 173 plants are already there, how many more plants are needed to reach that target exactly?
Correct answer: B
Square the map scale in kilometers and multiply by the mapped area. Apply the target rate to the actual planted area.
Two printed plans of the same garden have length scales 1:40 and 1:25 in model-to-actual order. A planting region covers 75 square centimeters on the 1:40 plan. What area does it cover on the 1:25 plan?
For a fixed actual length, the drawing with denominator 25 is larger than the one with denominator 40 by factor 40/2540/2540/25. Square that factor for area.
An actual rectangular bin has a similar interior to a 1:10 model. The model’s rim is 6 cm above the inside floor and its capacity to the rim is 180 cubic centimeters. The actual bin has an operating fill depth of 42 cm. What is its modeled operating capacity?
The rim is 60 cm above the actual floor. In a rectangular bin, the volume fraction equals the fill-depth fraction. Convert the scaled rim volume to liters first.
A uniform solid model of a display occupies 96 cubic centimeters and has mass 48 g. The actual solid display has similar shape with lengths five times as large and is made of a different uniform material. Its mass is 18 kg. How many times as dense is the actual material as the model material?
Compare both densities in grams per cubic centimeter. The actual volume is 96(53)=1200096(5^3)=1200096(53)=12000 cubic centimeters and 18 kg is 18,000 g.
An actual display and its solid model are similar. Both are made of the same homogeneous material under the same conditions. The model mass is 54 g and the actual mass is 1,458 g. A model edge measures 7 cm. How long is the actual edge?
The equal densities make the mass ratio the volume ratio. Take the positive cube root for lengths.
A rectangular filter chamber’s model interior has base 7 cm by 4 cm and a mark 3 cm above the inside floor. The full-size chamber has similar inside dimensions and marks at model-to-actual scale 1:20. A planned batch of filter granules has mass 390 kg and bulk density 650 kilograms per cubic meter. Does the batch fit below the mark, and how much volume remains if it fits?
Scale the volume below the model mark. Find the batch volume independently from its mass and bulk density.
The batch fits, leaving 72 liters below the mark.
A digital drawing is stretched horizontally to twice its previous width. Its vertical dimensions are unchanged, and it still represents the same 240 trees. What happens to the count of trees represented per square centimeter of drawing?
Only the horizontal dimension doubles. The drawing area doubles and the represented tree count stays fixed.
Two nonoverlapping plots form a 200-square-meter garden with 430 plants. Plot A has average density 3 plants per square meter, and Plot B has average density 1.5 plants per square meter. What area does Plot A cover?
If the full garden had Plot B’s rate, there would be 200(1.5)=300200(1.5)=300200(1.5)=300 plants. The 130 additional plants come from the extra 1.5 plants per square meter in Plot A.
A store has two blocks of the same outer volume and the same mass. One is solid throughout. The other has an empty sealed cavity. Which comparison follows from these facts?
Average density calculated using the outer volume is total mass divided by that outer volume. Both of those inputs agree. The sealed cavity changes how much of the outer volume is occupied by solid material.
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