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| 10 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
When modeling a real-life situation, it is sometimes impossible to represent an object or scenario using the original dimensions. In these situations, it is more convenient to work with manageable units while still being able to maintain the same properties as the original. This can be done by making a scale drawing.
A scale drawing is a two-dimensional drawing that is similar to an actual object or place. In a scale drawing, the ratio of any length on the drawing to the actual length is always the same. drawing→/actual→ l_1/L_1 = l_2/L_2 ←drawing/←actual Possible examples of scale drawing are floor plans, blueprints, and maps.
In the case of the original real-life situation involving a three-dimensional object, making a 3D model is more useful than a drawing. The idea behind a 3D model is the same as a scale drawing, but the model has three dimensions instead of two.
A scale model is a three-dimensional model that is similar to a three-dimensional object. The ratio of a linear measurement of a model to the corresponding linear measurement of the actual object is always the same. model→/actual→ l_1/L_1 = l_2/L_2 ←model/←actual Here is an example scale model of a building.
The scale of a model or drawing is the ratio between any length on the model or drawing and its corresponding length on the actual object or place.
lLength on the drawing : lCorresponding length on the actual object
Suppose a drawing has a scale of 1 in:100 ft. This means that 1 inch on the drawing represents 100 feet on the actual object. Apart from the colon notation, a scale can be expressed using an equals sign or as a fraction, as it is a ratio.
Denoting a Scale | |
---|---|
Ratio | 1 in : 100 ft |
Equals Sign | 1 in = 100 ft |
Fraction | 1 in/100 ft |
When a scale is written without specifying the units, it is understood that both numbers have the same unit of measure. For example, a scale of 1:2 means that the actual object is twice the size of the model. A scale of 1:0.5 means that the actual object is half the size of the model — whether it be in meters, inches, yards, and so on.
Multiply 1 300 000 000 cm by 1 m/100 cm*1 km/1000 m
Cross out common factors
Cancel out common factors
Multiply fractions
Calculate quotient
300km* 1= 20 000 000^(Original scale) * 300km ⇕ 300 km=6 000 000 000 km In conclusion, the distance from Pluto to the Sun is 6 billion kilometers.
ccc Model Height& &Actual Height 55.375 cm &:& 443 m According to this scale, 55.375 centimeters correspond to 443 meters. The scale can be simplified by dividing both numbers by 55.375. 55.375/55.375 cm : 443/55.375 m ⇕ 1 cm : 8 m In the scale used by Tom, 1 centimeter corresponds to 8 meters.
7.125* 1 cm= 8 m * 7.125 ⇕ 7.125 cm= 57 m Therefore, the Empire State Building is 57 meters wide.
The United States of America has an area of 9 147 420 square kilometers. In mid-2020, the total population was 331 923 317 people.
In the United States of America, the population density is 94 people per square mile. The population density measures the number of people per unit of area.
People per square mile can be written as peoplemi^2. This implies that the population density is expressed as a ratio. Remember that 1 square kilometer is equal to 0.386102 square mile.
Multiply 9 147 420 km^2 by 0.386102 mi^2/1 km^2
Cross out common factors
Multiply fractions
The population density of the United States of America is approximately 94 people per square mile.
This statement assumes that the total population is spread out evenly across the United States.
Mark wants to paint all of the walls of his bedroom except for the wall that contains the door. Each rectangular wall is 2.8 meters high. The paint he will use is sold in 5-liter cans — the price per liter is $ 25.
After painting for a few minutes, Mark noticed that 5 square meters can be covered with one liter of paint.
Dimensions | |
---|---|
Wall 1 | l_1 &= 4.75 m h &= 2.8 m |
Wall 2 | l_2 &= 4.75 m h &= 2.8 m |
Wall 3 | l_3 &= 3 m h &= 2.8 m |
l_1= 4.75 m, h= 2.8 m
Multiply
5 m^2/liter ⇔ 1 liter/5 m^2 Multiplying 15 liters/m^2 by the total area gives the number of liters that Mark needs to paint the three walls. 35 m^2 * 1 liter/5 m^2 = 7 liters
Price of2cans &= 10 liters* 25 dollars/liter [0.8em] &= 250 dollars If the paint was sold in 1-liter cans, Mark would need to buy 7 cans. In this case, he would pay 7* 25=$175. The amount of money that Mark could have saved can be found by calculating the difference between the price of 10 liters of paint and 7 liters of paint. 250-175=$ 75
On the applet, the model and the actual object are shown. Using the given information, find the scale or the size of either the model or the actual object.
Tadeo wants to buy a cheap phone plan for calling his friends and family. He asked Ali and Ramsha how much they pay per call.
Initially, Tadeo decided to take Ramsha's plan since she paid less. Later, he realized that this information is not helpful since he does not know the duration of each of the calls made by Ali and Ramsha.
Price per call is not an appropriate unit.
Therefore, Tadeo decided to ask Ali and Ramsha how long each phone call lasted.
Since the duration of the calls is different, Tadeo became confused and made the following diagram to think about the situation. Ali: & $10 → 40 minutes Ramsha: & $8 → 30 minutes After this, Tadeo realized that dividing each call's cost by its duration will give him the price per minute, which is an appropriate unit to compare the plans.
Person | Scale of the Plan |
---|---|
Ali | $10/40 min=$0.25 per minute |
Ramsha | $8/30 min≈ $0.27 per minute |
To determine the number of laps the Enterprise can make around the Earth in 1 second, we first need to determine how far the ship travels with the speed of Warp 9.
To find the speed, let's substitute the warp and speed of light in the given formula.
Now we know the speed in kilometers per hour. However, we need the speed in kilometers per second because we want to know how far the ship travels in 1 second. Since 1 hour equals 3600 seconds, we get the following conversion factor. 1 h/3600 s By multiplying the speed by this conversion factor, we get the speed in kilometers per second.
At Warp 9, the ship travels a distance of 218 700 000 kilometers in one second.
To determine how many laps around the Earth the found distance corresponds to, we need to figure out the distance of one lap around the earth. We know that the ship is traveling at an orbit of 150 kilometers above the surface of the earth.
By adding the radius of the Earth and the distance of 150 kilometers, we can calculate the radius of the circular orbit. 6371+150=6521 km With this information, we can calculate the circumference of the circle.
One lap around the Earth at the given orbit equals 13 042π kilometers. This is the distance of one lap. By dividing distance the ship travels in one second by the distance of one lap, we can find the number of laps around the Earth that Enterprise travels in one second. 218 700 000/13 042π=5337.706801... Since we want to know the whole number of laps, we need to round this number down. Therefore, in 1 second, at Warp 9, the ship travels 5337 whole laps around the Earth.