Unit 1 - Scale Drawings
A scale drawing shows a rectangle with a length of 4 cm and a width of 2 cm.
The scale factor is 1:5.
What are the actual dimensions of the rectangle?
Define what a ratio is and give an example from your everyday life.
A ratio is a comparison of two quantities using division or a colon.
Example answers may include: recipes (2 cups flour to 3 eggs), sports (wins to losses), or speed (miles per hour).
List the three main parts of a circle.
The three main parts are: center, radius, and diameter.
In the equation y=3x, what is the constant of proportionality?
The constant of proportionality is 3.
What is the relationship between the radius and diameter of a circle?
The radius is half the diameter. r = d/2
The diameter is double the radius. d = 2r
Define what a scale factor is in your own words.
A scale factor is the ratio between the dimensions on a scale drawing and the actual dimensions of the object. It tells you how many times larger (or smaller) the real object is compared to the drawing.
Acceptable answers should include that a scale factor shows the relationship between drawing measurements and real measurements.
If a recipe calls for 2 cups of flour for every 3 eggs, what is the ratio of flour to eggs?
The ratio is 2:3 (or 2 to 3).
Analyze this scenario: A student claims that a circle can have different lengths for different radii. Is this correct? Justify your answer with mathematical reasoning.
No, this is incorrect. All radii of the same circle must have equal length because a circle is defined as the set of all points equidistant from a center point. If radii had different lengths, the shape would not be a circle.
A classroom actual dimensions: 30 feet × 25 feet
With a scale factor: 1 inch = 5 feet what are the dimensions on the drawing?
Drawing dimensions:
30 feet/5 feet = 6 inches
25 feet/5 feet = 5 inches
A rectangular swimming pool has actual dimensions of 20 meters by 30 meters.
An architect creates a scale drawing using a scale factor of 1:100. Investigate what the dimensions of the pool would be on the scale drawing, and explain how you determined your answer.
Step 1: Understand the scale factor. A scale factor of 1:100 means that 1 unit on the drawing represents 100 units in reality. To find the drawing dimensions, divide the actual dimensions by 100.
Step 2: Calculate the drawing dimensions and convert meters to cm.
If 1 inch equals 2.54 centimeters, how many centimeters are in 5 inches?
5 inches×2.54 cm/inch=12.7 cm
Create a table showing the relationship between the number of hours worked and total pay earned if someone makes $12 per hour. Identify the constant of proportionality.
Hours Worked Total Pay
1 $12
2 $24
3 $36
4 $48
The constant of proportionality is 12 or k=12
Define circumference in your own words.
Circumference is the distance around the outside of a circle. (Accept variations: the perimeter of a circle, the total length around the edge, 1 full rotation around the circle.)
Solve this problem: A circular garden has a radius of 10 meters. What is the area of the garden?
Use π = 3.14 and round your answer to the nearest hundredth.
A = πr²
So
A = π(10)² = 100π ≈ 314.00 m²
A circular table has a diameter of 6 feet. If I want run lights around the table would I need to find the Circumference or Area? Explain your answer.
CIRCUMFERENCE because I am only putting the lights on the OUTSIDE of the table. If I wanted to put a tablecloth over the table I would need to find the area.
A blueprint of a house shows a room that is 6 cm by 8 cm.
If the scale is 1 cm = 2 meters, what are the actual dimensions of the room?
A baker uses 4.5 cups of sugar for every batch of cookies she makes.
How many cups of sugar does she need for 8 batches?
y=4.5x
where x=8
y = 4.5 × 8 = 36
She needs 36 cups of sugar.
If a circle has a circumference of 31.4 meters, what is the radius? Use π = 3.14 and round to the nearest whole number.
Using C = 2πr:
31.4 = 2πr so r = 31.4 ÷ (2π) = 31.4 ÷ 6.28 ≈ 5 meters.
Analyze the two different proportional relationships and critique which one represents a better deal or outcome. Justify your conclusion using mathematical reasoning.
Relationship 1: Pizza Place A charges $3.50 per slice.
Relationship 2: Pizza Place B charges $18 for 6 slices.
Relationship 2: Pizza Place B charges $18 for 6 slices.
Is the better deal because you are paying $3.00 for 1 slice.
Scenario: A car rental company charges $0.25 per mile driven. Create a table that shows the miles driven if the cost was: $0, $2.50, $5.00 and $12.50.
Miles Cost
0 $0
10 $2.50
20 $5.00
50 $12.50
Estimate the area of a triangular garden if a scale drawing shows the triangle with a base of 3 cm and height of 2 cm, using a scale of 1 cm = 4 meters.
Step 1: Find the actual dimensions.
Step 2: Calculate the actual area using the triangle area formula: A = 1/2bh
A=1/2 × 12 × 8 = 96/2 = 48 square meters
A car rental company charges $0.25 per mile driven. Write an equation to show the relationship between the total cost (C) and the miles driven (m).
C = 0.25m
where C = total cost and m = miles driven
Find the area of a circle with a diameter of 8 meters. Use π = 3.14 and round your answer to the nearest hundredth.
A = πr²
[Note: radius = diameter ÷ 2 = 8 ÷ 2 = 4 meters.]
A = π(4)² = 16π ≈ 50.24 m²
A basketball court has actual dimensions of 94 feet long and 50 feet wide.
A coach creates a scale drawing using a scale of 1 inch = 10 feet. What are the dimensions of the basketball court on the scale drawing?
Step 1: Set up the scale relationship.
Step 2: Convert actual dimensions to drawing dimensions.
A circular garden needs fencing around its perimeter and mulch to cover the ground. If the diameter is 14 feet, calculate both the circumference (for fencing) and the area (for mulch).
Use π = 3.14 and round your answers to the nearest hundredth.
Circumference (fencing):
C = πd so C = π(14) = 14π ≈ 43.96 feet.
Area (mulch): A = πr²
[Notice the diameter is 14 so our radius is 7.]
So A = π(7)² = 49π ≈ 153.86 square feet.
Approximately 43.98 feet of fencing is needed, and approximately 153.86 square feet of mulch is needed.