Mathematics
Grade 11
15 min
Which metric unit of length is appropriate?
Which metric unit of length is appropriate?
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Introduction & Learning Objectives
Learning Objectives
Relate the unit of a circle's radius to the resulting unit of its arc length.
Select an appropriate metric unit for heights and distances calculated using trigonometric ratios in real-world scenarios.
Determine the appropriate unit for wavelength when analyzing sinusoidal function models.
Justify the choice of a metric unit for the magnitude of a displacement vector.
Convert between metric units to maintain consistency within a single trigonometric calculation.
Analyze the scale of a problem to predict the most logical unit for the solution.
🛰️ If you use trigonometry to find the altitude of a satellite, should your answer be in millimeters, meters, or kilometers? Let's explore why the choice of unit is as critical as the calculation itself!
Th...
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Key Concepts & Vocabulary
TermDefinitionExample
ScaleThe relative size or extent of a quantity in a given context. The scale of a problem (e.g., a microchip vs. a solar system) determines the most practical and appropriate units for measurement and results.The scale of a human is best measured in meters (m), while the scale of an atom is best measured in picometers (pm).
Arc Length (s)The physical distance along the curved line making up an arc of a circle. Its unit is directly determined by the unit of the circle's radius.If a circle's radius is 5 meters, the arc length subtended by an angle is also measured in meters.
Wavelength (λ)The spatial period of a periodic wave, representing the distance over which the wave's shape repeats. The appropriate unit depends on the type of wave being modeled.The...
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Core Formulas
Arc Length Formula
s = rθ
Calculates the arc length 's' of a circle with radius 'r' subtended by a central angle 'θ'. For this formula to be valid, 'θ' MUST be in radians. The unit of 's' will always be the same as the unit of 'r'.
Trigonometric Ratios
sin(θ) = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad cos(θ) = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \quad tan(θ) = \frac{\text{Opposite}}{\text{Adjacent}}
Used to find unknown side lengths in right-angled triangles. All lengths (Opposite, Adjacent, Hypotenuse) must be in the same unit. The calculated length will have the same unit as the known length(s) used in the calculation.
Wavelength from a Sinusoidal Function
\lambda = \frac{2\pi}{|B|}
In a sin...
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Challenging
A function f(x) = 5 sin(Bx) models the ripple pattern in a specialized optical fiber, with x in meters. The distance between the first and fifth consecutive crests is 8 micrometers (μm). After calculating the wavelength λ and then the value of B, what is the most logical unit to associate with the amplitude, 5?
A.Meters, because x is in meters.
B.Micrometers, because the scale of the wavelength is in micrometers.
C.Radians, because it is part of a trigonometric function.
D.No units, because amplitude is a ratio.
Challenging
A precision milling machine etches a circular arc on a component. The radius is 20.00 mm and the arc must be exactly 31.40 mm long. The machine's control software requires the angle input in radians. After calculating θ using θ = s/r, an engineer needs to verify the radius measurement. A laser scanner measures the radius as 0.02000 meters. What must be done before this new measurement can be used to verify the original calculation's unit consistency?
A.Convert the arc length from 31.40 mm to 0.03140 m.
B.No conversion is needed as both 20.00 mm and 0.02000 m are the same length.
C.Convert the angle from radians to degrees.
D.Average the two radius measurements.
Challenging
The propagation of a low-frequency radio wave is modeled by a sinusoidal function where distance 'x' is in meters. The constant B in the function is 2.1 x 10⁻². Calculating the wavelength λ = 2π/|B| gives approximately 299.2 meters. How should this wavelength be most appropriately reported for a general audience?
A.299,200,000 micrometers
B.299.2 meters
C.0.2992 kilometers
D.29,920 centimeters
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