Mathematics
Grade 11
15 min
Compare and convert metric units of length
Compare and convert metric units of length
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1
Introduction & Learning Objectives
Learning Objectives
Apply dimensional analysis to convert between metric units of length (e.g., kilometers, meters, nanometers) within trigonometric contexts.
Calculate the arc length of a circle, ensuring the radius and final arc length units are consistent and correctly converted.
Solve right-triangle problems where side lengths are given in different metric units, requiring conversion before applying trigonometric ratios.
Determine the wavelength of a periodic function from its equation and convert it into appropriate metric units (e.g., meters to nanometers).
Analyze and solve word problems involving trigonometric functions where unit conversion is a critical intermediate step.
Verify the dimensional consistency of their answers in trigonometric calculations involving leng...
2
Key Concepts & Vocabulary
TermDefinitionExample
Metric PrefixesMultipliers that precede a basic unit to indicate a multiple or fraction of that unit. In trigonometry, we often encounter lengths from the very large (kilo-) to the very small (nano-).A satellite's altitude might be 35,786 kilometers (km), while the wavelength of red light is about 700 nanometers (nm). 1 km = 10^3 m, and 1 nm = 10^-9 m.
Dimensional AnalysisA problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value. It is used to convert units by multiplying with a conversion factor that equals one.To convert 2.5 kilometers (km) to meters (m), you multiply by the conversion factor (1000 m / 1 km): 2.5 km * (1000 m / 1 km) = 2500 m. The 'km' units cancel out.
Arc Length (s...
3
Core Formulas
Metric Conversion Formula
New Value = Old Value \times \frac{\text{Conversion Factor for New Unit}}{\text{Conversion Factor for Old Unit}}
Use this rule to convert any metric measurement. The conversion factor is the unit's value relative to the base unit (meter). For example, for kilometers, the factor is 10^3; for centimeters, it is 10^-2.
Arc Length Formula
s = r\theta
Calculates the arc length 's' of a circle with radius 'r' subtended by a central angle 'θ'. CRITICAL: 'θ' must be in radians, and the units for 's' will be the same as the units for 'r'.
Wavelength from Wave Number
\lambda = \frac{2\pi}{k}
In a trigonometric wave function like y = A \sin(kx), 'k' is the wave number. This form...
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Easy
According to the tutorial, a satellite's altitude might be 35,786 kilometers. How is this distance expressed in meters?
A.35.786 m
B.3,578,600 m
C.35,786,000 m
D.35,786 m
Easy
The wavelength of red light is about 700 nanometers (nm). Given that 1 nm = 10^-9 m, how would you express 700 nm in meters?
A.7 x 10^-7 m
B.7 x 10^-9 m
C.700 x 10^9 m
D.7 x 10^11 m
Easy
When using the arc length formula, s = rθ, the tutorial specifies a critical requirement for the central angle 'θ'. What is this requirement?
A.θ must be in degrees.
B.θ must be in radians.
C.θ must be less than 90.
D.θ must be a positive integer.
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