Mathematics
Grade 11
15 min
Multiplication word problems: find the missing factor
Multiplication word problems: find the missing factor
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1
Introduction & Learning Objectives
Learning Objectives
Identify the product and the known factor in word problems involving angle measures.
Set up an algebraic equation of the form A = B * x to represent a word problem.
Solve for a missing angular velocity when given the total angle of rotation and time.
Determine a missing angle in radians when given the arc length and radius.
Calculate the number of rotations or steps when given the total angle and the angle per step.
Convert between degrees and radians as required to solve for the missing factor.
Interpret the missing factor in the context of real-world scenarios like rotational mechanics and circular motion.
A high-performance car engine revs at 7200 RPM (revolutions per minute). How could you find the angle its crankshaft turns in just one millisecond?...
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Key Concepts & Vocabulary
TermDefinitionExample
Missing Factor ProblemA problem structure where the product of two quantities is known, along with one of the quantities (factors), and the goal is to find the other unknown quantity (the missing factor). It takes the form Product = (Known Factor) × (Missing Factor).If a wheel rotates a total of 3600° through 10 equal revolutions, the angle of each revolution is the missing factor. Equation: 3600° = 10 × θ. The missing factor θ is 360°.
Angular Displacement (θ)The total angle through which a point or line has been rotated in a specified direction about a specified axis. It can be measured in degrees or radians.The minute hand of a clock moving from the 12 to the 6 has an angular displacement of -180° or -π radians.
Angular Velocity (ω)The rate at which an object rota...
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Core Formulas
Total Angular Displacement
\theta_{total} = n \times \theta_{step}
Used when a total rotation is composed of 'n' identical, smaller rotations (steps). The missing factor could be 'n' (the number of steps) or 'θ_step' (the angle of each step).
Angular Velocity Formula
\theta = \omega \times t
Relates total angular displacement (θ), angular velocity (ω), and time (t). Use this to find a missing velocity or time when the other two quantities are known.
Arc Length Formula
s = r \times \theta
Connects arc length (s), radius (r), and the central angle (θ). This formula requires the angle θ to be in radians. The missing factor is often the angle θ.
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Challenging
A car is on a circular track with a radius of 200 meters. The car's tires have a radius of 0.4 meters and rotate at an angular velocity of 50 rad/s. How much time, in seconds, does it take for the car itself (not its tires) to travel through an angular displacement of π/2 radians on the track?
A.5π seconds
B.4π seconds
C.π/5 seconds
D.25π seconds
Challenging
A satellite corrects its orientation in 4 steps. The angular displacement of each step is half the previous one, forming a geometric sequence. The total angular displacement after 4 steps is 45π/8 radians. What was the angular displacement of the first step?
A.π radians
B.2π radians
C.3π radians
D.4π radians
Challenging
A motor's angular velocity increases linearly from 0 rad/s to a final velocity ω_f. The total angular displacement is 300π radians. For linear acceleration, the average angular velocity is half the final angular velocity (ω_avg = ω_f / 2). If the final angular velocity was 50π rad/s, what was the total time, t, of the acceleration?
A.6 seconds
B.12 seconds
C.24 seconds
D.6π seconds
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