Connected rates, geometry and time units
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| connected rates | connected rates |
A spherical object gains volume at a steady rate. Does its radius increase at a steady rate too?
- A spherical object gains volume at a steady rate. Does its radius increase at a steady rate too?
- This lesson studies connected rates 关联变化率: Rates linked by a relationship between quantities changing with the same time variable.
Choose the mathematical structure
- If y=f(x) and x varies with time t, dy/dt=(dy/dx)(dx/dt). Derive a relation between changing quantities, differentiate with respect to time, then substitute the measured values at that instant. Keep the time variable and each rate’s sign and units. For a sphere V=(4/3)πr³ and A=4πr², so dV/dt=4πr² dr/dt and dA/dt=8πr dr/dt.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines connected rates?
Rates linked by a relationship between quantities changing with the same time variable.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A sphere has radius 3 cm and gains volume at 18π cm³/s. Its instantaneous radius rate is 18π/(4π×9)=1/2 cm/s, and surface-area rate is 8π×3×(1/2)=12π cm²/s. This is a local rate; as r grows, the same volume rate gives a smaller radius rate. In a conical vessel whose full radius is half its full height, similarity gives r=h/2 at every water level. Thus V=πh³/12 and dV/dt=(πh²/4)dh/dt. At h=4 cm and inflow 8π cm³/s, dh/dt=2 cm/s. For a fixed 5 m ladder, x²+y²=25; at (x,y)=(3,4) m with dx/dt=0.2 m/s, dy/dt=−(3/4)×0.2=−0.15 m/s: the top descends.
Connected rates, geometry and time units
If y=f(x) and x varies with time t, dy/dt=(dy/dx)(dx/dt)
Identify the appropriate changing variable and validate the derivative denominator or local branch.
Find the worked sphere radius rate in cm/s.
Divide 18π by 4π×3² to obtain 1/2.
Test a tempting shortcut
- Do not substitute the current radius or height before differentiating: that would falsely turn the relation into a constant. dV/dt is not dV/dr. A changing cone radius must follow the similarity relation, not stay fixed at its current value. Convert a per-minute inflow to a per-second rate before combining it with second-based data. Negative rate means decrease, not a negative length.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Substitute a measured radius into the volume formula before differentiating it with respect to time. This claim is false. Explain which definition or assumption it violates.
Find the worked cone height rate in cm/s.
Similarity gives dV/dh=πh²/4; at h=4 divide 8π by 4π to obtain 2.
Substitute a measured radius into the volume formula before differentiating it with respect to time.
Do not substitute the current radius or height before differentiating: that would falsely turn the relation into a constant. dV/dt is not dV/dr. A changing cone radius must follow the similarity relation, not stay fixed at its current value. Convert a per-minute inflow to a per-second rate before combining it with second-based data. Negative rate means decrease, not a negative length.
Interpret a new situation
- Draw the geometry, identify constants and state the valid time/measurement units. Eliminate an extra changing variable using similarity or a constraint, then differentiate. Use the values for the same instant and describe the result’s direction. Physical assumptions, such as a fixed ladder length or a true sphere, limit the model; rates need not remain constant after that instant.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the ladder top rate in m/s.
Differentiate the fixed ladder constraint: dy/dt=−(3/4)×0.2=−0.15.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · G. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Rates linked by a relationship between quantities changing with the same time variable. Choose the relationship, show the method, check its assumptions and interpret the result.