Connected rates, geometry and time units
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| connected rates/kəˈnektɪd reɪts/ | tốc độ liên kết |
Does its radius increase at a steady rate too?
- A spherical object gains volume at a steady rate.
- This lesson studies connected rates 关联变化率: Rates linked by a relationship between quantities changing with the same time variable.
The method
- 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.
Why must the radius NOT be substituted into V=(4/3)πr³ before differentiating with respect to time?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The relation must be differentiated WHILE r is still a function of t. Differentiate first, then substitute the instant’s values: at r=3 with dV/dt=18π, dr/dt=18π/(36π)=1/2 cm/s.
A worked case
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Divide 18π by 4π×3² to obtain 1/2.
Where it goes wrong
- 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.
Not true: Substitute a measured radius into the volume formula before differentiating it with respect to time. Differentiate the formula with respect to time first, then substitute the measured radius; substituting first turns the radius into a constant.
Find the worked cone height rate in cm/s.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Similarity gives dV/dh=πh²/4; at h=4 divide 8π by 4π to obtain 2.
Differentiate the formula with respect to time first, then substitute the measured radius.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Differentiate the formula with respect to time first, then substitute the measured radius; substituting first turns the radius into a constant.
Applying it
- 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.
Find the ladder top rate in m/s.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Differentiate the fixed ladder constraint: dy/dt=−(3/4)×0.2=−0.15.
Match each connected-rates step to its purpose.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Signs carry direction - a negative dy/dt means the ladder top descends - and every rate must share the same time unit before combining.
In your course
- AQA A-level Mathematics (7357), specification section G.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Connected rates: Rates linked by a relationship between quantities changing with the same time variable.
- If y=f(x) and x varies with time t, dy/dt=(dy/dx)(dx/dt).
- Watch for: 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.