Math Bridge: Capstone Calculation Record

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Math BridgeCapstoneStruggle-friendly runway

How do several formulas become one reviewable capstone decision?

Carry units and assumptions through sampling, channel, link, filter, and battery checks without merging their claims.

Sammy, the capstone guideSammy guides
The one targetBuild a bounded multi-formula calculation record.
The chapter case360 Hz signal, 2 kHz sampling, 16 bits, radio, RC, and battery.
What it buys youA review trail that keeps unlike margins distinct.

A field team faces an unresolved physical question: How do several formulas become one reviewable capstone decision? They must answer it before changing sample rate on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is sample rate. The middle card applies this page's relationship. The green card is minimum rate. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Sample rate changes minimum rate An input card leads through the page relationship to the minimum rate result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Independent outputs should not move merely because one input changed; false coupling would make the record look responsive while breaking its meaning.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for sample rate is 2000.

  2. 2

    Name the relationship. fs,min=2x360=720 Hz; 2000/720=2.78x SNRideal=6.02x16+1.76=98.08 dB; data=32,000 bit/s C=125,000 log₂(11)=432,429 bit/s Prx=14-120=-106 dBm; margin=31 dB RC=1/(2πx400)=398 us; C=39.8 nF at 10 kohm Iavg=(160x2+0.8x58)/60=6.11 mA; life=13.7 days

  3. 3

    Substitute the chapter fixture. Set sample rate to 2000. The page ledger gives minimum rate as 720 Hz.

  4. 4

    Read the result. Keep Hz beside the value. Use it only inside the technical boundary on this page.

Predict, then change sample rate

Try Predict the direction of minimum rate. Move one control, calculate, then check your prediction.

2000
Chapter baseline
Minimum rate

Observe Independent outputs should not move merely because one input changed; false coupling would make the record look responsive while breaking its meaning. Reset the control to 2000 and compare minimum rate.

Explain Only sample rate moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only sample rate moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only sample rate. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

Each formula bounds a different failure: aliasing, quantisation, channel capacity, link reserve, filter cutoff, or energy. A capstone becomes reviewable when those boundaries stay named and unit-correct.

Sammy: A page of correct equations is not one proof; each result must return to the decision it can support.

2. Name every algebra move

1

SamplingDouble the highest signal frequency and compare the chosen rate.

2

QuantisationUse 6.02N+1.76 for the ideal full-scale sine SNR.

3

ChannelUse B log₂(1+SNR) for an upper bound.

4

LinkSubtract path loss, then sensitivity.

5

FilterUse RC=1/(2πfc).

6

BatteryWeight each current by its time, then divide capacity by average current.

3. Reproduce the chapter case

fs,min=2×360=720 Hz; 2000/720=2.78×
SNRideal=6.02×16+1.76=98.08 dB; data=32,000 bit/s
C=125,000 log₂(11)=432,429 bit/s
Prx=14−120=−106 dBm; margin=31 dB
RC=1/(2π×400)=398 µs; C=39.8 nF at 10 kΩ
Iavg=(160×2+0.8×58)/60=6.11 mA; life=13.7 days

Every number matches the chapter record, but each retains its own evidence boundary.

4. Try one real input

TryChange sample rate and identify which outputs should move and which must remain independent.

Sample rate
Minimum rate
Nyquist frequency
Sampling margin
Ideal ADC SNR
Raw data rate
Shannon ceiling
Received power
Link margin
RC product
Capacitance
Average current
Ideal life
Ideal life

ObserveSample rate changes Nyquist margin and raw data rate, while radio margin, filter values, and the fixed current-profile life stay put.

ExplainIndependent outputs should not move merely because one input changed; false coupling would make the record look responsive while breaking its meaning.

Technical boundaries.

This joins a review record, not the underlying physical systems.

Sampling
Nyquist assumes the input is band-limited before sampling.
Radio
Shannon and link margins do not predict delivered throughput or range.
Battery
Nameplate capacity divided by average current is an ideal ceiling.

Correct, not complete: each bound needs its own measured assumptions before a capstone decision closes.

5. Use the result in the design

Attach every result to one decision, one unit-consistent evidence source, one safe range, and one field check that can overturn it.

6. Record the evidence state

Keep formula version, units, input source, uncertainty, operating boundary, result, design consequence, owner, and retest trigger for every row.

7. Check yourself

Does 2 kHz prove a 360 Hz signal is safe to sample?
Answer: Only if the analogue input is band-limited and the filter performs as assumed.
Does 31 dB prove field range?
Answer: No. It is margin under the named path-loss and sensitivity assumptions.
Why does moving sample rate leave battery life fixed here?
Answer: The illustrative current profile is a separate fixed record; coupling it needs measured rate-dependent current.
Honesty boundary.

The arithmetic reproduces the chapter's illustrative sampling, radio, filter, and battery records.

Sampling
Nyquist assumes the input is band-limited before sampling.
Radio
Shannon and link margins do not predict delivered throughput or range.
Battery
Nameplate capacity divided by average current is an ideal ceiling.

Correct, not complete: each bound needs its own measured assumptions before a capstone decision closes.

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