Practise the method.
Four linked problems drawn from this topic’s subtopics. 12 parts · untimed · answers and worked solutions at the end.
Example 1
A sensor follows M(t)=B+Aekt for t≥0 minutes. M is measured in millivolts (mV). A,B and k are unknown, with A>0. The readings in the table are exact. Retain the exact logarithmic value of k for later parts.
Use all three readings to find B. You may first compare the two successive changes.
| Time t(min) | Reading M(t)(mV) |
|---|---|
| 0 | 72 |
| 3 | 128 |
| 6 | 240 |
Show worked answer
Answer: 16
- Let R=e3k. Successive changes are A(R−1) and AR(R−1), so their ratio is R.
- R=128−72240−128=2.
- M(3)−B=R[M(0)−B], so B=(1−2)[128−(2)72]=16 mV.
Example 2
Sound level is L=10log10(I0I) dB, with I0=m210−12W. Independent source intensities add. One source operating with background noise gives a total level of 40 dB. The background alone has I0I=1000. Treat all given values as exact.
a Find the intensity ratio I0I of the first source alone.
Find the level of that source alone. Give your answer to 3 decimal places.
Show worked answer
Answer: 39.542
- The isolated intensity ratio is 9000; hence L=10log10(9000).
- Decibel values cannot be directly added or subtracted to combine or remove independent intensities.
- Rounded answer: 39.542 dB.
Example 3
Consider y=a(x+3)2+4, for x∈ ℝ. P is the point on the curve with x=−2. The normal at P has gradient 21. Coordinates are dimensionless.
a Find the coefficient a.
b Find the y−intercept of the tangent at P.
Q=(−3,5) lies on the tangent at P. Find the x−coordinate of the other point on the curve whose tangent also passes through Q.
Show worked answer
Answer: −4
- Write a general contact point as x=−3+u. Its y−value is−1u2+4 and tangent gradient is−2u.
- At x=−3, that tangent has y=4−(−1)u2. For it to pass through Q,u2=1.
- The known point P uses u=1; the other uses u=−1, so x=−4.