Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 13) Variation and Rate of Change cover? A: Model direct, inverse, and joint variation, and interpret gradient as rate of change in real contexts.
Variation equations compress word problems into algebra. Identify whether quantities move together or in opposition, then encode that with proportionality constants and a positive constant k that captures the context (units, efficiency, scaling).
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Quick reference
Direct variation: y∝x⇒y=kx.
Power variation: y∝xn⇒y=kxn
for real
n
.
Inverse variation: y∝x1⇒y=xk.
Joint variation: y∝xz⇒y=kxz (extend to more variables similarly).
Combined variation: mix the patterns, e.g. y∝zx2⇒y=kzx2.
Rate of change from a graph: gradient =ΔxΔy; include units such as m⋅s−1 or $per hour.
"Varies as the square / cube / square root" ⇒ attach the stated power.
"Constant of variation" ⇒ solve for k using the given data before predicting new values.
Workflow for variation questions
Translate the English statement into a proportionality sentence (e.g. d∝v2).
Replace the proportionality symbol with an equation and constant k.
Substitute known values to solve for k (keep units consistent).
Substitute the new input(s) and compute the required output, rounding sensibly.
Check the answer against intuition (does the result get larger or smaller as expected?).
Worked example 1 - Fuel consumption model
The stopping distance d of a car is directly proportional to the square of its speed v. When v=40km⋅h−1, d=24m. Find the stopping distance when v=70km⋅h−1.
Model: d=kv2.
Substitute known values: 24=k(40)2=1600k⇒k=160024=0.015.
For v=70: d=0.015(70)2=0.015×4900=73.5.
Hence d≈73.5m.
Worked example 2 - Inverse variation (product stays constant)
The illuminance I at a sensor varies inversely with the square of the distance r from a point light source: I∝r21. When the sensor is 2.5m away, the illuminance is 180lx. Find the illuminance at 4.0m.
Model: I=r2k.
Solve for k: 180=(2.5m)2k=6.25k⇒k=1125.
At r=4.0m: I=(4.0)21125=161125=70.3125
Quote the answer to three significant figures: I≈70.3lx.
Inverse variation questions often include phrases like "constant product" or "work rate". Always state the restriction r=0.
Worked example 3 - Joint variation with rate of change
The volume of water V flowing through a pipe varies jointly with the cross-sectional area A and the flow speed u: V∝Au. When A=18cm2 and u=2.4m⋅s−1, the volume rate is 0.0432m3⋅s−1. Find the rate when A=25cm2 and u=2.9m⋅s−1.
Convert the area to m2: 18cm2=1.8×10−3m2.
Model: V=kAu.
Solve for k: 0.0432=k(1.8×10−3)(2.4)⇒k=10.
New area: 25cm2=2.5×10−3m2.
Substitute: V=10×(2.5×10−3)×2.9=0.0725.
Therefore V≈7.25×10−2m3⋅s−1.
Rate of change on graphs and tables
Average rate of change between x=a and x=b: b−af(b)−f(a).
Gradient of a line segment on a linear graph gives a constant rate (e.g. "cost per item").
Always pair the gradient with units: if y is measured in dollars and x in hours, the rate is $per hour.
For distance-time graphs, gradient gives speed; for velocity-time graphs, the area under the curve gives displacement.
Worked example 4 - Average rate from a data table
A technician records the volume of a substance in a tank every 10s.
Timet/s
VolumeV/L
0
120
10
147
20
171
30
195
Estimate the average rate of change between t=10s and t=30s.
ΔtΔV=30−10195−147=2048=2.4L⋅s−1.
State the meaning: the tank gains about 2.4L of fluid per second over that interval.
Practice Quiz
Model direct, inverse, and joint variation scenarios and interpret gradients with instant feedback.
Try this
The time T required to print a batch of photos varies inversely with the number of printers n operating and directly with the number of photos p. One printer processes 120 photos in 18min. Form the model and find the time needed for 3 printers to process 300 photos.
The brightness B of a screen varies directly with the voltage V and inversely with the square of the distance d. When V=24V and d=1.2m
A cyclist's position is recorded every 5s. At t=15s the displacement is 210m; at t=30s it is 420m
IP EMaths Notes (Upper Sec, Year 3-4): 13) Variation and Rate of Change
.
, the brightness is
540cd⋅m−2
. Predict
B
when
V=18V
and
d=0.9m
.
. Interpret the average rate of change and state its units.
