Unit 9
11 marks · HardA scientist records the cooling of a metal rod and the time it takes to drop temperature. The results are grouped below.
The grouped results appear below.
| Time, t (seconds) | Frequency |
|---|---|
| 0 < t ≤ 10 | 6 |
| 10 < t ≤ 20 | 14 |
| 20 < t ≤ 30 | 9 |
| 30 < t ≤ 50 | 11 |
Complete the table by finding the class width and frequency density for each group.
Use the information from (i) to draw a histogram representing the data (you may make a rough sketch).
A function $f$ is defined by $f(x) = e^{-x/10}$ for $0 \leq x \leq 50$. The curve $y = f(x)$ models the cooling process. On the axes where your histogram is drawn, sketch the curve $y = f(x)$, matching the scales.
Describe two key differences between the histogram and the curve $y = f(x)$ in the context of the cooling process.
The scientist shifts the experiment by 10 seconds, so she now measures $y = f(x-10)$. Describe, algebraically and using your sketches, the transformation applied to the curve.
Suppose all times in the experiment are doubled. Write down the new function (in terms of $x$) and describe the graphical effect.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a(i)) Widths: 10,10,10,20. Frequency densities: $0.6, 1.4, 0.9, 0.55$. · must show Width: 10,10,10,20, Frequency density as freq÷width
(a(ii)) Histogram: 4 bars with correct width/height, axes labelled, heights as calculated. · must show Bars drawn to calculated height, Labels match table
(b) Curve $y = e^{-x/10}$ decreasing, starting at 1 and decaying, same axes as histogram. · must show Curve passes (0,1), decays smoothly, Sketch overlays histogram axes
(c) Histogram is discrete/grouped, curve is continuous; histogram heights show frequency density, curve height shows value of function (temperature at each instant). · must show Discrete vs continuous, What height represents in each
(d) $y = f(x-10)$ is a translation 10 units to the right; graph is shifted right by 10s. · must show Translation right by 10
(e) New function is $y = e^{-x/20}$. Graph is stretched horizontally by scale factor 2. · must show Function $e^{-x/20}$, Horizontal stretch by 2
Must show: frequency density calculation, histogram correct widths, curve $y = e^{-x/10}$ sketched, translation for $f(x-10)$, stretch for $f(\frac{x}{2})$
Common mistakes: Mistaking histogram height for frequency; Incorrect translation direction; Wrong stretch factor
Show mark scheme and worked solution
Marking points
M1 (a(i)) · 1 mark · Correctly determines all class widths.
A1 (a(i)) · 1 mark · Correctly calculates and writes all frequency densities.
B1 (a(ii)) · 1 mark · Histogram drawn with correct heights, widths and labelled axes. (follow-through)
B2 (b) · 2 marks · Sketches $y = e^{-x/10}$ on same axes, correctly placed, well-matched to interval/scale.
B3 (c) · 2 marks · Two accurate differences explained, referencing discrete/continuous and meaning of height.
B4 (d) · 2 marks · Explains and identifies the transformation algebraically and graphically (shift right by 10s).
B5 (e) · 2 marks · Writes $y = e^{-x/20}$ and describes horizontal stretch by factor 2.
Worked solution
a(i) Class widths: 10, 10, 10, 20. Frequency densities: 6/10 = 0.6; 14/10 = 1.4; 9/10 = 0.9; 11/20 = 0.55.
a(ii) Histogram bars drawn for each interval, with heights as above. Bar 1: 0–10 width, height 0.6. Bar 2: 10–20 width, height 1.4. Bar 3: 20–30 width, height 0.9. Bar 4: 30–50 width, height 0.55.
b) Curve $y = e^{-x/10}$ starts at (0,1), decreases smoothly. Sketch overlays histogram axes, curve decays.
c) Histogram is grouped/discrete, bars represent frequency density of intervals; curve is continuous, heights represent modeled temperature at every $x$.
d) $y = f(x-10)$ is a translation of $y=f(x)$ right by 10 units; algebraically: replace $x$ with $x-10$.
e) New function $y = e^{-x/20}$; horizontally, graph is stretched by factor 2 (all $x$ values double).