Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.
A model for the water level in a test tank is
where is the water level in centimetres and is the time in hours after the start of a test. The model is valid for
Show that
Hence determine the intervals in the model domain for which and State when the water level is increasing and when it is decreasing. Give boundary times to 2 decimal places.
A manufacturer models its profit, in thousands of dollars, when producing thousand units, by
The model is shown in Figure 1.
Using a GDC, determine the values of at which the gradient of the curve is zero. Give your answers to 1 decimal place.
Hence, find the coordinates of both stationary points and state whether each is a local maximum or a local minimum. Give the profit coordinates to 2 decimal places.
The owner claims that the local maximum gives the greatest profit in the model’s domain. Use Figure 1 to decide whether this claim is correct. Justify your answer.
A surveyor estimates the area of an irregular lake inlet. A straight survey line of length is laid along one edge of the inlet. At equally spaced stations, the perpendicular distance from the line to the opposite shoreline is measured. The width increases continuously between consecutive stations.
| Distance along survey line (m) | 0 | 55 | 110 | 165 | 220 |
|---|---|---|---|---|---|
| Width (m) | 0 | 15 | 28 | 42 | 55 |
Use the trapezoidal rule to estimate the area of the inlet. Give your answer to the nearest
Use rectangular strips to find lower and upper estimates for the area. Hence state bounds for the true area and whether your estimate in part (a) is consistent with these bounds.
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