Two more 30-mark Paper 032 style questions with tagged marks — a calculus optimisation of a cylindrical can, and a lighthouse problem using a circle, a line and the discriminant.
4 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
A can that uses the least metal. A ship whose path just touches a lighthouse’s range. Two real situations, two full 30-mark practice questions.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Copy these into your notebook in your own words — that is how they stick.
V = πr²h; S = 2πr² + 2πrh; minimum S for fixed V when h = 2r.
One variable → dS/dr = 0 → second derivative → answer with units.
Percentage more = (b − a)/a × 100.
x² + y² = r²; line + circle → quadratic → discriminant.
Discriminant zero; path just touches.
Distance between the intersection points.
Perpendicular from O: gradient −1/m.
Quote numbers when recommending.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (60 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [30 marks]
A company makes closed cylindrical cans of volume 500 cm³. Let the radius be r cm and the height h cm. A calculator may be used.
(a) (a) Write an expression for h in terms of r. [2]
(b) (b) Show that the total surface area is S = 2πr² + 1000/r. [3]
(c) (c) Find dS/dr. [2]
(d) (d) Find the value of r for which S is a minimum, giving your answer to 3 significant figures. [3]
(e) (e) Show that this value of r gives a minimum. [2]
(f) (f) Calculate the corresponding height and the minimum surface area. [3]
(g) (g) Show that h = 2r at the minimum. [2]
(h) (h) A tall thin can has r = 3 cm. Calculate its surface area and the percentage more metal it uses. [4]
(i) (i) Metal costs $0.002 per cm². Find the metal cost of one minimum can and of 10 000 minimum cans. [3]
(j) (j) Recommend the dimensions for the can and give ONE reason why the company may choose a different shape. [2]
(k) (k) State one limitation of this model. [1]
(l) (l) The company also considers an open-top can of the same volume. Write S in terms of r, find the radius that minimises S, and state how h compares with r. [3]
Question 2 [30 marks]
A lighthouse is at the origin O. Its light can be seen up to 10 km from O. A ship sails along the straight line with equation 3x − 4y = 40 (distances in km).
(a) (a) Write the equation of the boundary of the lighthouse’s range. [2]
(b) (b) Show that the ship’s path meets the boundary at (0, −10) and (9.6, −2.8). [5]
(c) (c) Calculate the distance the ship travels within range. [2]
(d) (d) The ship travels at 24 km/h. For how many minutes is it within range? [2]
(e) (e) Another path has equation 3x − 4y = c. Show that it is a tangent to the boundary when c = 50 (c > 0). [4]
(f) (f) Find the coordinates of the point of contact for c = 50. [3]
(g) (g) The captain wants to avoid being within range. State the condition on c for the path 3x − 4y = c (c > 0) to stay out of range and give a reason. [3]
(h) (h) Find the shortest distance from O to the path 3x − 4y = 40 and comment. [5]
(i) (i) State one assumption in this model. [1]
(j) (j) Write a short recommendation for the captain. [3]
Everything in this lesson is based on these parts of the CXC syllabus and papers.