|
|
Exam-Style Question on OptimisationA mathematics exam-style question with a worked solution that can be revealed gradually |
Question id: 657. This question is similar to one that appeared on an IB AA Higher paper in 2023. The use of a calculator is allowed.
A circle with equation \(x^2+y^2=25 \) has centre \((0,0)\) and radius 5.
A trapezium, ABCD, is inscribed in the circle with its vertices at \( A(x,y), B(4,-3), C(-4,-3) \text{ and } D(-x,y)\), where A and D are variable points in the first and second quadrants respectively. This is shown in the following sketch.
(a) For point A, show that \(y = \sqrt{25-x^2} \).
(b) Hence, find an expression for T, the area of trapezium ABCD, in terms of x.
(c) Show that \( \dfrac{dT}{dx} = 3 - \dfrac{2x^2 +4x -25}{\sqrt{25-x^2}} \).
(d) Hence or otherwise, find the value of \(x\) such that T is a maximum.
|
The worked solutions to these exam-style questions are only available to those who have a Transum Subscription. Subscribers can drag down the panel to reveal the solution line by line. This is a very helpful strategy for the student who does not know how to do the question but given a clue, a peep at the beginnings of a method, they may be able to make progress themselves. This could be a great resource for a teacher using a projector or for a parent helping their child work through the solution to this question. The worked solutions also contain screen shots (where needed) of the step by step calculator procedures. A subscription also opens up the answers to all of the other online exercises, puzzles and lesson starters on Transum Mathematics and provides an ad-free browsing experience. |
|
Drag this panel down to reveal the solution
The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
The solutions to the questions on this website are only available to those who have a Transum Subscription.
Exam-Style Questions Main Page
To search the entire Transum website use the search box in the grey area below.
Do you have any comments about these exam-style questions? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.
©1997 - 2024 Transum Mathematics :: For more exam-style questions and worked solutions go to Transum.org/Maths/Exam/
