Largest rectangle under a sloping line
Largest rectangle under a sloping line
IP 1
7 marks
The line \(y=4-\dfrac12x\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). \(O\) is the origin.
- Find the coordinates of \(A\) and \(B\).[2]
- Find the area of triangle \(OAB\).[1]
- \(P\left(t,4-\dfrac12t\right)\), where \(0<t<8\), is a point on \(AB\). A rectangle has opposite corners \(O\) and \(P\) and sides along the axes. Show that its area is \(4t-\dfrac12t^2\).[2]
- By showing that \(4t-\dfrac12t^2=8-\dfrac12(t-4)^2\), find the largest possible area of the rectangle and the corresponding point \(P\).[2]
Solution:
Solution locked
680+ questions here come with free worked solutions. A free account saves your questions and progress; this one needs tuition enrolment.
Create a free account to practise similar questions.
Answer:(a) \(A(8,0)\), \(B(0,4)\) (b) \(16\text{ units}^2\) (c) \(t\left(4-\dfrac12t\right)=4t-\dfrac12t^2\) (d) largest area \(8\text{ units}^2\) at \(P(4,2)\)
Need help? Join our IP Math tuition programme.
Learn more