| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Optimise perimeter or area of 2D region |
| Difficulty | Moderate -0.3 This is a standard C1 optimization problem with a straightforward constraint (perimeter = 20m) leading to a simple quadratic area function. Part (i) requires basic algebraic manipulation to express area in terms of one variable, and part (ii) involves routine differentiation of a quadratic and finding the maximum—both are textbook exercises requiring no novel insight, making it slightly easier than average. |
| Spec | 1.02z Models in context: use functions in modelling1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Length \(= 20 - 2x\) | M1 | Expression for length of enclosure in terms of \(x\) |
| Area \(= x(20-2x) = 20x - 2x^2\) | A1 (2) | Correctly shows area \(= 20x - 2x^2\) AG |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dA}{dx} = 20 - 4x\) | M1 | Differentiates area expression |
| For max, \(20 - 4x = 0\) | M1 | Uses \(\frac{dy}{dx} = 0\) |
| \(x = 5\) only | A1 | |
| Area \(= 50\) | A1 (4) |
# Question 5:
**(i)**
Length $= 20 - 2x$ | M1 | Expression for length of enclosure in terms of $x$
Area $= x(20-2x) = 20x - 2x^2$ | A1 (2) | Correctly shows area $= 20x - 2x^2$ **AG**
**(ii)**
$\frac{dA}{dx} = 20 - 4x$ | M1 | Differentiates area expression
For max, $20 - 4x = 0$ | M1 | Uses $\frac{dy}{dx} = 0$
$x = 5$ only | A1 |
Area $= 50$ | A1 (4) |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{581ef815-59f0-434e-a7ec-9128e74c0323-2_256_1113_1366_516}
The diagram shows a rectangular enclosure, with a wall forming one side. A rope, of length 20 metres, is used to form the remaining three sides. The width of the enclosure is x metres.\\
(i) Show that the enclosed area, $\mathrm { Am } ^ { 2 }$, is given by
$$A = 20 x - 2 x ^ { 2 } .$$
(ii) Use differentiation to find the maximum value of A .
\hfill \mbox{\textit{OCR C1 2007 Q5 [6]}}