Question

A rectangular sheet of paper has its area \(32\) meters\({}^{2}\). The margins at the top and bottom are \(50\) cm each and at the sides \(25\) cm each. If the area of the printed space is maximum, then the dimensions of the paper are

• A. \(16\) meters and \(2\) meters
• B. \(30\) meters and \(2\) meters
• C. \(8\) meters and \(4\) meters
• D. \(25\) meters and \(7\) meters

Hint:

Express the printed area in terms of one variable using the fixed paper area and maximise it.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Let the width be \(x\) m and the length \(y\) m. Then \(xy=32\).

Step 2: Printed Area
Margins: top and bottom \(0.5\) m each, sides \(0.25\) m each.
Printed area \(P=(x-0.5)(y-1)=xy-x-0.5y+0.5\).

Step 3: Express in One Variable
With \(y=\dfrac{32}{x}\): \(P=32-x-\dfrac{16}{x}+0.5\).
\(\dfrac{dP}{dx}=-1+\dfrac{16}{x^{2}}=0\Rightarrow x=4\).

Step 4: Check and Conclude
\(\dfrac{d^{2}P}{dx^{2}}=-\dfrac{32}{x^{3}}<0\), so it is a maximum.
Then \(y=\dfrac{32}{4}=8\). The dimensions are 8 m and 4 m.

Final Answer:
The paper is 8 meters by 4 meters. \[ \boxed{\text{(C) }8 \text{ meters and } 4 \text{ meters}} \]