Question

The area of the region bounded by the lines $x + 2y = 12$, $x = 2$, $x = 6$, and the $x$-axis is:

• A. $34 \text{ sq units}$
• B. $20 \text{ sq units}$
• C. $24 \text{ sq units}$
• D. $16 \text{ sq units}$

Answer 1

The Correct Option is D

To find the area of the region bounded by \( x + 2y = 12 \), \( x = 2 \), \( x = 6 \), and the x-axis, we start by expressing \( y \) in terms of \( x \) from the equation \( x + 2y = 12 \):

\[ y = \frac{12 - x}{2} \]

The area between \( x = 2 \) and \( x = 6 \) under the line \( y = \frac{12 - x}{2} \) is given by:

\[ \text{Area} = \int_{2}^{6} \frac{12 - x}{2} \, dx \]

Evaluating this integral:

\[= \int_{2}^{6} \frac{12 - x}{2} \, dx = \frac{1}{2} \int_{2}^{6} (12 - x) \, dx\]

\[= \frac{1}{2} \left[ 12x - \frac{x^2}{2} \right]_{2}^{6}\]

\[= \frac{1}{2} \left[ \left( 12 \times 6 - \frac{6^2}{2} \right) - \left( 12 \times 2 - \frac{2^2}{2} \right) \right]\]

\[= \frac{1}{2} \left[ (72 - 18) - (24 - 2) \right]\]

\[= \frac{1}{2} [54 - 22] = \frac{1}{2} \times 32 = 16\]

Therefore, the area of the region is 16 sq units.