Solve the problem.
-A rectangle with sides parallel to the axes is inscribed in the region bounded by the axes and the line x + 2y = 2. Find the maximum area of this rectangle.
A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 2 }$

Question

Solve the problem.
-A rectangle with sides parallel to the axes is inscribed in the region bounded by the axes and the line x + 2y = 2. Find the maximum area of this rectangle. 
A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 2 }$

Quizplus · Accepted Answer

The equation of the line is $x + 2y = 2$. Let the coordinates of the top right corner of the rectangle be $(x, y)$. The area of the rectangle is $A = xy$. Since the rectangle is bounded by the line, we have $y = \frac{2 - x}{2}$. Substituting this into the area formula gives $A = x\left(\frac{2 - x}{2}\right) = \frac{2x - x^2}{2}$. To find the maximum area, we differentiate $A$ with respect to $x$ and set the derivative equal to zero: $\frac{dA}{dx} = 1 - x = 0$, which gives $x = 1$. Substituting $x = 1$ back into the equation of the line gives $y = \frac{1}{2}$. Therefore, the maximum area is $A = 1 \times \frac{1}{2} = \frac{1}{2}$.

Solve the Problem 14\frac { 1 } { 4 }41​ B) 13\frac { 1 } { 3 }31​

Solve the Problem $\frac { 1 } { 4 }$ B) $\frac { 1 } { 3 }$