Solve the problem.
-A farmer has 1,600 yards of fencing to enclose a rectangular garden. Express the area A of the rectangle as a function of the width x of the rectangle. What is the domain of A?
A) $A ( x ) = - x ^ { 2 } + 1,600 x ; \{ x \mid 0

Question

Solve the problem.
-A farmer has 1,600 yards of fencing to enclose a rectangular garden. Express the area A of the rectangle as a function of the width x of the rectangle. What is the domain of A?
A) $A ( x ) = - x ^ { 2 } + 1,600 x ; \{ x \mid 0 < x < 1,600 \}$
B) $\mathrm { A } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 800 \mathrm { x } ; \{ \mathrm { x } \mid 0 < \mathrm { x } < 1,600 \}$
C) $\mathrm { A } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 800 \mathrm { x } ; \{ \mathrm { x } \mid 0 < \mathrm { x } < 800 \}$
D) $\mathrm { A } ( \mathrm { x } ) = \mathrm { x } ^ { 2 } + 800 \mathrm { x } ; \{ \mathrm { x } \mid 0 < \mathrm { x } < 800 \}$

Quizplus · Accepted Answer

The perimeter of the rectangle is given as 1,600 yards, which can be expressed as $2x + 2y = 1,600$, where $x$ is the width and $y$ is the length. Solving for $y$, we get $y = 800 - x$. The area $A$ of the rectangle is $x \times y$, which gives $A(x) = x(800 - x) = 800x - x^2$. The domain of $A$ is determined by the physical constraints of the problem: $x$ must be greater than 0 but less than 800, because if $x$ is 800, then $y$ would be 0 (resulting in no area), and $x$ cannot be more than 800 because it would make $y$ negative, which is not possible for the dimensions of a rectangle.

Solve the Problem A(x)=−x2+1,600x;{x∣0<x<1,600}A ( x ) = - x ^ { 2 } + 1,600 x ; \{ x \mid 0 < x < 1,600 \}A(x)=−x2+1,600x;{x∣0<x<1,600}

Solve the Problem $A ( x ) = - x ^ { 2 } + 1,600 x ; \{ x \mid 0 < x < 1,600 \}$