A rectangle has its base on the x-axis and its upper two vertices on the parabola $y = 27 - x ^ { 2 }$ The largest area of the rectangle is

A)100
B)105
C)108
D)125
E)128

Question

A rectangle has its base on the x-axis and its upper two vertices on the parabola $y = 27 - x ^ { 2 }$ The largest area of the rectangle is

A)100
B)105
C)108
D)125
E)128

Quizplus · Accepted Answer

The vertices of the rectangle on the parabola can be represented as $(x, 27 - x^2)$ and $(-x, 27 - x^2)$, since the rectangle is symmetric about the y-axis. The width of the rectangle is $2x$ and the height is $27 - x^2$. Thus, the area of the rectangle is $A = 2x(27 - x^2) = 54x - 2x^3$. To find the maximum area, we differentiate $A$ with respect to $x$ and set the derivative equal to zero: $\frac{dA}{dx} = 54 - 6x^2 = 0$. Solving for $x$, we get $x^2 = 9$, so $x = 3$ (considering the positive value as we are dealing with distances). Substituting $x = 3$ into the area formula, we get $A = 54(3) - 2(3)^3 = 162 - 54 = 108$.

A Rectangle Has Its Base on the X-Axis and Its y=27−x2y = 27 - x ^ { 2 }y=27−x2

A Rectangle Has Its Base on the X-Axis and Its $y = 27 - x ^ { 2 }$