Questions: A rectangle is bounded by the parabola defined by y=x^2, the x-axis, and the line x=5 as shown in the figure. If the area of the rectangle is 4 in ^2 determine the dimensions of the rectangle. The rectangle is either in by in or in by in.

Solution Steps

The rectangle has a vertex $(x, y)$ on the parabola $y = x^2$. The width of the rectangle is given as $5 - x$, and the height is given by $y$. Since $y = x^2$, the height can also be written as $x^2$.

The area of the rectangle is given by width multiplied by height. We are given that the area is 4 in². So we have:

$(5 - x) * x^2 = 4$

Expand and rearrange the equation to form a cubic equation: $5x^2 - x^3 = 4$ $x^3 - 5x^2 + 4 = 0$

By observation or by using the Rational Root Theorem and testing possible rational roots, we find that $x = 1$ is a root, since $(1)^3 - 5(1)^2 + 4 = 1 - 5 + 4 = 0$.

Now we perform polynomial division to factor out $(x-1)$: $(x-1)(x^2 - 4x - 4) = 0$ We can find the other roots of $x^2 - 4x - 4 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(-4)}}{2(1)} = \frac{4 \pm \sqrt{16 + 16}}{2} = \frac{4 \pm \sqrt{3### Step 1: Define the Variables Let the coordinates of the top-right corner of the rectangle be \((x, y)\). The height of the rectangle is \(y\) and the width is \(5 - x\).

The area \(A\) of the rectangle is given by: \[ A = \text{width} \times \text{height} = (5 - x) \times y \]

Since the rectangle is bounded by the parabola \(y = x^2\), substitute \(y = x^2\) into the area equation: \[ A = (5 - x) \times x^2 \]

Given that the area of the rectangle is 4 square inches: \[ (5 - x) \times x^2 = 4 \]

Solve the equation for \(x\): \[ 5x^2 - x^3 = 4 \] \[ x^3 - 5x^2 + 4 = 0 \]

Factor the cubic equation: \[ (x - 1)(x^2 - 4x + 4) = 0 \] \[ (x - 1)(x - 2)^2 = 0 \]

Solve for \(x\): \[ x - 1 = 0 \quad \text{or} \quad (x - 2)^2 = 0 \] \[ x = 1 \quad \text{or} \quad x = 2 \]

For \(x = 1\): \[ y = 1^2 = 1 \] For \(x = 2\): \[ y = 2^2 = 4 \]

For \(x = 1\): \[ \text{Width} = 5 - 1 = 4 \] \[ \text{Height} = 1 \] Dimensions: \(4 \text{ in} \times 1 \text{ in}\)

For \(x = 2\): \[ \text{Width} = 5 - 2 = 3 \] \[ \text{Height} = 4 \] Dimensions: \(3 \text{ in} \times 4 \text{ in}\)

Final Answer