Question

1. given the following rectangle and circle, at what approximate value of x are the two areas equal?

rectangle dimensions: height is \\(3x - 1\\), width is \\(x + 6\\)
circle dimension: radius is \\(x + 1\\)

\\(x = 0.91\\)
\\(x = 0.44\\)
\\(x = 0.86\\)
the curves do not intersect, so there is no solution.

Explanation:

Express the area of the rectangle

Using the Algebraic Simplification knowledge point

$$ A_{\text{rect}} = (3x - 1)(x + 6) = 3x^2 + 17x - 6 $$

Express the area of the circle

The radius of the circle is given as \(r = x + 1\).

$$ A_{\text{circle}} = \pi r^2 = \pi (x + 1)^2 = \pi(x^2 + 2x + 1) $$

Set the two areas equal

We equate the two area expressions to find the value of \(x\) where they are equal:

$$ 3x^2 + 17x - 6 = \pi(x^2 + 2x + 1) $$

Solve the quadratic equation

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK0
$$

Applying the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

$$
LATEXBLOCK1
$$

Calculating the two possible values:

$$ x_1 \approx \frac{21.1892}{0.28318} \approx 74.82 $$

$$ x_2 \approx \frac{0.2444}{0.28318} \approx 0.863 $$

Verify the solutions

For the dimensions of the rectangle to be positive, we must have:

$$ 3x - 1 > 0 \implies x > \frac{1}{3} \approx 0.33 $$

Both \(x \approx 74.82\) and \(x \approx 0.86\) are greater than \(0.33\). Comparing with the given choices, \(x \approx 0.86\) is listed.

Answer:

• x = 0.91
• x = 0.44
• x = 0.86 (Correct answer)
• The curves do not intersect, so there is no solution.