Question

an aboriginal dot painting artwork is sold at $10 per unit area in \text{cm}^2.

the artwork is in rectangular shape with length ax - b and width cx - d, where x is a positive integer.

if the price, p, in dollars of the artwork is calculated by the formula:
\\p = \\$(20x^2 - 150x + 180),\\

(a) determine the value of a, b, c and d.
(b) calculate the ratio between the length and width when x = 12.

Explanation:

Relate price to area

The price \(P\) is \$10 per \(\text{cm}^2\).

$$ P = 10 \times \text{Area} $$

$$ 20x^2 - 150x + 180 = 10 \times (ax - b)(cx - d) $$

Factor out the constant

Using Quadratic Factoring:
Divide both sides by 10 to find the area.

$$ 2x^2 - 15x + 18 = (ax - b)(cx - d) $$

Factor the quadratic expression

Using Quadratic Factoring:
Find two numbers that multiply to \(2 \times 18 = 36\) and add to \(-15\).
These numbers are \(-12\) and \(-3\).

$$ 2x^2 - 12x - 3x + 18 = 2x(x - 6) - 3(x - 6) = (2x - 3)(x - 6) $$

Thus, we have:

$$ (ax - b)(cx - d) = (2x - 3)(x - 6) $$

Comparing coefficients (assuming \(ax-b\) is the longer side or matching the standard order):

$$ a = 2, \quad b = 3, \quad c = 1, \quad d = 6 $$

Calculate length and width

Using Algebraic Modeling:
Substitute \(x = 12\) into the length and width expressions.

$$ \text{Length} = 2(12) - 3 = 21 $$

$$ \text{Width} = 1(12) - 6 = 6 $$

Find the ratio

Using Algebraic Modeling:
Calculate the ratio of length to width.

$$ \text{Ratio} = \frac{21}{6} = \frac{7}{2} = 3.5 $$

Answer:

Question (a)

The values are:
\(a = 2\), \(b = 3\), \(c = 1\), and \(d = 6\) (or equivalently \(a = 1\), \(b = 6\), \(c = 2\), and \(d = 3\)).

Question (b)

The ratio between the length and width when \(x = 12\) is:
\(\frac{7}{2}\) (or \(3.5 : 1\)).