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 artwork is sold at $\$10$ per unit area in \(\text{cm}^2\).
The area \(A\) of the rectangular artwork is:

$$A = \text{length} \times \text{width} = (ax - b)(cx - d)$$

The price \(P\) is given by:

$$P = 10 \times A = 10(ax - b)(cx - d)$$

Expand the price formula

We expand the algebraic expression for the price \(P\):

$$P = 10(acx^2 - (ad + bc)x + bd)$$

$$P = 10acx^2 - 10(ad + bc)x + 10bd$$

Compare coefficients to find parameters

We compare this expanded form with the given price formula:

$$P = 20x^2 - 150x + 180$$

By equating the coefficients:

1. \(10ac = 20 \implies ac = 2\)
2. \(10bd = 180 \implies bd = 18\)
3. \(10(ad + bc) = 150 \implies ad + bc = 15\)

Since \(a, b, c, d\) are integers from the linear dimensions \(ax-b\) and \(cx-d\):
From \(ac = 2\), the positive integer factors are either \((a=2, c=1)\) or \((a=1, c=2)\).
From the diagram, the length is labeled \(ax-b\) and the width is \(cx-d\). Usually, length \(\ge\) width, so we test \(a=2, c=1\).
If \(a=2\) and \(c=1\):

$$2d + b = 15$$

Since \(bd = 18\), we substitute \(b = \frac{18}{d}\):

$$2d + \frac{18}{d} = 15 \implies 2d^2 - 15d + 18 = 0$$

Factoring this quadratic equation:

$$(2d - 3)(d - 6) = 0$$

Since \(d\) must be an integer for standard school algebra problems of this type:

$$d = 6$$

Then:

$$b = \frac{18}{6} = 3$$

Thus, the parameters are:

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

This gives:
Length \(= 2x - 3\)
Width \(= x - 6\)

Calculate length and width at x = 12

We substitute \(x = 12\) into the expressions:

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

$$\text{Width} = 12 - 6 = 6$$

Determine the ratio

We find the ratio of length to width:

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

Answer:

Question (a)

The values are:

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

Question (b)

The ratio between the length and width when \(x = 12\) is:

$$\frac{7}{2} \quad (\text{or } 7:2)$$