Q6:

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was $3$ more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ $90$, find the number of articles produced and the cost of each article.

Given:

1. The total cost of production on a particular day = ₹ $90$.

2. The cost of production of each article is $3$ more than twice the number of articles produced.

To Find:

1. The number of articles produced on that day.

2. The cost of each article.

Step 1: Defining Variables

Let the number of pottery articles produced on that day be $x$.

According to the problem, the cost of production of each article is $3$ more than twice the number of articles produced.

Therefore, the cost of each article = $(2x + 3)$ rupees.

Step 2: Formulating the Quadratic Equation

The total cost of production is calculated by multiplying the number of articles by the cost per article.

Total Cost = (Number of articles) $\times$ (Cost of each article)

$90 = x(2x + 3)$ [Given that total cost is ₹ 90]

$90 = 2x^2 + 3x$ [Distributive property of multiplication]

Rearranging the terms to form a standard quadratic equation $ax^2 + bx + c = 0$:

$2x^2 + 3x - 90 = 0$

Step 3: Solving the Quadratic Equation by Factorization

To factorize $2x^2 + 3x - 90 = 0$, we need to find two numbers whose product is $a \times c = 2 \times (-90) = -180$ and whose sum is $b = 3$.

The factors of $-180$ that add up to $3$ are $15$ and $-12$.

Splitting the middle term:

$2x^2 + 15x - 12x - 90 = 0$

Grouping the terms:

$(2x^2 + 15x) - (12x + 90) = 0$

$x(2x + 15) - 6(2x + 15) = 0$ [Factoring out common terms]

$(2x + 15)(x - 6) = 0$

Step 4: Finding the Roots

Setting each factor to zero:

1. $2x + 15 = 0 \implies 2x = -15 \implies x = -7.5$

2. $x - 6 = 0 \implies x = 6$

Since the number of articles produced cannot be negative, we discard $x = -7.5$.

Therefore, the number of articles produced is $x = 6$.

Step 5: Calculating the Cost of Each Article

Cost of each article = $2x + 3$

Substitute $x = 6$ into the expression:

Cost = $2(6) + 3$

Cost = $12 + 3 = 15$

The cost of each article is ₹ $15$.

Final Answer: The number of articles produced is 6 and the cost of each article is ₹ 15.