Quadratic Equations | Exercise 4.2

Question 6

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.

Solution

Understand the Question

Let the number of pottery articles produced on that day be $x$ .
The cost of production of each article is ₹ $(2x + 3)$ (3 more than twice the number of articles).
$\text{Total cost} = (\text{Number of articles}) \times (\text{Cost of each article}) = 90$ .
Setting up this relation gives a quadratic equation in $x$ , which we solve to determine the number of articles and the cost per article.

Step 1 · Set up the Quadratic Equation

Let the number of articles produced be $x$ .

Then, the cost of production of each article $= \text{₹}(2x + 3)$ . Diagram 1

Given that the total cost of production is ₹ $90$ $x(2x + 3) = 90$

$2x^2 + 3x = 90$

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

Step 2 · Solve for the Number of Articles

Solve $2x^2 + 3x - 90 = 0$ by factorisation (splitting the middle term using $15 \times (-12) = -180$ and $15 - 12 = 3$ )

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

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

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

Setting each factor to zero $2x + 15 = 0 \implies x = -\dfrac{15}{2}$

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

Since the number of articles cannot be negative, we reject $x = -\dfrac{15}{2}$ .

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

Step 3 · Calculate the Cost of Each Article

Substitute $x = 6$ into the cost expression $(2x + 3)$

\begin{aligned} \text{Cost of each article} &= 2(6) + 3 \\ &= 12 + 3 \\ &= 15 \end{aligned}

So, the cost of each article is ₹ $15$ .

Answer

Number of articles produced $= 6$ , and the cost of each article $= \text{₹}15$

Common Mistakes

Sign Error in Cost Setup: Writing the cost as $2x - 3$ instead of $2x + 3$ .
Not Rejecting Negative Root: Keeping $x = -\dfrac{15}{2}$ without realizing that physical counts of articles must be positive integers.
Incomplete Answer: Stopping after finding the number of articles ( $x = 6$ ) and forgetting to compute the cost of each article.

Question 6

More questions in Exercise 4.2