Question 6 of 11intermediate🔧 ApplyLong Answer4 marks

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.

Correct Answer

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. Therefore, the cost of each article is ₹ (2x + 3).

The total cost of production is given as ₹ 90.

We can form the quadratic equation as follows:

Number of articles × Cost of each article = Total Cost

x(2x+3)=90x(2x + 3) = 90 2x2+3x−90=02x^2 + 3x - 90 = 0

To find the value of x, we solve this equation using the factorisation method:

2x2+15x−12x−90=02x^2 + 15x - 12x - 90 = 0 x(2x+15)−6(2x+15)=0x(2x + 15) - 6(2x + 15) = 0 (2x+15)(x−6)=0(2x + 15)(x - 6) = 0

This gives two possible values for x:

x=−152orx=6x = -\frac{15}{2} \quad \text{or} \quad x = 6

Since the number of articles cannot be negative, we reject x=−152x = -\frac{15}{2}.

Thus, the number of articles produced is 6.

Now, we find the cost of each article:

Cost=2x+3=2(6)+3=12+3=₹ 15\text{Cost} = 2x + 3 = 2(6) + 3 = 12 + 3 = \text{₹ } 15

∴ The number of articles produced is 6 and the cost of each article is ₹ 15.

Exercise: EXERCISE 4.2 | Q: 6 | (Page 7)
For More Understanding

Explanation

This solution follows the method shown in Example 1(ii) and Example 6 of the provided text. First, a variable xx is defined to represent the unknown quantity (number of articles). The relationship 'cost is 3 more than twice the number' is translated into the algebraic expression (2x+3)(2x + 3). The equation is formed by equating the product of quantity and cost to the total cost (90). The resulting quadratic equation is solved by splitting the middle term. Finally, the negative root is discarded because a count of articles cannot be negative, consistent with the logic in Example 6.

Solution Steps

  1. Step 1: Define the variable. Let the number of articles be xx.

  2. Step 2: Formulate the equation. Cost per article =2x+3= 2x + 3. Total Cost =x(2x+3)=90= x(2x + 3) = 90.

  3. Step 3: Simplify to standard form.

    2x2+3x−90=02x^2 + 3x - 90 = 0

  4. Step 4: Factorise.

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

  5. Step 5: Solve for xx. x=−152x = -\frac{15}{2} or x=6x = 6. Reject negative value.

  6. Step 6: Calculate cost. Cost =2(6)+3=15= 2(6) + 3 = 15.

PreviousNext