Introduction to Linear Polynomials | Exercise 2.2

Question 7

If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle?

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Solution
Understand the Question
  • Let the width of the rectangle be x cmx\text{ cm}.
  • The length is three more than twice the width, which can be expressed as (2x+3) cm(2x + 3)\text{ cm}.
  • The formula for the perimeter of a rectangle is Perimeter=2×(length+width)\text{Perimeter} = 2 \times (\text{length} + \text{width}).
  • We can set up a linear equation equating the perimeter formula to the given value of 24 cm24\text{ cm} and solve for xx to find the dimensions.

Step 1 · Set up equation and solve for width

Let the width of the rectangle be x cmx\text{ cm}.

Then, Length=(2x+3) cm\text{Length} = (2x + 3)\text{ cm}.Diagram 1

Given Perimeter=24 cm\text{Perimeter} = 24\text{ cm}:

2((2x+3)+x)=242(3x+3)=246x+6=246x=24−66x=18x=186x=3 cm\begin{aligned} 2((2x + 3) + x) &= 24 \\ 2(3x + 3) &= 24 \\ 6x + 6 &= 24 \\ 6x &= 24 - 6 \\ 6x &= 18 \\ x &= \dfrac{18}{6} \\ x &= 3\text{ cm} \end{aligned}

Step 2 · Calculate dimensions

Substituting x=3x = 3 into the dimensions: Width=3 cm\text{Width} = 3\text{ cm}

Length=2(3)+3=6+3=9 cm\begin{aligned} \text{Length} &= 2(3) + 3 \\ &= 6 + 3 \\ &= 9\text{ cm} \end{aligned}
Answer

Width=3 cm,  Length=9 cm\text{Width} = 3\text{ cm},\; \text{Length} = 9\text{ cm}

Common Mistakes
  • Translating Algebraic Expressions: Misinterpreting "three more than twice its width" as 3x+23x + 2 or 2(x+3)2(x + 3) instead of 2x+32x + 3.
  • Perimeter Formula Error: Forgetting to multiply by 22 in the perimeter formula, writing length+width=24\text{length} + \text{width} = 24 instead of 2(length+width)=242(\text{length} + \text{width}) = 24.

More questions in Exercise 2.2

Q1

Find the value of the linear polynomial 5x−35x - 3 if:

(i) x=0x = 0

(ii) x=−1x = -1

(iii) x=2x = 2

Q2

Find the value of the quadratic polynomial 7s2−4s+67s^2 - 4s + 6 if:

(i) s=0s = 0

(ii) s=−3s = -3

(iii) s=4s = 4

Q3

The present age of Salil's mother is three times Salil's present age. After 5 years, their ages will add up to 70 years. Find their present ages.

Q4

The difference between two positive integers is 63. The ratio of the two integers is 2:52 : 5. Find the two integers.

Q5

Ruby has 3 times as many two-rupee coins as she has five rupee-coins. If she has a total ₹88, how many coins does she have of each type?

Q6

A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?

Q7

If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle?

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