Introduction to Linear Polynomials | Exercise 2.3

Question 4

Suppose the length of a rectangular box is 7 cm7\text{ cm} and breadth is 11 cm11\text{ cm}. Find the volume if the height is (i) 5 cm5\text{ cm}, (ii) 9 cm9\text{ cm}, (iii) 13 cm13\text{ cm}. Find the linear pattern representing the volume of the rectangular box.

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Solution
Understand the Question
  • The volume of a rectangular box is given by the formula: Volume=Length×Breadth×Height=Base Area×Height\text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height} = \text{Base Area} \times \text{Height}
  • Given length =7 cm= 7\text{ cm} and breadth =11 cm= 11\text{ cm}, the base area is: Base Area=7×11=77 cm2\text{Base Area} = 7 \times 11 = 77\text{ cm}^2
  • The volume for any height hh is found by multiplying the fixed base area by hh, giving a linear relation V=77hV = 77h.

(i) Find the volume if the height is 5 cm5\text{ cm}.

Step 1 · Calculate Volume for Height 5 cm5\text{ cm}

Diagram 1

Base Area=Length×Breadth=7×11=77 cm2\begin{aligned} \text{Base Area} &= \text{Length} \times \text{Breadth} \\[0.6em] &= 7 \times 11 \\[0.6em] &= 77\text{ cm}^2 \end{aligned} Volume=Base Area×Height=77×5=385 cm3\begin{aligned} \text{Volume} &= \text{Base Area} \times \text{Height} \\[0.6em] &= 77 \times 5 \\[0.6em] &= 385\text{ cm}^3 \end{aligned}
Answer

(i) 385 cm3385\text{ cm}^3

(ii) Find the volume if the height is 9 cm9\text{ cm}.

Step 1 · Calculate Volume for Height 9 cm9\text{ cm}

Volume=77×9=693 cm3\begin{aligned} \text{Volume} &= 77 \times 9 \\[0.6em] &= 693\text{ cm}^3 \end{aligned}
Answer

(ii) 693 cm3693\text{ cm}^3

(iii) Find the volume if the height is 13 cm13\text{ cm}.

Step 1 · Calculate Volume for Height 13 cm13\text{ cm}

Diagram 2

Volume=77×13=1001 cm3\begin{aligned} \text{Volume} &= 77 \times 13 \\[0.6em] &= 1001\text{ cm}^3 \end{aligned}
Answer

(iii) 1001 cm31001\text{ cm}^3

Find the linear pattern representing the volume of the rectangular box.

Step 1 · Determine the Linear Pattern

Let the height of the rectangular box be h cmh\text{ cm}.

Given the constant base area is 77 cm277\text{ cm}^2, the volume VV is:

V=77×hV = 77 \times h

V=77hV = 77h

Answer

V=77hV = 77h

Common Mistakes
  • Unit Confusion: Using cm2\text{cm}^2 instead of cm3\text{cm}^3 for volume.
  • Incorrect Formula: Adding the dimensions instead of multiplying (Volume=length×breadth×height\text{Volume} = \text{length} \times \text{breadth} \times \text{height}).
  • Linear Pattern Form: Omitting the variable hh when stating the general algebraic pattern.

More questions in Exercise 2.3

Q1

Solve the following:

  1. A student has ₹500 in her savings bank account. She gets ₹150 every month as pocket money. How much money will she have at the end of every month from the second month onwards? Find a linear expression to represent the amount she will have in the nthn^{\text{th}} month.
Q2

A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after 1,2,3,…1, 2, 3, \dots hours? Find a linear expression to represent the number of members at the end of the nthn^{\text{th}} hour.

Q3

Suppose the length of a rectangle is 13 cm13\text{ cm}. Find the area if the breadth is (i) 12 cm12\text{ cm}, (ii) 10 cm10\text{ cm}, (iii) 8 cm8\text{ cm}. Find the linear pattern representing the area of the rectangle.

Q4

Suppose the length of a rectangular box is 7 cm7\text{ cm} and breadth is 11 cm11\text{ cm}. Find the volume if the height is (i) 5 cm5\text{ cm}, (ii) 9 cm9\text{ cm}, (iii) 13 cm13\text{ cm}. Find the linear pattern representing the volume of the rectangular box.

Q5

Sarita is reading a book of 500 pages. She reads 20 pages every day. How many pages will be left after 15 days? Express this as a linear pattern.

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