Rational Numbers | FIO

Question 4

Find the length of the side of a square whose area is $441 \text{ m}^2$ .

Solution

Understand the Question

The area of a square is given by the formula $\text{Area} = \text{side}^2$ .
Given the area is $441 \text{ m}^2$ , we need to find the side length $s$ by taking the square root: $s = \sqrt{441}$ .
We find the square root of $441$ by prime factorisation.

Step 1 · Set up the Equation

Let $s$ be the length of the side of the square in metres. Diagram 1

\begin{aligned} s \times s &= 441 \\ s^2 &= 441 \end{aligned}

Step 2 · Find the Side Length

To find $s$ , find the square root of $441$ using prime factorisation:

\begin{aligned} 441 &= 3 \times 147 \\ 147 &= 3 \times 49 \\ 49 &= 7 \times 7 \end{aligned}

$441 = 3 \times 3 \times 7 \times 7$

Grouping prime factors into pairs to find the square root:

\begin{aligned} s &= \sqrt{3 \times 3 \times 7 \times 7} \\[0.6em] &= \sqrt{(3 \times 3) \times (7 \times 7)} \\[0.6em] &= \sqrt{3^2 \times 7^2} \\[0.6em] &= 3 \times 7 \\[0.6em] &= 21 \text{ m} \end{aligned}

Answer

$21\text{ m}$

Common Mistakes

Dividing by 4 instead of Square Root: Conflating perimeter with area. Area is $\text{side}^2$ , so we find $\sqrt{441}$ , not $\dfrac{441}{4}$ .
Omitting Units: Forgetting to write the unit of length ( $\text{m}$ ) in the final answer when the area is in $\text{m}^2$ .

More questions in FIO

Which of the following numbers are not perfect squares?

(i) 2032 (ii) 2048 (iii) 1027 (iv) 1089

Which one among $64^2$ , $108^2$ , $292^2$ , $36^2$ has last digit 4?

Given $125^2 = 15625$ , what is the value of $126^2$ ?

(i) $15625 + 126$

(ii) $15625 + 26^2$

(iii) $15625 + 253$

(iv) $15625 + 251$

(v) $15625 + 51^2$

Find the length of the side of a square whose area is $441 \text{ m}^2$ .

Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.

Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.

How many numbers lie between the squares of the following numbers?

(i) 16 and 17 (ii) 99 and 100

In the following pattern, fill in the missing numbers:

\begin{aligned} 1^2 + 2^2 + 2^2 &= 3^2 \\ 2^2 + 3^2 + 6^2 &= 7^2 \\ 3^2 + 4^2 + 12^2 &= 13^2 \end{aligned}

(a) $4^2 + 5^2 + 20^2 = (\underline{\quad})^2$

(b) $9^2 + 10^2 + (\underline{\quad})^2 = (\underline{\quad})^2$

How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny squares.

Q10

Find the cube roots of $27000$ and $10648$ .

Q11

What number will you multiply by 1323 to make it a cube number?

Q12

State true or false. Explain your reasoning.

(i) The cube of any odd number is even.

(ii) There is no perfect cube that ends with 8.

(iii) The cube of a 2-digit number may be a 3-digit number.

(iv) The cube of a 2-digit number may have seven or more digits.

(v) Cube numbers have an odd number of factors.

Q13

You are told that $1331$ is a perfect cube. Can you guess without factorisation what its cube root is? Similarly, guess the cube roots of $4913$ , $12167$ , and $32768$ .

Q14

Which of the following is the greatest? Explain your reasoning.

(i) $67^3 - 66^3$

(ii) $43^3 - 42^3$

(iii) $67^2 - 66^2$

(iv) $43^2 - 42^2$