Given that [tex]$\sqrt{5.2}=2.2004$[/tex] and [tex]$\sqrt{52}=[/tex] 7.2111. Find the value [tex]$\sqrt{520}$[/tex] correct to 3 s.f.
A. 721
B. 22.8
C. 2.804
D. 7,21.
Answer (1)
Express 520 as 52 × 10 .
Substitute the given value of 52 = 7.2111 and approximate 10 ≈ 3.162 .
Calculate the product: 7.2111 × 3.162 ≈ 22.803 .
Round the result to 3 significant figures: 22.8 .
Explanation
Understanding the Problem We are given that 5.2 = 2.2004 and 52 = 7.2111 . We want to find the value of 520 correct to 3 significant figures.
Expressing the Square Root We can express 520 as 52 × 10 = 52 × 10 . We know the value of 52 , so we need to find the value of 10 .
Approximating the Value We can approximate 10 using a calculator or other methods. The result of this approximation is 10 ≈ 3.1622776601683795 .
Substituting the Values Now, we can substitute the given value of 52 and the approximated value of 10 into the expression: 520 = 52 × 10 ≈ 7.2111 × 3.1622776601683795 ≈ 22.803500435240203 .
Rounding to 3 Significant Figures We need to round the result to 3 significant figures. The first three significant figures are 2, 2, and 8. The next digit is 0, so we don't need to round up. Therefore, the value of 520 correct to 3 significant figures is 22.8.
Alternative Method Alternatively, we can express 520 as 5.2 × 100 = 5.2 × 100 = 10 × 5.2 . We are given that 5.2 = 2.2004 , so 520 = 10 × 2.2004 = 22.004 . Rounding this to 3 significant figures gives 22.0. However, the options provided do not contain 22.0. The previous method gives 22.8, which is one of the options.
Examples
Square roots are used in many fields, such as physics, engineering, and computer science. For example, when calculating the distance between two points in a coordinate plane, we use the distance formula, which involves square roots. Also, in physics, the period of a pendulum is calculated using a formula that involves the square root of the length of the pendulum. Understanding how to estimate and calculate square roots is therefore essential in many practical applications.
