Which of the following is the best approximation of 6 times the positive square root of 42

Which of the following is the best approximation of 6 times the positive square root of 42?

A.
60.3

B.
44.7

C.
30.0

D.
38.9

The Correct Answer and Explanation is:

To determine the best approximation of 6 times the positive square root of 42, let’s break down the problem step by step.

1. Calculate the Square Root of 42:
   • First, we need to find the square root of 42. Using a calculator or estimation techniques: 42≈6.4807\sqrt{42} \approx 6.480742​≈6.4807
2. Multiply by 6:
   • Now, we multiply this result by 6 to find 6×426 \times \sqrt{42}6×42​: 6×6.4807≈38.88426 \times 6.4807 \approx 38.88426×6.4807≈38.8842
3. Compare with the Given Options:
   • We now compare our calculated result with the provided options:
     • A. 60.3
     • B. 44.7
     • C. 30.0
     • D. 38.9
   The closest approximation to 38.8842 among the choices is 38.9.

Detailed Explanation

To ensure accuracy, let’s verify each step:

1. Square Root of 42:
   • Square roots are often approximated or calculated using numerical methods. The exact square root of 42 falls between 6.4 and 6.5. More precisely, 42\sqrt{42}42​ is around 6.4807. This value can be verified using a scientific calculator or mathematical software.
2. Multiplication:
   • Multiplying 6 by the approximate square root value: 6×6.4807=38.88426 \times 6.4807 = 38.88426×6.4807=38.8842
   This value is rounded to four decimal places.
3. Option Comparison:
   • Comparing our result with the options, we notice:
     • 60.3 is much higher than 38.8842.
     • 44.7 is also higher but closer than 60.3.
     • 30.0 is lower and further away.
     • 38.9 is the nearest to our calculated value of 38.8842.

Thus, the closest approximation to 6 times the positive square root of 42 is D. 38.9.