Approximate $\sqrt{46}$ to the tenths place.
Answer (1)
Approximate 46 by recognizing it lies between 36 = 6 and 49 = 7 .
Test values to the tenths place: 6. 7 2 = 44.89 , 6. 8 2 = 46.24 , 6. 9 2 = 47.61 , 7. 0 2 = 49.0 .
Find the closest square to 46: 6. 8 2 = 46.24 is the closest.
Therefore, 46 ≈ 6.8 .
Explanation
Problem Analysis We are asked to approximate the square root of 46 to the tenths place.
Estimating the Whole Number We know that 6 2 = 36 and 7 2 = 49 . Since 46 is between 36 and 49, 46 is between 6 and 7. Also, 46 is closer to 49 than to 36, so 46 is closer to 7 than to 6.
Finding the Tenths Place Now we need to find the tenths place. We can test values between 6 and 7, such as 6.7, 6.8, 6.9, and 7.0.
Calculating Squares We calculate the squares of these values: 6. 7 2 = 44.89 6. 8 2 = 46.24 6. 9 2 = 47.61 7. 0 2 = 49.0
Finding the Closest Value Now we find the absolute difference between each square and 46: ∣46 − 44.89∣ = 1.11 ∣46 − 46.24∣ = 0.24 ∣46 − 47.61∣ = 1.61 ∣46 − 49.0∣ = 3.0
Approximating the Square Root The value 6. 8 2 = 46.24 is the closest to 46. Therefore, 46 ≈ 6.8 .
Final Answer The approximate value of 46 to the tenths place is 6.8 .
Examples
Estimating square roots is useful in many real-world situations. For example, if you need to find the length of the side of a square with an area of 46 square meters, you would need to calculate 46 . Approximating this value to 6.8 meters allows you to make practical decisions about the dimensions of the square without needing an exact calculation.
