Select the correct answer.
Which expression has a value closest to 1? (Use 3.14 as the value of [tex]$\pi$[/tex].)
A. [tex]$\frac{7 \pi}{\sqrt{10}}$[/tex]
B. [tex]$\sqrt{75}-\sqrt{74}$[/tex]
C. [tex]$\sqrt{37}-\sqrt{26}$[/tex]
D. [tex]$\frac{3 \sqrt{5}-\sqrt{20}}{\sqrt{2}}$[/tex]
Answer (2)
Evaluate each expression using the given value of π .
Calculate the approximate value of each expression.
Find the absolute difference between each value and 1.
The expression with the smallest absolute difference is the closest to 1, which is 37 − 26 .
Explanation
Understanding the Problem We are given four expressions and we need to find the one closest to 1. We are given that π = 3.14 .
Evaluating the Expressions First, let's evaluate each expression using the given value of π :
A. 10 7 π = 10 7 × 3.14
B. 75 − 74
C. 37 − 26
D. 2 3 5 − 20 = 2 3 5 − 2 5 = 2 5
Calculating Approximate Values Now, let's calculate the approximate value of each expression:
A. 10 7 × 3.14 ≈ 3.16 21.98 ≈ 6.95
B. 75 − 74 ≈ 8.66 − 8.60 ≈ 0.06
C. 37 − 26 ≈ 6.08 − 5.10 ≈ 0.98
D. 2 5 ≈ 1.41 2.24 ≈ 1.58
Finding the Absolute Differences Next, we find the absolute difference between each value and 1:
A. ∣6.95 − 1∣ = 5.95
B. ∣0.06 − 1∣ = 0.94
C. ∣0.98 − 1∣ = 0.02
D. ∣1.58 − 1∣ = 0.58
Determining the Closest Value The expression with the smallest absolute difference is the closest to 1. Comparing the absolute differences, we see that expression C has the smallest difference (0.02).
Final Answer Therefore, the expression with a value closest to 1 is 37 − 26 .
Examples
Understanding how to approximate values and compare them to a target number is useful in many real-life situations. For example, if you are estimating the cost of items in your shopping cart to stay within a budget, you are approximating and comparing values. Similarly, in science, you might need to determine which measurement is closest to a theoretical value. This skill helps in making quick and informed decisions.
After evaluating all four expressions using the given value of π , the expression closest to 1 is 37 − 26 with an approximate value of 0.98.
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