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QuestionApproximate 37\sqrt{37} using perfect squares, referencing 36=6\sqrt{36}=6 and 49=7\sqrt{49}=7.

Studdy Solution

STEP 1

Assumptions1. We are asked to find a rational approximation of 37\sqrt{37}. . We are to use the square roots of36 and49 as reference points.
3. We know that 36=6\sqrt{36}=6 and 49=7\sqrt{49}=7.

STEP 2

Since37 is between36 and49, we know that 37\sqrt{37} will be between 36\sqrt{36} and 49\sqrt{49}, that is between6 and7.
6<37<76 < \sqrt{37} <7

STEP 3

We can express 37\sqrt{37} as a weighted average of6 and7. The weights are determined by how close37 is to36 and49.Let's denote the weight for6 as w1w1 and the weight for7 as w2w2.37≈w1⋅6+w2⋅7\sqrt{37} \approx w1 \cdot6 + w2 \cdot7

STEP 4

The weights w1w1 and w2w2 should add up to1, because they represent proportions of the total distance between6 and7.
w1+w2=1w1 + w2 =1

STEP 5

The weight for, w1w1, is determined by how close37 is to49, and the weight for7, w2w2, is determined by how close37 is to36.
w1=49−3749−36=1213w1 = \frac{49 -37}{49 -36} = \frac{12}{13}w2=37−3649−36=113w2 = \frac{37 -36}{49 -36} = \frac{1}{13}

STEP 6

Now, we can substitute the weights w1w1 and w2w2 into the weighted average equation.
37≈w1⋅6+w2⋅=1213⋅6+113⋅\sqrt{37} \approx w1 \cdot6 + w2 \cdot = \frac{12}{13} \cdot6 + \frac{1}{13} \cdot

STEP 7

Calculate the rational approximation of 37\sqrt{37}.
37≈1213⋅6+113⋅7=7213+713=7913=6.07692308\sqrt{37} \approx \frac{12}{13} \cdot6 + \frac{1}{13} \cdot7 = \frac{72}{13} + \frac{7}{13} = \frac{79}{13} =6.07692308So, the rational approximation of 37\sqrt{37} using perfect squares is approximately6.08.

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