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The area of a rectangle is 30 meters squared. If the length is the sq. root of 75, what is the...

Question:

The area of a rectangle is 30 meters squared. If the length is the sq. root of 75, what is the width?

Rationalizing a Denominator:

Sometimes in mathematics, we come across a division problem, or fraction, in which we have a radical in the denominator. When this is the case, we can simplify the fraction by multiplying both the numerator and denominator by that radical. This is often necessary when we are trying to find exact answers to a given problem.

Answer and Explanation:

If the area of a rectangle is 30 square meters and the length is √(75), then the width is 2√(3) meters.

To find this, we start by using the formula for the area of a rectangle, which is A = l × w. Plugging our area and length into this formula, and then solving for w gives the following:

  • A = l × w

Plug in A = 30 and l = √(75)

  • 30 = √(75) × w

Divide both sides by √(75).

  • 30 / √(75) = w

We get that the width of the rectangle is equal to 30 ÷ √(75). Plugging this into a calculator gives an irrational answer or approximately 3.4641016, but this is not an exact answer. To get an exact answer, we'll need to rationalize the denominator of 30 / √(75) by multiplying the numerator and denominator by √(75).

  • (30 × √(75)) / (√(75) × √(75)) = 30√(75) / 75

We can simplify this further by simplifying √(75) as follows:

  • √(75) = √(25 × 3) = √(25) × √(3) = 5√(3)

Plugging 5√(3) in for √(75) and simplifying gives the following:

  • 30√(75) / 75 = (30(5√(3)) / 75 = 150√(3) / 75 = 2√(3)

All together, this gives that the width of the rectangle is 2√(3) meters long.


Learn more about this topic:

How to Rationalize the Denominator with a Radical Expression

from Algebra II: High School

Chapter 1 / Lesson 6
8.8K

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