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Question Video: Solving Word Problems Involving Square Roots Mathematics • First Year of Preparatory School

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Which of the following can be the area of a square if the measure of its side length is a whole number? [A] 247 ft² [B] 489 ft² [C] 531 ft² [D] 868 ft² [E] 1,764 ft²

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Video Transcript

Which of the following can be the area of a square if the measure of its side length is a whole number? Option (A) 247 square feet. Option (B) 489 square feet. Option (C) 531 square feet. Option (D) 868 square feet. Or is it option (E) 1,764 square feet?

In this question, we are told that the sides in a square have a measure that is a whole number of feet. And we need to determine which of five possibilities could be the area of the square.

To answer this question, let’s start by sketching our square. We will say that the sides of the square are 𝑙 feet long, so 𝑙 is a whole number. We can then recall that the area of a square is the square of its side length. So this square will have an area of 𝑙 squared square feet. This then allows us to consider the five possible areas. For these to be the area of the square, they must be equal to 𝑙 squared, where 𝑙 is a whole number.

If we take the square root of both sides of the equation, where we note that 𝑙 must be positive — so we only consider the positive root — then we see that the square root of the area is equal to 𝑙, which is a whole number. For the square root of an integer to be a whole number, we must be taking the square root of a perfect square. So we need to determine which of the five options are perfect squares.

There are a few different ways to do this. We will factor each of the areas into primes to see if each prime factor is raised to even exponents. First, we can calculate that 247 is 13 times 19, so this is not a perfect square. This means that root 247 is not a whole number, so it cannot be the area of the square.

We can follow a similar process for option (B). We can note that 489 has a single factor of three, so it is not a perfect square. In fact, its prime factorization is three times 163. Once again, since this is not a perfect square, its square root is not a whole number. So it cannot be the area of the square.

We can follow a similar process for options (C) and (D). We can find that 531 is equal to three squared times 59 and 868 is equal to two squared times seven times 31. Neither of these are perfect squares. So both of these cannot be the area of the square.

This only leaves option (E). We can factor 1,764 into primes to obtain two squared times three squared times seven squared. This confirms that 1,764 is a perfect square. In fact, we can show that it is equal to 42 squared. This means that the square root of 1,764 is 42, which is a whole number.

We can verify this answer by considering the area of a square with sides of length 42 feet. We see that its area is 42 times 42, which we can calculate is 1,764. Hence, our answer is option (E). Of the options listed, only 1,764 square feet is a possible area of a square with a whole number as a side length.

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