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.
