Question

to the nearest whole number, the length of the diagonal of a square is 7 inches. which measure could be the actual length, in inches, of the diagonal of the square? √21 √40 √13 √50

Explanation:

Step1: Calculate the value of each square root

• For $\sqrt{21}$: We know that $4^2 = 16$ and $5^2=25$, so $\sqrt{16}<\sqrt{21}<\sqrt{25}$, i.e., $4 < \sqrt{21}<5$. Calculate $\sqrt{21}\approx 4.58$.
• For $\sqrt{40}$: We know that $6^2 = 36$ and $7^2 = 49$, so $\sqrt{36}<\sqrt{40}<\sqrt{49}$, i.e., $6 < \sqrt{40}<7$. Calculate $\sqrt{40}\approx 6.32$.
• For $\sqrt{13}$: We know that $3^2=9$ and $4^2 = 16$, so $\sqrt{9}<\sqrt{13}<\sqrt{16}$, i.e., $3<\sqrt{13}<4$. Calculate $\sqrt{13}\approx 3.61$.
• For $\sqrt{50}$: We know that $7^2 = 49$ and $8^2=64$, so $\sqrt{49}<\sqrt{50}<\sqrt{64}$, i.e., $7 < \sqrt{50}<8$. Calculate $\sqrt{50}\approx 7.07$.

Step2: Find the value nearest to 7

We need to find which of the values $\approx 4.58$, $\approx 6.32$, $\approx 3.61$, $\approx 7.07$ is nearest to 7.

• The difference between $4.58$ and $7$ is $7 - 4.58=2.42$.
• The difference between $6.32$ and $7$ is $7 - 6.32 = 0.68$.
• The difference between $3.61$ and $7$ is $7 - 3.61=3.39$.
• The difference between $7.07$ and $7$ is $7.07 - 7=0.07$.

Since $0.07$ is the smallest difference, $\sqrt{50}\approx 7.07$ is nearest to 7. But wait, there is a mistake above. Wait, the problem says "to the nearest whole number, the length of the diagonal of a square is 7 inches". Let's re - check the calculations:

Wait, $\sqrt{49}=7$, $\sqrt{50}\approx7.07$, which is 7 when rounded to the nearest whole number. $\sqrt{40}\approx6.32$, which is 6 when rounded to the nearest whole number. $\sqrt{21}\approx4.58$, rounded to 5? No, 4.58 is closer to 5? Wait, 4.58 is 5 when rounded to the nearest whole number? Wait, no, 4.58: the tenths place is 5, so we round up, 4.58 $\approx$ 5. $\sqrt{13}\approx3.61\approx4$. $\sqrt{50}\approx7.07\approx7$. $\sqrt{40}\approx6.32\approx6$. So the one that rounds to 7 is $\sqrt{50}$. But wait, maybe I made a mistake in the first calculation of differences. Let's recalculate the differences correctly:

• For $\sqrt{21}\approx4.58$: $|7 - 4.58| = 2.42$
• For $\sqrt{40}\approx6.32$: $|7 - 6.32|=0.68$
• For $\sqrt{13}\approx3.61$: $|7 - 3.61| = 3.39$
• For $\sqrt{50}\approx7.07$: $|7 - 7.07| = 0.07$

So $\sqrt{50}$ is the closest to 7. But wait, the options are $\sqrt{21}$, $\sqrt{40}$, $\sqrt{13}$, $\sqrt{50}$. Wait, maybe I misread the options. Wait, the last option is $\sqrt{50}$? Wait, maybe there is a typo, but assuming the options are as given. Wait, but let's check again. Wait, the problem says "to the nearest whole number, the length of the diagonal of a square is 7 inches". So we need a number that when rounded to the nearest whole number is 7.

$\sqrt{49}=7$, $\sqrt{50}\approx7.07$, which rounds to 7. $\sqrt{48}\approx6.93$, which also rounds to 7, but among the given options, $\sqrt{50}\approx7.07$ is the one that rounds to 7. Wait, but in the options, $\sqrt{40}\approx6.32$ (rounds to 6), $\sqrt{21}\approx4.58$ (rounds to 5), $\sqrt{13}\approx3.61$ (rounds to 4), $\sqrt{50}\approx7.07$ (rounds to 7). So the answer should be $\sqrt{50}$. But wait, maybe I made a mistake. Wait, let's check the calculation of $\sqrt{50}$ again: $\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\approx5\times1.414 = 7.07$, which is correct. And 7.07 rounded to the nearest whole number is 7.

Answer:

$\sqrt{50}$