Question

what is the length of the hypotenuse? if necessary, round to the nearest tenth.
5.3 km
2.7 km
c =
kilometers

Explanation:

Step1: Identify the formula

For a right - triangle, we use the Pythagorean theorem, which is $c^{2}=a^{2}+b^{2}$, where $a$ and $b$ are the lengths of the legs of the right - triangle and $c$ is the length of the hypotenuse. Here, let $a = 5.3$ km and $b=2.7$ km.

Step2: Substitute the values into the formula

First, calculate $a^{2}$ and $b^{2}$.
$a^{2}=(5.3)^{2}=5.3\times5.3 = 28.09$
$b^{2}=(2.7)^{2}=2.7\times2.7=7.29$
Then, find $a^{2}+b^{2}$: $a^{2}+b^{2}=28.09 + 7.29=35.38$

Step3: Solve for c

Since $c^{2}=35.38$, we take the square root of both sides to find $c$. So, $c=\sqrt{35.38}\approx5.95$ (rounded to the nearest hundredth) or if we round to the nearest tenth, we look at the hundredth place. The hundredth digit of $5.95$ is $5$, so we round up the tenth place. $c\approx6.0$ (rounded to the nearest tenth) or $c\approx5.9$ (if we consider the original calculation more precisely, $\sqrt{35.38}\approx5.948\approx5.9$ when rounded to the nearest tenth? Wait, no. Wait, $5.948$: the tenths place is $9$, the hundredths place is $4$. Wait, I made a mistake earlier. Wait, $5.3^{2}=28.09$, $2.7^{2}=7.29$, $28.09 + 7.29 = 35.38$. $\sqrt{35.38}\approx5.948$, which is approximately $5.9$ when rounded to the nearest tenth (because the digit in the hundredths place is $4$, which is less than $5$). Wait, no, $5.948$: the tenths digit is $9$, the hundredths digit is $4$. So, to round to the nearest tenth, we look at the hundredths digit. Since $4\lt5$, we keep the tenths digit as it is. So $\sqrt{35.38}\approx5.9$ (rounded to the nearest tenth) or if we round to the nearest whole number, since the tenths digit is $9\gt5$, we round up to $6$. But let's check the calculation again. Wait, maybe I misread the legs. Wait, the vertical leg is $5.3$ km and the horizontal leg is $2.7$ km. So $a = 5.3$, $b = 2.7$. Then $c=\sqrt{5.3^{2}+2.7^{2}}=\sqrt{28.09 + 7.29}=\sqrt{35.38}\approx5.95\approx6.0$ (if we round to the nearest tenth, because $5.95$ has a hundredth digit of $5$, so we round up the tenth digit. $5.9 + 0.1=6.0$). Wait, the confusion is between different levels of rounding. Let's do it properly.

$\sqrt{35.38}\approx5.948$, which is:

• Rounded to the nearest whole number: $6$ (because $0.948\gt0.5$)
• Rounded to the nearest tenth: $5.9$ (because the hundredth digit is $4\lt5$)
• Rounded to the nearest hundredth: $5.95$

Answer:

If we round to the nearest tenth, the length of the hypotenuse $c\approx5.9$ km. If we round to the nearest whole number, $c\approx6$ km. If we round to the nearest hundredth, $c\approx5.95$ km. (The most appropriate rounding might depend on the problem's requirement. If we assume rounding to the nearest tenth, the answer is approximately $5.9$ km or if to the nearest whole number, $6$ km. But let's recalculate $\sqrt{35.38}$:

$5.9^{2}=34.81$, $6.0^{2}=36.0$. $35.38 - 34.81 = 0.57$, $36.0 - 35.38 = 0.62$. So $35.38$ is closer to $34.81$ than to $36.0$? No, $35.38-34.81 = 0.57$, $36.0 - 35.38=0.62$. Wait, no, $5.9^{2}=34.81$, $5.95^{2}=(6 - 0.05)^{2}=36-0.6 + 0.0025 = 35.4025$, which is very close to $35.38$. So $\sqrt{35.38}\approx5.95$, which is approximately $6.0$ when rounded to the nearest tenth (because the hundredth digit is $5$). So the correct rounding to the nearest tenth is $6.0$ (or $6$ when rounded to the nearest whole number).

Wait, let's use a calculator for $\sqrt{35.38}$. $\sqrt{35.38}\approx5.948$, which is $5.9$ when rounded to the nearest tenth (since the digit in the hundredths place is $4$) and $6$ when rounded to the nearest whole number.

So, depending on the rounding requirement:

• Nearest whole number: $6$ km
• Nearest tenth: $5.9$ km
• Nearest hundredth: $5.95$ km

Since the problem says "if necessary, round to the nearest tenth" (from the image, it says "round to the nearest tenth"? Wait, the original problem says "round to the nearest tenth"? Let me check the image again. The user's image says "round to the nearest tenth"? Wait, the text in the image: "round to the nearest tenth"? Wait, the OCR might be a bit off. But assuming we need to round to the nearest tenth, $\sqrt{35.38}\approx5.9$ km (because $5.948$: the tenths digit is $9$, the hundredths digit is $4$, so we don't round up the tenths digit). Wait, no, $5.948$: the number is $5.948$. To round to the nearest tenth, we look at the digit in the hundredths place, which is $4$. Since $4\lt5$, we keep the tenths digit as it is. So $5.9$ km. But if we round to the nearest whole number, it's $6$ km.

But let's do the calculation again:

$a = 5.3$, $b = 2.7$

$c=\sqrt{a^{2}+b^{2}}=\sqrt{5.3^{2}+2.7^{2}}=\sqrt{28.09 + 7.29}=\sqrt{35.38}\approx5.9$ (nearest tenth) or $6$ (nearest whole number)

So the answer is approximately $\boldsymbol{5.9}$ kilometers (rounded to the nearest tenth) or $\boldsymbol{6}$ kilometers (rounded to the nearest whole number). If we follow the standard of rounding to the nearest tenth, the answer is $5.9$ km.