Question

what is the volume of a hemisphere with a radius of 34.1 in, rounded to the nearest tenth of a cubic inch?

Explanation:

Step1: Recall the formula for the volume of a hemisphere

The volume \( V \) of a hemisphere is given by the formula \( V=\frac{2}{3}\pi r^{3} \), where \( r \) is the radius of the hemisphere.

Step2: Substitute the given radius into the formula

We are given that \( r = 34.1 \) inches. Substitute \( r = 34.1 \) into the formula:
\[ V=\frac{2}{3}\pi(34.1)^{3} \]

First, calculate \( (34.1)^{3} \):
\( 34.1\times34.1 = 1162.81 \), then \( 1162.81\times34.1=1162.81\times(34 + 0.1)=1162.81\times34+1162.81\times0.1 = 39535.54+116.281 = 39651.821 \)

Then, multiply by \( \frac{2}{3}\pi \):
\[ V=\frac{2}{3}\times\pi\times39651.821 \]
\( \frac{2}{3}\times39651.821=\frac{79303.642}{3}\approx26434.547 \)

Then, multiply by \( \pi \) (using \( \pi\approx3.14159 \)):
\( V\approx26434.547\times3.14159\approx26434.547\times3 + 26434.547\times0.14159\)
\( 26434.547\times3 = 79303.641 \)
\( 26434.547\times0.14159\approx26434.547\times0.1+26434.547\times0.04 + 26434.547\times0.00159\)
\( 26434.547\times0.1 = 2643.4547 \)
\( 26434.547\times0.04 = 1057.38188 \)
\( 26434.547\times0.00159\approx42.031 \)
Adding these together: \( 2643.4547+1057.38188 + 42.031\approx3742.8676 \)
Then, \( 79303.641+3742.8676\approx83046.5086 \)

(Alternatively, using a calculator for a more accurate calculation: \( (34.1)^3=34.1\times34.1\times34.1 = 34.1\times1162.81=39651.821 \), then \( \frac{2}{3}\times\pi\times39651.821=\frac{2\times39651.821\times\pi}{3}=\frac{79303.642\pi}{3}\approx\frac{79303.642\times3.1415926535}{3}\)

\( 79303.642\times3.1415926535\approx79303.642\times3 + 79303.642\times0.1415926535\)
\( 79303.642\times3 = 237910.926 \)
\( 79303.642\times0.1415926535\approx79303.642\times0.1+79303.642\times0.04+79303.642\times0.0015926535\)
\( = 7930.3642+3172.14568+126.234\approx7930.3642 + 3172.14568=11102.50988+126.234 = 11228.74388\)
Total \( 237910.926+11228.74388 = 249139.66988\)
Then divide by 3: \( \frac{249139.66988}{3}\approx83046.5566 \)

Rounded to the nearest tenth, we look at the hundredth place. The number is approximately \( 83046.5566 \), the hundredth digit is 5, so we round up the tenth digit. So \( 83046.6 \) (wait, wait, no, let's check the calculation again with a more accurate method using a calculator - like directly computing \( \frac{2}{3}\pi(34.1)^3 \))

Using a calculator:
\( 34.1^3 = 34.1\times34.1\times34.1 = 39651.821 \)
\( \frac{2}{3}\times\pi\times39651.821=\frac{2\times39651.821\times\pi}{3}\approx\frac{79303.642\times3.14159265}{3}\approx\frac{249139.67}{3}\approx83046.56 \)

Rounded to the nearest tenth, since the hundredth digit is 6 (wait, 83046.56, the tenths place is 5, hundredths is 6, so we round up the tenths place: 83046.6? Wait, no, 83046.56, the tenths digit is 5, hundredths is 6, so 83046.6? Wait, no, 83046.56 rounded to the nearest tenth: the rule is if the digit in the hundredth place is 5 or more, we round up the tenths place. So 5 (tenths) + 1 = 6, so 83046.6? Wait, but let's use a calculator for precise calculation.

Using a calculator:
\( V=\frac{2}{3}\pi r^{3}=\frac{2}{3}\times\pi\times(34.1)^{3} \)

Calculate \( (34.1)^{3}=34.1\times34.1\times34.1 = 39651.821 \)

Then \( \frac{2}{3}\times39651.821 = 26434.547333\)

Then \( 26434.547333\times\pi\approx26434.547333\times3.1415926535\approx83046.56 \)

Rounded to the nearest tenth, we look at the digit in the hundredth place, which is 6. Since 6 > 5, we round up the tenths place. So the tenths place digit is 5, so 5 + 1 = 6. So the volume is approximately \( 83046.6 \) cubic inches.

Wait, but let's check with another approach. Maybe I made a mistake in the manual calculation. Let's use the formula directly with a calculator:

\( r = 34.1 \)

Volume of hemisphere \( V=\frac{2}{3}\pi r^{3} \)

\( r^{3}=34.1\times34.1\times34.1 = 34.1\times1162.81 = 39651.821 \)

\( \frac{2}{3}\times39651.821 = 26434.547333 \)

\( 26434.547333\times\pi\approx26434.547333\times3.1415926535 = 83046.556 \)

Rounded to the nearest tenth: the number is 83046.556, the tenths digit is 5, hundredths digit is 5, so we round up the tenths digit: 83046.6

Answer:

\( 83046.6 \)