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

Answer

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 )