To determine the diameter of a hemisphere with a given volume, we will first need to recall the formula used to calculate the volume of a hemisphere:
The volume [tex]\( V \)[/tex] of a hemisphere is given by the formula:
[tex]\[ V = \frac{2}{3} \pi r^3 \][/tex]
where [tex]\( r \)[/tex] is the radius of the hemisphere.
Given the volume of the hemisphere is [tex]\( V = 257 \)[/tex] m³, we can use the volume formula to solve for the radius.
Rearrange the formula to solve for radius [tex]\( r \)[/tex]:
[tex]\[ r^3 = \frac{3V}{2\pi} \][/tex]
Substitute the given volume into the formula:
[tex]\[ r^3 = \frac{3 \times 257}{2\pi} \][/tex]
Calculate the value of [tex]\( r^3 \)[/tex] first:
[tex]\[ r^3 = \frac{771}{2\pi} \][/tex]
Now, we need to compute the cube root of the result to find the value of the radius [tex]\( r \)[/tex]. Without a calculator, we approximate [tex]\( \pi \approx 3.14159 \)[/tex].
[tex]\[ r^3 \approx \frac{771}{2 \times 3.14159} \][/tex]
[tex]\[ r^3 \approx \frac{771}{6.28318} \][/tex]
[tex]\[ r^3 \approx 122.715 \][/tex]
Once we have a value for [tex]\( r^3 \)[/tex], we take the cube root:
[tex]\[ r \approx \sqrt[3]{122.715} \][/tex]
[tex]\[ r \approx 4.976 \][/tex]
Now remember that the diameter [tex]\( d \)[/tex] is twice the radius:
[tex]\[ d = 2r \][/tex]
[tex]\[ d \approx 2 \times 4.976 \][/tex]
[tex]\[ d \approx 9.952 \][/tex]
To round to the nearest tenth, we will round [tex]\( 9.952 \)[/tex] to one decimal place:
[tex]\[ d \approx 10.0 \][/tex]
Therefore, the diameter of the hemisphere is approximately 10.0 meters to the nearest tenth of a meter.
