Question

What is the radius of a hemisphere with a volume of

757
cm
3
,
757 cm
3
, to the nearest tenth of a centimeter?

Answer 1

The radius of the hemisphere is approximately 9.0 cm to the nearest tenth of a centimeter.

Step-by-step explanation:

To find the radius of a hemisphere with a given volume, we will start with the formula for the volume of a sphere and then adjust it for a hemisphere.

The volume V of a sphere with radius r is given by the formula:


`\[V = (4)/(3) \pi r^3\]`

Since a hemisphere is half of a sphere, its volume is half this amount, so the volume \(V_h\) of a hemisphere is:


`\[V_h = (1)/(2) \cdot (4)/(3) \pi r^3\\\\\[V_h = (2)/(3) \pi r^3\]`

We are given that the volume of the hemisphere is 757 cm³, so let's set our formula equal to this value and solve for r:


`\[757 = (2)/(3) \pi r^3\]`

To find r, we can follow these steps:

1. Multiply both sides of the equation by
`\((3)/(2)\)` to isolate
`\(\pi r^3\)` on one side:


`\[(3)/(2) \cdot 757 = \pi r^3\]`

2. Divide both sides by \(\pi\) to isolate \(r^3\):


`\[r^3 = ((3)/(2) \cdot 757)/(\pi)\]`

3. Take the cube root of both sides to solve for r:


`\[r = \left(((3)/(2) \cdot 757)/(\pi)\right)^{(1)/(3)}\]`

Let's complete the calculation:


`\[r = \left((3)/(2) \cdot 757 / \pi\right)^{(1)/(3)}\]`

Approximating \(\pi\) to 3.14159:


`\[r = \left((3)/(2) \cdot 757 / 3.14159\right)^{(1)/(3)}\\\\\[r = \left((2271)/(3.14159)\right)^{(1)/(3)}\\\\\[r = \left(722.9529\right)^{(1)/(3)}\\\\\[r \approx 9.009\]`

To find the radius to the nearest tenth of a centimeter, we round the calculated radius:


`\[r \approx 9.0 \text{ cm}\]`

Therefore, the radius of the hemisphere is approximately 9.0 cm to the nearest tenth of a centimeter.

Answer 2

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