What is the radius of a sphere with a volume of $$ 7961 \mathrm{~cm}^{3} $$, to the nearest tenth of a centimeter?

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UpStudy AI · Accepted Answer

To find the radius of a sphere given its volume, we can use the formula for the volume of a sphere:

$$
V = \frac{4}{3} \pi r^3
$$

where $$ V $$ is the volume and $$ r $$ is the radius. We can rearrange this formula to solve for $$ r $$:

$$
r^3 = \frac{3V}{4\pi}
$$

Now, substituting the given volume $$ V = 7961 \, \text{cm}^3 $$ into the equation:

$$
r^3 = \frac{3 \times 7961}{4\pi}
$$

Next, we will calculate $$ r^3 $$ and then take the cube root to find $$ r $$. Let's perform the calculation.

First, we will calculate $$ \frac{3 \times 7961}{4\pi} $$.

Now, I will perform the calculation.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3\times 7961}{4\pi }$$
- step1: Multiply the numbers:
  $$\frac{23883}{4\pi }$$
Calculate or simplify the expression $$ 1900.548753^(1/3) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1900.548753^{\frac{1}{3}}$$
- step1: Convert the expressions:
  $$\left(\frac{1900548753}{1000000}\right)^{\frac{1}{3}}$$
- step2: Use the properties of exponents:
  $$\frac{1900548753^{\frac{1}{3}}}{1000000^{\frac{1}{3}}}$$
- step3: Evaluate the power:
  $$\frac{1900548753^{\frac{1}{3}}}{100}$$
- step4: Simplify:
  $$\frac{\sqrt[3]{1900548753}}{100}$$
We calculated $$ r^3 $$ as approximately $$ 1900.55 $$. Now, taking the cube root of this value gives us:

$$
r \approx 12.39 \, \text{cm}
$$

Rounding to the nearest tenth of a centimeter, the radius of the sphere is:

$$
\boxed{12.4 \, \text{cm}}
$$
The radius of the sphere is approximately 12.4 cm.

What is the radius of a sphere with a volume of \( 7961 \mathrm{~cm}^{3} \), to the nearest tenth of a centimeter?

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