Solved on Jan 28, 2024

Find the radius of a sphere with a surface area of $324 \pi \mathrm{cm}^{2}$ , given the formula $\text{surface area} = 4 \pi r^{2}$ , where $r$ is the radius. Round your answer to 1 decimal place.

STEP 1

Assumptions
1. The formula for the surface area of a sphere is $4\pi r^2$ .
2. The given surface area of the sphere is $324\pi \mathrm{cm}^2$ .
3. We need to find the radius $r$ of the sphere.
4. The answer should be rounded to one decimal place if it is not an integer.

STEP 2

We will use the formula for the surface area of a sphere to find the radius.
$Surface\, area = 4\pi r^2$

STEP 3

Substitute the given surface area into the formula.
$324\pi = 4\pi r^2$

STEP 4

To solve for $r^2$ , we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by $4\pi$ .
$\frac{324\pi}{4\pi} = r^2$

STEP 5

Simplify the equation by canceling out the $\pi$ on both sides and dividing 324 by 4.
$\frac{324}{4} = r^2$
$81 = r^2$

STEP 6

Now, take the square root of both sides to solve for $r$ .
$r = \sqrt{81}$

STEP 7

Calculate the square root of 81.
$r = 9$
The radius of the sphere is $9 \mathrm{cm}$ .

Was this helpful?

Find the radius of a sphere with a surface area of $324 \pi \mathrm{cm}^{2}$ , given the formula $\text{surface area} = 4 \pi r^{2}$ , where $r$ is the radius. Round your answer to 1 decimal place.

STEP 1

Assumptions
1. The formula for the surface area of a sphere is $4\pi r^2$ .
2. The given surface area of the sphere is $324\pi \mathrm{cm}^2$ .
3. We need to find the radius $r$ of the sphere.
4. The answer should be rounded to one decimal place if it is not an integer.

STEP 2

We will use the formula for the surface area of a sphere to find the radius.
$Surface\, area = 4\pi r^2$

STEP 3

Substitute the given surface area into the formula.
$324\pi = 4\pi r^2$

STEP 4

To solve for $r^2$ , we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by $4\pi$ .
$\frac{324\pi}{4\pi} = r^2$

STEP 5

Simplify the equation by canceling out the $\pi$ on both sides and dividing 324 by 4.
$\frac{324}{4} = r^2$
$81 = r^2$

STEP 6

Now, take the square root of both sides to solve for $r$ .
$r = \sqrt{81}$

STEP 7

Calculate the square root of 81.
$r = 9$
The radius of the sphere is $9 \mathrm{cm}$ .

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the radius of a sphere with a surface area of 324πcm2324 \pi \mathrm{cm}^{2}324πcm2, given the formula surface area=4πr2\text{surface area} = 4 \pi r^{2}surface area=4πr2, where rrr is the radius. Round your answer to 1 decimal place.

STEP 1

Assumptions1. The formula for the surface area of a sphere is 4πr24\pi r^24πr2.2. The given surface area of the sphere is 324πcm2324\pi \mathrm{cm}^2324πcm2.3. We need to find the radius rrr of the sphere.4. The answer should be rounded to one decimal place if it is not an integer.

STEP 2

We will use the formula for the surface area of a sphere to find the radius.Surface area=4πr2Surface\, area = 4\pi r^2Surfacearea=4πr2

STEP 3

Substitute the given surface area into the formula.324π=4πr2324\pi = 4\pi r^2324π=4πr2

STEP 4

To solve for r2r^2r2, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 4π4\pi4π.324π4π=r2\frac{324\pi}{4\pi} = r^24π324π​=r2

STEP 5

Simplify the equation by canceling out the π\piπ on both sides and dividing 324 by 4.3244=r2\frac{324}{4} = r^24324​=r281=r281 = r^281=r2

STEP 6

Now, take the square root of both sides to solve for rrr.r=81r = \sqrt{81}r=81​

STEP 7

Calculate the square root of 81.r=9r = 9r=9The radius of the sphere is 9cm9 \mathrm{cm}9cm.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the radius of a sphere with a surface area of $324 \pi \mathrm{cm}^{2}$ , given the formula $\text{surface area} = 4 \pi r^{2}$ , where $r$ is the radius. Round your answer to 1 decimal place.

Assumptions
1. The formula for the surface area of a sphere is $4\pi r^2$ .
2. The given surface area of the sphere is $324\pi \mathrm{cm}^2$ .
3. We need to find the radius $r$ of the sphere.
4. The answer should be rounded to one decimal place if it is not an integer.

We will use the formula for the surface area of a sphere to find the radius.
$Surface\, area = 4\pi r^2$

Substitute the given surface area into the formula.
$324\pi = 4\pi r^2$

To solve for $r^2$ , we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by $4\pi$ .
$\frac{324\pi}{4\pi} = r^2$

Simplify the equation by canceling out the $\pi$ on both sides and dividing 324 by 4.
$\frac{324}{4} = r^2$
$81 = r^2$

Now, take the square root of both sides to solve for $r$ .
$r = \sqrt{81}$

Calculate the square root of 81.
$r = 9$
The radius of the sphere is $9 \mathrm{cm}$ .