Math / Geometry

Questionh/public/activity/3006003/assessment 〔 3.6.3 Qulz: Spheres
Question 1 of 10 The area of a circle of radius 10 units is equal to the surface area of a sphere of radius 5 units. A. True B. False SUBMIT - PREVIOUS

Studdy Solution

STEP 1

What is this asking? Is the area of a circle with radius $10$ the same as the surface area of a sphere with radius $5$ ? Watch out! Don't mix up area of a circle and surface area of a sphere!
Also, don't forget those squared terms!

STEP 2

1. Calculate Circle Area
2. Calculate Sphere Surface Area
3. Compare Results

STEP 3

Alright, let's calculate the area of this circle!
The formula for the area of a circle is $\pi \cdot r^2$ , where $r$ is the radius.
Remember, the radius is the distance from the center of the circle to any point on the edge.

STEP 4

We're given that the circle has a radius of $10$ units.
So, plugging that into our formula, we get: $\text{Area}_{\text{circle}} = \pi \cdot (10)^2$

STEP 5

Squaring $10$ gives us $100$ , so the area of the circle is: $\text{Area}_{\text{circle}} = 100 \cdot \pi$ We'll keep it in terms of $\pi$ for now to avoid rounding errors.

STEP 6

Now, let's tackle the surface area of the sphere!
The formula for the surface area of a sphere is $4 \cdot \pi \cdot r^2$ , where $r$ is the radius of the sphere.

STEP 7

We know the sphere's radius is $5$ units.
Let's plug that in: $\text{Area}_{\text{sphere}} = 4 \cdot \pi \cdot (5)^2$

STEP 8

Squaring $5$ gives us $25$ , so: $\text{Area}_{\text{sphere}} = 4 \cdot \pi \cdot 25$

STEP 9

Multiplying $4$ and $25$ gives us $100$ , so the surface area of the sphere is: $\text{Area}_{\text{sphere}} = 100 \cdot \pi$

STEP 10

Let's compare our results.
The area of the circle is $100 \cdot \pi$ , and the surface area of the sphere is $100 \cdot \pi$ .
Are they the same?

STEP 11

Yes they are! $100 \cdot \pi$ is equal to $100 \cdot \pi$ .

STEP 12

The statement is true.
The area of the circle and the surface area of the sphere are both $100 \cdot \pi$ square units.

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Questionh/public/activity/3006003/assessment 〔 3.6.3 Qulz: Spheres
Question 1 of 10 The area of a circle of radius 10 units is equal to the surface area of a sphere of radius 5 units. A. True B. False SUBMIT - PREVIOUS

Studdy Solution

STEP 1

What is this asking? Is the area of a circle with radius $10$ the same as the surface area of a sphere with radius $5$ ? Watch out! Don't mix up area of a circle and surface area of a sphere!
Also, don't forget those squared terms!

STEP 2

1. Calculate Circle Area
2. Calculate Sphere Surface Area
3. Compare Results

STEP 3

Alright, let's calculate the area of this circle!
The formula for the area of a circle is $\pi \cdot r^2$ , where $r$ is the radius.
Remember, the radius is the distance from the center of the circle to any point on the edge.

STEP 4

We're given that the circle has a radius of $10$ units.
So, plugging that into our formula, we get: $\text{Area}_{\text{circle}} = \pi \cdot (10)^2$

STEP 5

Squaring $10$ gives us $100$ , so the area of the circle is: $\text{Area}_{\text{circle}} = 100 \cdot \pi$ We'll keep it in terms of $\pi$ for now to avoid rounding errors.

STEP 6

Now, let's tackle the surface area of the sphere!
The formula for the surface area of a sphere is $4 \cdot \pi \cdot r^2$ , where $r$ is the radius of the sphere.

STEP 7

We know the sphere's radius is $5$ units.
Let's plug that in: $\text{Area}_{\text{sphere}} = 4 \cdot \pi \cdot (5)^2$

STEP 8

Squaring $5$ gives us $25$ , so: $\text{Area}_{\text{sphere}} = 4 \cdot \pi \cdot 25$

STEP 9

Multiplying $4$ and $25$ gives us $100$ , so the surface area of the sphere is: $\text{Area}_{\text{sphere}} = 100 \cdot \pi$

STEP 10

Let's compare our results.
The area of the circle is $100 \cdot \pi$ , and the surface area of the sphere is $100 \cdot \pi$ .
Are they the same?

STEP 11

Yes they are! $100 \cdot \pi$ is equal to $100 \cdot \pi$ .

STEP 12

The statement is true.
The area of the circle and the surface area of the sphere are both $100 \cdot \pi$ square units.

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Questionh/public/activity/3006003/assessment 〔 3.6.3 Qulz: SpheresQuestion 1 of 10 The area of a circle of radius 10 units is equal to the surface area of a sphere of radius 5 units. A. True B. False SUBMIT - PREVIOUS

Studdy Solution

STEP 1

What is this asking? Is the area of a circle with radius 101010 the same as the surface area of a sphere with radius 555? Watch out! Don't mix up area of a circle and surface area of a sphere!Also, don't forget those squared terms!

STEP 2

1. Calculate Circle Area2. Calculate Sphere Surface Area3. Compare Results

STEP 3

Alright, let's **calculate the area** of this circle!The formula for the area of a circle is π⋅r2\pi \cdot r^2π⋅r2, where rrr is the **radius**.Remember, the radius is the distance from the center of the circle to any point on the edge.

STEP 4

We're given that the circle has a radius of 101010 units.So, plugging that into our formula, we get: Areacircle=π⋅(10)2 \text{Area}_{\text{circle}} = \pi \cdot (10)^2 Areacircle​=π⋅(10)2

STEP 5

Squaring 101010 gives us 100100100, so the area of the circle is: Areacircle=100⋅π \text{Area}_{\text{circle}} = 100 \cdot \pi Areacircle​=100⋅π We'll keep it in terms of π\piπ for now to avoid rounding errors.

STEP 6

Now, let's tackle the **surface area** of the sphere!The formula for the surface area of a sphere is 4⋅π⋅r24 \cdot \pi \cdot r^24⋅π⋅r2, where rrr is the **radius** of the sphere.

STEP 7

We know the sphere's radius is 555 units.Let's plug that in: Areasphere=4⋅π⋅(5)2 \text{Area}_{\text{sphere}} = 4 \cdot \pi \cdot (5)^2 Areasphere​=4⋅π⋅(5)2

STEP 8

Squaring 555 gives us 252525, so: Areasphere=4⋅π⋅25 \text{Area}_{\text{sphere}} = 4 \cdot \pi \cdot 25 Areasphere​=4⋅π⋅25

STEP 9

Multiplying 444 and 252525 gives us 100100100, so the surface area of the sphere is: Areasphere=100⋅π \text{Area}_{\text{sphere}} = 100 \cdot \pi Areasphere​=100⋅π

STEP 10

Let's compare our results.The area of the circle is 100⋅π100 \cdot \pi100⋅π, and the surface area of the sphere is 100⋅π100 \cdot \pi100⋅π.Are they the same?

STEP 11

Yes they are! 100⋅π100 \cdot \pi100⋅π is equal to 100⋅π100 \cdot \pi100⋅π.

STEP 12

The statement is **true**.The area of the circle and the surface area of the sphere are both 100⋅π100 \cdot \pi100⋅π square units.

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Start learning now

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

Questionh/public/activity/3006003/assessment 〔 3.6.3 Qulz: Spheres
Question 1 of 10 The area of a circle of radius 10 units is equal to the surface area of a sphere of radius 5 units. A. True B. False SUBMIT - PREVIOUS

What is this asking? Is the area of a circle with radius $10$ the same as the surface area of a sphere with radius $5$ ? Watch out! Don't mix up area of a circle and surface area of a sphere!
Also, don't forget those squared terms!

1. Calculate Circle Area
2. Calculate Sphere Surface Area
3. Compare Results

Alright, let's calculate the area of this circle!
The formula for the area of a circle is $\pi \cdot r^2$ , where $r$ is the radius.
Remember, the radius is the distance from the center of the circle to any point on the edge.

We're given that the circle has a radius of $10$ units.
So, plugging that into our formula, we get: $\text{Area}_{\text{circle}} = \pi \cdot (10)^2$

Squaring $10$ gives us $100$ , so the area of the circle is: $\text{Area}_{\text{circle}} = 100 \cdot \pi$ We'll keep it in terms of $\pi$ for now to avoid rounding errors.

Now, let's tackle the surface area of the sphere!
The formula for the surface area of a sphere is $4 \cdot \pi \cdot r^2$ , where $r$ is the radius of the sphere.

We know the sphere's radius is $5$ units.
Let's plug that in: $\text{Area}_{\text{sphere}} = 4 \cdot \pi \cdot (5)^2$

Squaring $5$ gives us $25$ , so: $\text{Area}_{\text{sphere}} = 4 \cdot \pi \cdot 25$

Multiplying $4$ and $25$ gives us $100$ , so the surface area of the sphere is: $\text{Area}_{\text{sphere}} = 100 \cdot \pi$

Let's compare our results.
The area of the circle is $100 \cdot \pi$ , and the surface area of the sphere is $100 \cdot \pi$ .
Are they the same?

Yes they are! $100 \cdot \pi$ is equal to $100 \cdot \pi$ .

The statement is true.
The area of the circle and the surface area of the sphere are both $100 \cdot \pi$ square units.