Math / Geometry

QuestionGiven a cone with a height or radius of $18 \, \text{yd}$ , find the surface area and volume of the cone. Use the following formulas:
1. Surface Area ( $A$ ) of a cone: $ A = \pi r (r + \sqrt{h^2 + r^2}) \] where $r$ is the radius of the base and $h$ is the height.
2. Volume ( $V$ ) of a cone: $ V = \frac{1}{3} \pi r^2 h \]
Determine whether the given measurement of $18 \, \text{yd}$ is the radius or the height, and use it to calculate the surface area and volume. If only one measurement is given, assume it is the radius and provide the formulas for both scenarios.

Studdy Solution

STEP 1

What is this asking? We need to find the surface area and volume of a sphere, given its diameter. Watch out! The image shows a diameter, not a radius!
Don't forget to divide the diameter by two to get the radius.
Also, don't mix up the formulas for surface area and volume!

STEP 2

1. Calculate the radius.
2. Calculate the surface area.
3. Calculate the volume.

STEP 3

The diameter is given as $18$ yd.
Since the radius is half the diameter, we divide the diameter by $2$ to get the radius.
This is because the diameter spans the entire circle, while the radius is the distance from the center to the edge, which is half the distance across.

STEP 4

$\text{Radius} = \frac{\text{Diameter}}{2}$ $r = \frac{18 \text{ yd}}{2} = \mathbf{9 \text{ yd}}$ So, our radius, $r$ , is $\mathbf{9}$ yd.

STEP 5

Now, let's calculate the surface area of the sphere.
The formula for the surface area of a sphere is $A = 4 \pi r^2$ , where $r$ is the radius.
This formula tells us how much "skin" it would take to perfectly cover the entire sphere.

STEP 6

We plug in our radius, $\mathbf{9}$ yd, into the formula: $A = 4 \cdot \pi \cdot (9 \text{ yd})^2$ $A = 4 \cdot \pi \cdot 81 \text{ yd}^2$ $A = 324\pi \text{ yd}^2$ So, the surface area is $\mathbf{324\pi}$ square yards.

STEP 7

Finally, let's calculate the volume!
The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$ , where $r$ is the radius.
This formula tells us how much space is inside the sphere.

STEP 8

We plug in our radius, $\mathbf{9}$ yd, into the formula: $V = \frac{4}{3} \cdot \pi \cdot (9 \text{ yd})^3$ $V = \frac{4}{3} \cdot \pi \cdot 729 \text{ yd}^3$ $V = \frac{4 \cdot 729}{3} \pi \text{ yd}^3$ $V = \frac{2916}{3} \pi \text{ yd}^3$ $V = 972\pi \text{ yd}^3$ So, the volume of the sphere is $\mathbf{972\pi}$ cubic yards.

STEP 9

The surface area of the sphere is $324\pi$ square yards, and the volume is $972\pi$ cubic yards.

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Studdy Solution

STEP 1

What is this asking? We need to find the surface area and volume of a sphere, given its diameter. Watch out! The image shows a diameter, not a radius!
Don't forget to divide the diameter by two to get the radius.
Also, don't mix up the formulas for surface area and volume!

STEP 2

1. Calculate the radius.
2. Calculate the surface area.
3. Calculate the volume.

STEP 3

The diameter is given as $18$ yd.
Since the radius is half the diameter, we divide the diameter by $2$ to get the radius.
This is because the diameter spans the entire circle, while the radius is the distance from the center to the edge, which is half the distance across.

STEP 4

$\text{Radius} = \frac{\text{Diameter}}{2}$ $r = \frac{18 \text{ yd}}{2} = \mathbf{9 \text{ yd}}$ So, our radius, $r$ , is $\mathbf{9}$ yd.

STEP 5

Now, let's calculate the surface area of the sphere.
The formula for the surface area of a sphere is $A = 4 \pi r^2$ , where $r$ is the radius.
This formula tells us how much "skin" it would take to perfectly cover the entire sphere.

STEP 6

We plug in our radius, $\mathbf{9}$ yd, into the formula: $A = 4 \cdot \pi \cdot (9 \text{ yd})^2$ $A = 4 \cdot \pi \cdot 81 \text{ yd}^2$ $A = 324\pi \text{ yd}^2$ So, the surface area is $\mathbf{324\pi}$ square yards.

STEP 7

Finally, let's calculate the volume!
The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$ , where $r$ is the radius.
This formula tells us how much space is inside the sphere.

STEP 8

STEP 9

The surface area of the sphere is $324\pi$ square yards, and the volume is $972\pi$ cubic yards.

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Studdy Solution

STEP 1

What is this asking? We need to find the surface area and volume of a sphere, given its diameter. Watch out! The image shows a diameter, not a radius!Don't forget to divide the diameter by two to get the radius.Also, don't mix up the formulas for surface area and volume!

STEP 2

1. Calculate the radius.2. Calculate the surface area.3. Calculate the volume.

STEP 3

The **diameter** is given as 181818 yd.Since the radius is half the diameter, we **divide** the diameter by 222 to get the radius.This is because the diameter spans the entire circle, while the radius is the distance from the center to the edge, which is half the distance across.

STEP 4

Radius=Diameter2 \text{Radius} = \frac{\text{Diameter}}{2} Radius=2Diameter​ r=18 yd2=9 yd r = \frac{18 \text{ yd}}{2} = \mathbf{9 \text{ yd}} r=218 yd​=9 yd So, our **radius**, rrr, is 9\mathbf{9}9 yd.

STEP 5

Now, let's **calculate** the surface area of the sphere.The formula for the surface area of a sphere is A=4πr2A = 4 \pi r^2A=4πr2, where rrr is the radius.This formula tells us how much "skin" it would take to perfectly cover the entire sphere.

STEP 6

STEP 7

Finally, let's **calculate** the volume!The formula for the volume of a sphere is V=43πr3V = \frac{4}{3} \pi r^3V=34​πr3, where rrr is the radius.This formula tells us how much space is inside the sphere.

STEP 8

STEP 9

The surface area of the sphere is 324π324\pi324π square yards, and the volume is 972π972\pi972π cubic yards.

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Studdy solves anything!

Start learning now

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

What is this asking? We need to find the surface area and volume of a sphere, given its diameter. Watch out! The image shows a diameter, not a radius!
Don't forget to divide the diameter by two to get the radius.
Also, don't mix up the formulas for surface area and volume!

1. Calculate the radius.
2. Calculate the surface area.
3. Calculate the volume.

The diameter is given as $18$ yd.
Since the radius is half the diameter, we divide the diameter by $2$ to get the radius.
This is because the diameter spans the entire circle, while the radius is the distance from the center to the edge, which is half the distance across.

$\text{Radius} = \frac{\text{Diameter}}{2}$ $r = \frac{18 \text{ yd}}{2} = \mathbf{9 \text{ yd}}$ So, our radius, $r$ , is $\mathbf{9}$ yd.

Now, let's calculate the surface area of the sphere.
The formula for the surface area of a sphere is $A = 4 \pi r^2$ , where $r$ is the radius.
This formula tells us how much "skin" it would take to perfectly cover the entire sphere.

Finally, let's calculate the volume!
The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$ , where $r$ is the radius.
This formula tells us how much space is inside the sphere.

The surface area of the sphere is $324\pi$ square yards, and the volume is $972\pi$ cubic yards.