Math / Geometry

QuestionWhat is the volume of the cone below? A. $56 \pi$ units $^{3}$ B. $336 \pi$ units $^{3}$ C. $112 \pi$ units $^{3}$ D. $168 \pi$ units $^{3}$

Studdy Solution

STEP 1

What is this asking? We need to find how much space this cone takes up, given its height and the size of its circular base. Watch out! Don't mix up the formulas for volume of a cone and a cylinder!

STEP 2

1. Recall the formula
2. Plug in the values
3. Calculate the volume

STEP 3

Alright, let's kick things off by remembering the formula for the volume of a cone.
It's $V = \frac{1}{3} \cdot \pi \cdot r^2 \cdot h$ , where $V$ is the volume, $r$ is the radius of the base, and $h$ is the height of the cone.
Why is there a $\frac{1}{3}$ you ask?
Well, a cone takes up one-third of the space of a cylinder with the same base and height!
Pretty neat, huh?

STEP 4

Now, let's plug in what we know!
We're given that the radius, $r$ , is $\textbf{4}$ units and the height, $h$ , is $\textbf{21}$ units.
Substituting these values into our formula, we get $V = \frac{1}{3} \cdot \pi \cdot (\textbf{4})^2 \cdot (\textbf{21})$ .

STEP 5

Time to crunch those numbers!
First, let's square the radius: $4^2 = 4 \cdot 4 = \textbf{16}$ .
So, our formula becomes $V = \frac{1}{3} \cdot \pi \cdot \textbf{16} \cdot 21$ .

STEP 6

Now, let's multiply $16$ by $21$ : $16 \cdot 21 = \textbf{336}$ .
Our formula now looks like this: $V = \frac{1}{3} \cdot \pi \cdot \textbf{336}$ .

STEP 7

Finally, let's multiply by $\frac{1}{3}$ , which is the same as dividing by $3$ : $\frac{336}{3} = \textbf{112}$ .
So, our final volume is $V = \textbf{112}\pi$ cubic units.
Boom!

STEP 8

The volume of the cone is $112\pi$ cubic units, which corresponds to answer choice C.

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QuestionWhat is the volume of the cone below? A. $56 \pi$ units $^{3}$ B. $336 \pi$ units $^{3}$ C. $112 \pi$ units $^{3}$ D. $168 \pi$ units $^{3}$

Studdy Solution

STEP 1

What is this asking? We need to find how much space this cone takes up, given its height and the size of its circular base. Watch out! Don't mix up the formulas for volume of a cone and a cylinder!

STEP 2

1. Recall the formula
2. Plug in the values
3. Calculate the volume

STEP 3

STEP 4

Now, let's plug in what we know!
We're given that the radius, $r$ , is $\textbf{4}$ units and the height, $h$ , is $\textbf{21}$ units.
Substituting these values into our formula, we get $V = \frac{1}{3} \cdot \pi \cdot (\textbf{4})^2 \cdot (\textbf{21})$ .

STEP 5

Time to crunch those numbers!
First, let's square the radius: $4^2 = 4 \cdot 4 = \textbf{16}$ .
So, our formula becomes $V = \frac{1}{3} \cdot \pi \cdot \textbf{16} \cdot 21$ .

STEP 6

Now, let's multiply $16$ by $21$ : $16 \cdot 21 = \textbf{336}$ .
Our formula now looks like this: $V = \frac{1}{3} \cdot \pi \cdot \textbf{336}$ .

STEP 7

Finally, let's multiply by $\frac{1}{3}$ , which is the same as dividing by $3$ : $\frac{336}{3} = \textbf{112}$ .
So, our final volume is $V = \textbf{112}\pi$ cubic units.
Boom!

STEP 8

The volume of the cone is $112\pi$ cubic units, which corresponds to answer choice C.

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QuestionWhat is the volume of the cone below? A. 56π56 \pi56π units 3^{3}3 B. 336π336 \pi336π units 3^{3}3 C. 112π112 \pi112π units 3^{3}3 D. 168π168 \pi168π units 3^{3}3

Studdy Solution

STEP 1

What is this asking? We need to find how much space this cone takes up, given its height and the size of its circular base. Watch out! Don't mix up the formulas for volume of a cone and a cylinder!

STEP 2

1. Recall the formula2. Plug in the values3. Calculate the volume

STEP 3

STEP 4

STEP 5

Time to **crunch those numbers**!First, let's square the radius: 42=4⋅4=164^2 = 4 \cdot 4 = \textbf{16}42=4⋅4=16.So, our formula becomes V=13⋅π⋅16⋅21V = \frac{1}{3} \cdot \pi \cdot \textbf{16} \cdot 21V=31​⋅π⋅16⋅21.

STEP 6

Now, let's multiply 161616 by 212121: 16⋅21=33616 \cdot 21 = \textbf{336}16⋅21=336.Our formula now looks like this: V=13⋅π⋅336V = \frac{1}{3} \cdot \pi \cdot \textbf{336}V=31​⋅π⋅336.

STEP 7

Finally, let's multiply by 13\frac{1}{3}31​, which is the same as dividing by 333: 3363=112\frac{336}{3} = \textbf{112}3336​=112.So, our **final volume** is V=112πV = \textbf{112}\piV=112π cubic units.Boom!

STEP 8

The volume of the cone is 112π112\pi112π cubic units, which corresponds to answer choice C.

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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.

QuestionWhat is the volume of the cone below? A. $56 \pi$ units $^{3}$ B. $336 \pi$ units $^{3}$ C. $112 \pi$ units $^{3}$ D. $168 \pi$ units $^{3}$

1. Recall the formula
2. Plug in the values
3. Calculate the volume

Time to crunch those numbers!
First, let's square the radius: $4^2 = 4 \cdot 4 = \textbf{16}$ .
So, our formula becomes $V = \frac{1}{3} \cdot \pi \cdot \textbf{16} \cdot 21$ .

Now, let's multiply $16$ by $21$ : $16 \cdot 21 = \textbf{336}$ .
Our formula now looks like this: $V = \frac{1}{3} \cdot \pi \cdot \textbf{336}$ .

Finally, let's multiply by $\frac{1}{3}$ , which is the same as dividing by $3$ : $\frac{336}{3} = \textbf{112}$ .
So, our final volume is $V = \textbf{112}\pi$ cubic units.
Boom!

The volume of the cone is $112\pi$ cubic units, which corresponds to answer choice C.