Math / Geometry

Question7) $94.6 \mathrm{~m}^{3}$ 9) $3801.3 \mathrm{ft}^{3}$ 8) $30 \mathrm{mi}^{3}$ 10)
270 km ${ }^{3}$

Studdy Solution

STEP 1

What is this asking? We need to find the volume of four different 3D shapes: a trapezoidal prism, a pentagonal pyramid, a cylinder, and another pentagonal pyramid! Watch out! Don't mix up the formulas for different shapes!
Also, be careful with the units – we've got meters, miles, feet, and kilometers!

STEP 2

1. Trapezoidal Prism Volume
2. Pentagonal Pyramid Volume
3. Cylinder Volume
4. Pentagonal Pyramid Volume

STEP 3

Alright, for the trapezoidal prism, the formula for the volume is $V = \frac{1}{2} (b_1 + b_2) \cdot h_1 \cdot h_2$ , where $b_1$ and $b_2$ are the lengths of the parallel bases of the trapezoid, $h_1$ is the height of the trapezoid, and $h_2$ is the height or depth of the prism.
It's like finding the area of the trapezoid and then multiplying by how deep the prism goes!

STEP 4

Let's plug in the values!
We have $b_1 = 4$ m, $b_2 = 8$ m, $h_1 = 3$ m, and $h_2 = 4.3$ m.
So, $V = \frac{1}{2} (4 + 8) \cdot 3 \cdot 4.3$ .

STEP 5

First, add those bases: $4 + 8 = 12$ .
Now we have $V = \frac{1}{2} \cdot 12 \cdot 3 \cdot 4.3$ .

STEP 6

Multiply $1/2$ by $12$ to get $6$ .
Now, $V = 6 \cdot 3 \cdot 4.3$ .

STEP 7

Multiply $6$ and $3$ to get $18$ .
Finally, $V = 18 \cdot 4.3 = 77.4 \text{ m}^3$ .
So, the volume of the trapezoidal prism is $\mathbf{77.4} \text{ }\mathbf{m^3}$ .

STEP 8

The volume of a pyramid is $V = \frac{1}{3} \cdot B \cdot h$ , where $B$ is the area of the base and $h$ is the height of the pyramid.
Here, the base is a pentagon, but they've already given us the area!

STEP 9

We're given that the area of the pentagonal base is $5 \cdot 5 = 25$ mi $^2$ and the height is $h = 3$ mi.
Plugging these values into the formula, we get $V = \frac{1}{3} \cdot 25 \cdot 3$ .

STEP 10

Multiplying $\frac{1}{3}$ by $3$ gives us $1$ , so $V = 25 \cdot 1 = 25$ mi $^3$ .
The volume of this pentagonal pyramid is $\mathbf{25} \text{ }\mathbf{mi^3}$ .

STEP 11

The volume of a cylinder is $V = \pi r^2 h$ , where $r$ is the radius and $h$ is the height.
We're given the diameter, which is $22$ ft, so the radius is half of that, $r = 11$ ft.
The height is given as $h = 10$ ft.

STEP 12

Let's plug in the values: $V = \pi \cdot 11^2 \cdot 10$ .

STEP 13

Squaring the radius gives $11^2 = 121$ , so $V = \pi \cdot 121 \cdot 10$ .

STEP 14

Multiplying $121$ by $10$ gives $1210$ , so $V = \pi \cdot 1210$ .
Using $\pi \approx 3.14$ , we get $V \approx 3.14 \cdot 1210 = 3799.4$ ft $^3$ .
The volume of the cylinder is approximately $\mathbf{3799.4} \text{ }\mathbf{ft^3}$ .

STEP 15

Just like the previous pentagonal pyramid, the volume is $V = \frac{1}{3}Bh$ .
We're given that the area of the base is $9 \cdot 10 = 90$ km $^2$ and the height is $6$ km.

STEP 16

Plugging in the values, we get $V = \frac{1}{3} \cdot 90 \cdot 6$ .

STEP 17

Multiplying $\frac{1}{3}$ by $90$ gives $30$ , so $V = 30 \cdot 6$ .

STEP 18

Finally, $V = 180$ km $^3$ .
The volume of this pentagonal pyramid is $\mathbf{180} \text{ }\mathbf{km^3}$ .

STEP 19

7) $77.4 \text{ m}^3$ 8) $25 \text{ mi}^3$ 9) $3799.4 \text{ ft}^3$ 10) $180 \text{ km}^3$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

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

Parents Influencer program Contact Policy Terms