Math, asked by pastynub, 5 days ago

Write an equation that represents the following statements:
The height (H) of a shape is 6 less than 3 times its volume (V).


The weight (W) of an object is 4 less than twice its height (H).​

Answers

Answered by mp40loverspy
0

Step-by-step explanation:

Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in the image below has length

4

units, width

2

units, and height

3

units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This

4

by

2

by

3

rectangular solid has

24

cubic units.

A rectangular solid is shown. Each layer is composed of 8 cubes, measuring 2 by 4. The top layer is pink. The middle layer is orange. The bottom layer is green. Beside this is an image of the top layer that says

Altogether there are

24

cubic units. Notice that

24

is the

length

×

width

×

height

.

The top line says V equals L times W times H. Beneath the V is 24, beneath the equal sign is another equal sign, beneath the L is a 4, beneath the W is a 2, beneath the H is a 3.

The volume,

V

, of any rectangular solid is the product of the length, width, and height.

V

=

L

W

H

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base,

B

, is equal to

length

×

width

.

B

=

L

W

We can substitute

B

for

L

W

in the volume formula to get another form of the volume formula.

The top line says V equals red L times red W times H. Below this is V equals red parentheses L times W times H. Below this is V equals red capital B times h.

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the

4

×

2

×

3

rectangular solid we started with. See the image below.

An image of a rectangular solid is shown. It is made up of cubes. It is labeled as 2 by 4 by 3. Beside the solid is V equals Bh. Below this is V equals Base times height. Below Base is parentheses 4 times 2. The next line says V equals parentheses 4 times 2 times 3. Below that is V equals 8 times 3, then V equals 24 cubic units.

To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

A

front

=

L

×

W

A

side

=

L

×

W

A

top

=

L

×

W

A

front

=

4

3

A

side

=

2

3

A

top

=

4

2

A

front

=

12

A

side

=

6

A

top

=

8

Notice for each of the three faces you see, there is an identical opposite face that does not show.

S

=

(

front

+

back

)

+

(

left side

+

right side

)

+

(

top

+

bottom

)

S

=

(

2

front

)

+

(

2

left side

)

+

(

2

top

)

S

=

2

12

+

2

6

+

2

8

S

=

24

+

12

+

16

S

=

52

sq. units

The surface area

S

of the rectangular solid shown above is

52

square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see the image below). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

S

=

2

L

H

+

2

L

W

+

2

W

H

For each face of the rectangular solid facing you, there is another face on the opposite side. There are

6

faces in all.

A rectangular solid is shown. The sides are labeled L, W, and H. One face is labeled LW and another is labeled WH.

VOLUME AND SURFACE AREA OF A RECTANGULAR SOLID

For a rectangular solid with length

L

, width

W

, and height

H

:

A rectangular solid is shown. The sides are labeled L, W, and H. Beside it is Volume: V equals LWH equals BH. Below that is Surface Area: S equals 2LH plus 2LW plus 2WH.

Doing the Manipulative Mathematics activity “Painted Cube” will help you develop a better understanding of volume and surface area.

EXAMPLE

For a rectangular solid with length

14

cm, height

17

cm, and width

9

cm, find the 1. volume and 2. surface area.

Solution

Step 1 is the same for both 1. and 2., so we will show it just once.

Step 1. Read the problem. Draw the figure and

label it with the given information.

.

1.

Step 2. Identify what you are looking for. the volume of the rectangular solid

Step 3. Name. Choose a variable to represent it. Let

V

= volume

Step 4. Translate.

Write the appropriate formula.

Substitute.

V

=

L

W

H

V

=

14

9

17

Step 5. Solve the equation.

V

=

2

,

142

Step 6. Check

We leave it to you to check your calculations.

Step 7. Answer the question. The surface area is

1,034

square centimeters.

2.

Step 2. Identify what you are looking for. the surface area of the solid

Step 3. Name. Choose a variable to represent it. Let

S

= surface area

Step 4. Translate.

Write the appropriate formula.

Substitute.

S

=

2

L

H

+

2

L

W

+

2

W

H

S

=

2

(

14

17

)

+

2

(

14

9

)

+

2

(

9

17

)

Step 5. Solve the equation.

S

=

1

,

034

Step 6. Check: Double-check with a calculator.

Step 7. Answer the question. The surface area is

1

,

034

square centimeters.

Similar questions