8. write a formula for the surface area s of a cube in which each side measures x units.

9. a company will make a cereal box with whole number dimensions and a volume of 100 cubic centimeters. if cardboard costs $0.05 per 100 square centimeters, what is the least cost to make 100 boxes?

h.o.t. problems higher - order thinking
10. evaluate determine if the following statement is true or false. explain your reasoning.
if you double one of the dimensions of a rectangular prism, the surface area will double.

11. analyze a prism with a base that is a regular hexagon is shown. how would you determine the surface area of the hexagonal prism if the area of the base of the prism is x square centimeters?

12. evaluate the figure at the right is made by placing a cube with 12 - centimeter sides on top of another cube with 15 - centimeter sides. is the surface area more or less than 2,214 square centimeters? explain.

13. create draw and label a rectangular prism that has a total surface area between 100 and 200 square units. then find the surface area of your prism.

Question

8. write a formula for the surface area s of a cube in which each side measures x units.

9. a company will make a cereal box with whole number dimensions and a volume of 100 cubic centimeters. if cardboard costs $0.05 per 100 square centimeters, what is the least cost to make 100 boxes?

h.o.t. problems higher - order thinking
10. evaluate determine if the following statement is true or false. explain your reasoning.
    if you double one of the dimensions of a rectangular prism, the surface area will double.

11. analyze a prism with a base that is a regular hexagon is shown. how would you determine the surface area of the hexagonal prism if the area of the base of the prism is x square centimeters?

12. evaluate the figure at the right is made by placing a cube with 12 - centimeter sides on top of another cube with 15 - centimeter sides. is the surface area more or less than 2,214 square centimeters? explain.

13. create draw and label a rectangular prism that has a total surface area between 100 and 200 square units. then find the surface area of your prism.

Accepted Answer

### Question 8
# Explanation:
## Step1: Recall the surface area of a cube
A cube has 6 square faces, and the area of one square face with side length $ x $ is $ x^2 $.
## Step2: Calculate the total surface area
To find the total surface area $ S $ of the cube, we multiply the area of one face by 6. So the formula is $ S = 6x^2 $.
# Answer: $ S = 6x^2 $

### Question 9
# Explanation:
## Step1: Recall the volume and surface area of a rectangular prism
The volume of a rectangular prism is $ V = l \times w \times h $, and the surface area is $ SA = 2(lw + lh + wh) $. We need to find whole - number dimensions $ l, w, h $ such that $ l\times w\times h=100 $ and the surface area is minimized.
First, factorize 100: $ 100 = 1\times1\times100=1\times2\times50 = 1\times4\times25=1\times5\times20 = 1\times10\times10=2\times2\times25 = 2\times5\times10=4\times5\times5$
## Step2: Calculate the surface area for each set of dimensions
- For $ l = 1,w = 1,h = 100 $: $ SA=2(1\times1 + 1\times100+1\times100)=2(1 + 100 + 100)=402$
- For $ l = 1,w = 2,h = 50 $: $ SA=2(1\times2+1\times50 + 2\times50)=2(2 + 50+100)=304$
- For $ l = 1,w = 4,h = 25 $: $ SA=2(1\times4 + 1\times25+4\times25)=2(4 + 25 + 100)=258$
- For $ l = 1,w = 5,h = 20 $: $ SA=2(1\times5+1\times20 + 5\times20)=2(5 + 20+100)=250$
- For $ l = 1,w = 10,h = 10 $: $ SA=2(1\times10+1\times10 + 10\times10)=2(10 + 10+100)=240$
- For $ l = 2,w = 2,h = 25 $: $ SA=2(2\times2+2\times25 + 2\times25)=2(4 + 50+50)=208$
- For $ l = 2,w = 5,h = 10 $: $ SA=2(2\times5+2\times10 + 5\times10)=2(10 + 20+50)=160$
- For $ l = 4,w = 5,h = 5 $: $ SA=2(4\times5+4\times5 + 5\times5)=2(20 + 20+25)=130$

The minimum surface area for one box is 130 square centimeters when the dimensions are $ 4\times5\times5 $.
## Step3: Calculate the surface area for 100 boxes
The surface area for 100 boxes is $ 100\times130 = 13000$ square centimeters.
## Step4: Calculate the cost
Cardboard costs $ \$0.05 $ per 100 square centimeters. The number of 100 - square - centimeter units in 13000 square centimeters is $ \frac{13000}{100}=130 $.
The cost is $ 130\times0.05=\$6.5$
# Answer: $\$6.5$

### Question 10
# Explanation:
## Step1: Recall the surface area formula of a rectangular prism
The surface area of a rectangular prism with length $ l $, width $ w $, and height $ h $ is $ SA=2(lw + lh+wh) $
## Step2: Double one dimension and calculate the new surface area
Let's double the length, so the new length is $ 2l $. The new surface area $ SA'=2((2l)w+(2l)h + wh)=2(2lw + 2lh+wh)=4lw + 4lh+2wh$
The original surface area $ SA = 2lw+2lh + 2wh$
Now, find the ratio of $ SA' $ to $ SA $:
$\frac{SA'}{SA}=\frac{4lw + 4lh+2wh}{2lw + 2lh+2wh}$
Let's take an example. Let $ l = 1,w = 1,h = 1 $
Original surface area $ SA=2(1\times1 + 1\times1+1\times1)=6$
Double the length: new $ l = 2 $, new surface area $ SA'=2(2\times1+2\times1 + 1\times1)=2(2 + 2+1)=10$
$ 10
eq2\times6 $
In general, when we double one dimension, say $ l $ to $ 2l $, the change in surface area is not a simple doubling because the cross - terms (like $ lw,lh,wh $) do not all double. The surface area will increase, but not by a factor of 2.
# Answer: The statement is false. When one dimension of a rectangular prism is doubled, the surface area does not double. For example, if a rectangular prism has dimensions $ l = 1,w = 1,h = 1 $ (surface area $ SA = 6 $), and we double the length to $ l = 2 $, the new surface area is $ 10$, which is not double of $ 6 $. Mathematically, from the surface area formula $ SA = 2(lw+lh + wh) $, doubling one dimension (e.g., $ l$ to $ 2l$) gives $ SA'=2(2lw + 2lh+wh)$, and $ SA'
eq2SA $ in general.

### Question 11
# Explanation:
## Step1: Recall the surface area formula of a prism
The surface area of a prism is the sum of the areas of the two bases and the lateral (side) surface area. The area of each base is $ x $ square centimeters, so the area of the two bases is $ 2x $ square centimeters.
## Step2: Determine the lateral surface area
The base is a regular hexagon. From the diagram, the side length of the hexagon base is 8 cm and the height (length) of the prism is $ y $ cm. A regular hexagon has 6 equal - length sides. The lateral surface area of a prism is the perimeter of the base times the height of the prism. The perimeter of the hexagonal base $ P=6\times8 = 48$ cm. So the lateral surface area is $ 48y $ square centimeters.
## Step3: Calculate the total surface area
The total surface area of the hexagonal prism is the sum of the areas of the two bases and the lateral surface area, so it is $ 2x+48y $ square centimeters.
# Answer: The surface area of the hexagonal prism is $ 2x + 48y $ square centimeters, where $ x $ is the area of one base and $ y $ is the height (length) of the prism. We find it by adding the area of the two bases ($ 2x $) and the lateral surface area (perimeter of the base $ 48 $ cm times the height $ y $ cm, i.e., $ 48y $).

### Question 12
# Explanation:
## Step1: Recall the surface area formula of a cube
The surface area of a cube with side length $ s $ is $ SA = 6s^2 $
## Step2: Calculate the surface area of the two cubes without considering the overlapping area
The surface area of the lower cube (side length $ 15 $ cm) is $ 6\times15^2=6\times225 = 1350$ square centimeters.
The surface area of the upper cube (side length $ 12 $ cm) is $ 6\times12^2=6\times144 = 864$ square centimeters.
The total surface area of the two cubes without considering the overlap is $ 1350 + 864=2214$ square centimeters.
## Step3: Consider the overlapping area
When we place the upper cube on the lower cube, the overlapping area is the area of the base of the upper cube, which is $ 12\times12 = 144$ square centimeters. This area is subtracted twice (once from the lower cube's top surface and once from the upper cube's bottom surface) from the total surface area of the two non - overlapping cubes.
The actual surface area $ SA=2214-2\times144=2214 - 288 = 1926$ square centimeters.
Since $ 1926<2214 $, the surface area is less than 2214 square centimeters.
# Answer: The surface area is less than 2214 square centimeters. When we place the smaller cube on the larger cube, we have to subtract the overlapping area (the area of the base of the smaller cube) twice from the sum of the surface areas of the two separate cubes. The sum of the surface areas of the two separate cubes is $ 6\times15^{2}+6\times12^{2}=2214 $, and after subtracting the overlapping area ($ 2\times12\times12 = 288 $), the actual surface area is $ 2214 - 288=1926<2214 $.

### Question 13
# Explanation:
## Step1: Choose dimensions for the rectangular prism
Let's choose length $ l = 4 $, width $ w = 3 $, height $ h = 5 $ (these are whole numbers and we will check the surface area).
## Step2: Calculate the surface area
Using the surface area formula for a rectangular prism $ SA = 2(lw+lh + wh) $
$ lw=4\times3 = 12$, $ lh=4\times5 = 20$, $ wh=3\times5 = 15$
$ SA=2(12 + 20+15)=2(47)=94$ (this is too small). Let's try $ l = 5 $, $ w = 4 $, $ h = 6 $
$ lw = 5\times4=20$, $ lh=5\times6 = 30$, $ wh=4\times6 = 24$
$ SA=2(20 + 30+24)=2(74)=148$ (between 100 and 200)
## Step3: Draw and label the prism
Draw a rectangular prism. Label the length as 5 units, the width as 4 units, and the height as 6 units.
# Answer:
- **Draw and label**: Draw a rectangular prism. Mark the length as $ 5 $ units, the width as $ 4 $ units, and the height as $ 6 $ units.
- **Surface area calculation**: The surface area of the rectangular prism with $ l = 5 $, $ w = 4 $, $ h = 6 $ is $ 148 $ square units (calculated as $ 2(5\times4 + 5\times6+4\times6)=148 $).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 8

Step1: Recall the surface area of a cube

Step2: Calculate the total surface area

Step1: Recall the volume and surface area of a rectangular prism

Step2: Calculate the surface area for each set of dimensions

Step3: Calculate the surface area for 100 boxes

Step4: Calculate the cost

Step1: Recall the surface area formula of a rectangular prism

Step2: Double one dimension and calculate the new surface area

Answer:

Question 9