A box is formed by cutting squares from the four corners of a sheet of paper and folding up the sides.
Suppose the paper is 7"-wide by 9"-long.
a. Estimate the maximum volume for this box?
b. What cutout length produces the maximum volume?


Answer 1



Step-by-step explanation:

Answer 2

To answer this question it is necessary to find the volume of the box as a function of "x", and apply the concepts of a maximum of a function.

The solution is:

a) V (max) = 36.6 in³

b) x = 1.3 in

The volume of a cube is:

V(c) = w×L×h  ( in³)

In this case, cutting the length  "x" from each side, means:

wide of the box    ( w - 2×x )   equal to  ( 7 - 2×x )

Length of the box ( L - 2×x )   equal to  ( 9 - 2×x )

The height  is  x

Then the volume of the box,  as a function of x is:

V(x) = ( 7 - 2×x ) × ( 9 -2×x ) × x

V(x) = ( 63 - 14×x - 18×x + 4×x²)×x

V(x) = 4×x³ - 32×x² + 63×x

Tacking derivatives,  on both sides of the equation

V´(x) = 12×x² - 64 ×x + 63

If   V´(x) = 0      then      12×x² - 64 ×x + 63 = 0

This expression is a second-degree equation, solving for x

x₁,₂ = [ 64 ± √ (64)² - 4×12*63

x₁ =  ( 64 + 32.74 )/ 24

x₁ = 4.03     this value  will bring us an unfeasible solution,  since it is not possible to cut 2×4 in from a piece of paper of 7 in ( therefore we dismiss that value)

x₂ = ( 64 - 32.74)/24

x₂ = 1.30 in

The  maximum volume of the box is:

V(max) = ( 7 - 2.60) × ( 9 - 2.60)×1.3

V(max) = 4.4 × 6.4 × 1.3

V(max) = 36.60 in³

To chek for maximum value of V when x = 1.3

we find the second derivative of V  V´´,  and substitute the value of x = 1.3,    if the relation is smaller than 0,  we have a maximum value of V

V´´(x) = 24×x - 64 for x = 1.3

V´´(x) = 24× 1.3 - 64            ⇒   V´´(x) < 0

Then the value  x = 1.3 will bring maximum value for V

Related Link: https://brainly.com/question/13581879

Related Questions

what are the slopes of the two lines?


Step-by-step explanation:

The slopes of two lines are equal

you just purchased a new hooded sweatshirt for $24. the total cleaning cost is $2 total because it will be washed with other sweatshirts. you will wear the sweatshirt 18 times this winter. calculate the cost per wearing.



Step-by-step explanation:

espera y termino y te ayudo vale que me falta poco vale

Dominique has $13.75 in her piggy bank. All of the money are quarters. How many quarters are in the piggy bank?


Answer: I think its 0.55

Step-by-step explanation: if you do 25 times 0.55 because quarter's are 25 cents it equals 13.75

The number of quarters in the piggy bag will be 55.

What is unit conversion?

Multiplication or division by a numerical factor, selection of the correct number of significant figures, and unit conversion are all steps in a multi-step procedure. Unit conversion is the process of changing a quantity's measurement between different units, frequently using multiplicative conversion factors.

Given that Dominique has $13.75 in her piggy bank. All of the money is quarters.

The number of the quarters will be calculated as:-

1 dollar ⇒ 4 quarters

13.75 dollars ⇒ 13.75 x 4 quarters

13.75 dollars ⇒ 55 Quarters

Therefore, the number of quarters in the piggy bag will be 55.

To know more about unit conversion follow



There are 800 students who attend Park Middle School. There are 1.5 times as many boys as there are girls.
x is boys
y is girls
Determine how many boys and girls attend Park Middle School. Round to the nearest whole number if necessary. Show your work.



320 boys; 480 girls

Step-by-step explanation:

Here is the correct system of equations you can write for this equation:








Solve for x:

20- 3x= -3(x-8)​



[tex]20 - 3x = - 3( x - 8) \\ 20 - 3x = - 3x + 24 \\ - 3x + 3x = 24 - 20 \\ 0 = 4[/tex]

It does not have any answer.

What is pls help meeee6x+4




Step-by-step explanation:

can someone tell me what my GPA is.



probably 5.00

Step-by-step explanation:

maybe a 4.0 or higher cuz those r really good grades

7. Make a box and whisker plot of the following data. Show your work as to how you calculated the different key points on the box-and-whisker plot.

21, 25, 30, 27, 22, 23, 23, 24, 26, 28, 25, 22, 23



Oh boy this is a lot of work

Step-by-step explanation:

So 1st thing 1st, order the numbers from least to greatest. Then, find the median (which is the middle number). After that find the 1st Quartile and 2nd Quarile (you have to find the middle of the first half and the middle of the second half if it is two numbers, add them up and divide it by 2). Also, find the greatest and the least numbers. Then make the box and whisker plot. Since i can't make the full plot in brainly i will give you the steps to make it.

Make a number line. Label it from the least number which is 21 to the greatest number which is 30.Make a box from the 1st quartile which is from 33.5 to 39.5.Split the box on number 24.Make the whiskers which mean make a straight line from the left of the box to the least number which is 21 and make another straight line from the right of the box to the greatest number which is 30.That's all the step to make a box-and-whisker plot!

Hope this helped.

Danielle put 4.902 gallons of gasoline in a gas tank. What is 4.902 written in expanded form?



4= ones
9= tenths
0= hundredths
2= thousandths

Consider the diagram.
What is the length of segment AB?
A) 7
B) 9
C) 18
D) 25



the answer is 9

Step-by-step explanation:

Answer: the correct answer is 9!!!!!

Please help! I will give brainliest




Step-by-step explanation:

as x is bigger than 0in which x is maximum and 0 is minumum

What’s the slope of (-8,2) (1,9)


Step-by-step explanation:

equation of the line passing through two points







y=mx+b m is the slope

so m=7/9

there is a short way showing slope shown in the picture

all drop down menus are 0-9
use drop down menus to complete each equation so the statement about its solution is true.


Answer: 5-4+7x+1 = [7]x+[not 2]

5-4+7x+1 = [not 7]x+[any number]

5-4+7x+1 = [7]x+[2]

Helppppppp lol idk what I’m doing




Step-by-step explanation:

Step 1: Translate word to math

"Five" = 5

"times" means multiplication

"a number" = n

"plus" means addition

"3" = 3

"is" means equal

"12" = 12

Step 2: Set up equation (combine)

5n + 3 = 12



Step-by-step explanation:

a number represents n

is represents =

times represents multiplication

plus represents addition

5 times n plus 3 is 12

so this means


14) What is 50% of 180?
15) What is 25% of 36?
16) What is 100% of 166?
17) What is 150% of 126?
18) What is 10% of 100?
19) What is 200% of 133?
20) What is 25% of 172?



14. 90%

15. 9%

16. 166%

17. 189%

18. 10%

19. 266%

20. 40%

Please Solve as Fast as You Can
The coordinates of Point A are (-1, 4). If Point A is reflected across the x-axis AND then across the y-axis, what are the coordinates of Point A now?




Step-by-step explanation:

If you reflect it over the x it would equal (-1,-4), then if you reflect it over the y it would equal (1,-4)

Read the blog entry below and answer the question that follows:

At first I dreaded going to sixth grade camp. I was sure the food would be poisonous, I would freeze to death in my sleeping bag, and the other kids in my cabin would never talk to me. But was I wrong! In fact, it was the best week ever! I went horseback riding and participated in a ropes activity with a zip line course. After the course, I climbed to the top of a pole using only ropes and carabiner clips, and then I rappelled off the top. I even made a new best friend. Oh, by the way, the food was delicious. It tasted better than my mom's! (But don't tell her.)

Review the first sentence:

At first I dreaded going to sixth grade camp.

What is the connotation showing here?

A The author was impatient and could not wait to get to camp.
B The author was worried he/she might not enjoy camp.
C The author welcomed the opportunity to go to camp.
D The author was excited about going to camp.


The answer might be B

It costs $19.50 for postage and $5 for insurance to send a one-pound box by Express Mail.

How much does it cost to send 12 one-pound boxes by Express Mail?


294 dollars

Okay so you would take 19.50 and add the 5 dollar insurance giving you 24.50. Then you’d take the 24.50 and multiply it by 12 giving you 294 dollars

Really hope I got that right

What's another way to express the expression [4(17) + 3(25)] x 5?




Step-by-step explanation:

[4(17) + 3(25)] x 5





What is the slope of a line perpendicular to the line whose equation is
2x – 2y = 20.




Step-by-step explanation:

Solve for y into y=mx+b





the slope is m, so it’s 1

A line perpendicular to that would be -1

PLEASE HELP!!! i’ll give brainliest



x = [tex]\sqrt{164[/tex]

Step-by-step explanation:

In order to find the hypothenuse, you need to do [tex]8^{2} + 10^{2}[/tex]. Which equals 164. Therefore, since [tex]a^{2} +b^{2} =c^{2}[/tex], a=8, b=10, and c=[tex]\sqrt{164[/tex].

If the quarter note equals one beat, how many beats would this note be worth?



4 beats

Step-by-step explanation:

that is a whole note and 4 quarters are equal to one whole

If we double the sum of the numbers x and -21, we get the number -100. What is x?​


Step-by-step explanation:

2{x + (-21) }= –100

= 2x – 42 = –100

= 2x= –100 + 42

= 2x= –58

= x = –58/2

x= –29

What are the slopes of the perpendicular lines?

1: y=-5/2x-8
2: y=5x
3: y=3x-8



1. 2/5

2. -1/5

3. -1/3

Step-by-step explanation:

the parpendicular slope is found by taking the original slope and finding the opposite reciprocal. so just flip the slope and make it the opposite sign

-5/2  --->    2/5

+5/1  --->    -1/5

+3/1  --->    -1/3

1. 2/5
2. -1/5
3. -1/3

There are two key concepts that we have to understand in order to answer this question:

1) The given equations are organized in slope-intercept form: y=mx+b. Here’s what the variables mean:

Y= the y coordinate
M=the slope
X= the coordinate
B= the y intercept

From the given equations we can now identify the slopes as -5/2, 5 and 3.

2) Perpendicular lines have slopes that are NEGATIVE RECIPROCALS of each other.

Example: Line A has the slope 8. A line that is perpendicular to line A would have a slope of -1/8.

Example #2: Line B has the slope -2/3. A line that is perpendicular to line B would have a slope of 3/2 (two negatives make a positive).

Now that we understand these two concepts, we need to look at the slopes of the given lines and determine their negative reciprocals to find the slopes of their perpendicular lines.

1. y=-5/2x-8
We can see that -5/2 is in the place of “m” in y=mx+b. The negative reciprocal of -5/2 is 2/5, so therefore, the slope of the line perpendicular to y=-5/2x-8 is 2/5.

2. y=5x
There is no “b” value in this equation, but we can still identify 5 as “m”. The negative reciprocal of 5 is -1/5, so therefore, the slope of the line perpendicular to y=5c is -1/5.

3. y=3x-8
The negative reciprocal of 3 is -1/3, so therefore, the slope of the line perpendicular to line y=3x-8 is -1/3.

I hope this helps! Please comment if you have any questions.

If a Ferris wheel takes 45 seconds to make a complete rotation, how many rotations will it complete in 3 hours?


Answer:240 rotations

Step-by-step explanation: 1 rotation/45 seconds * 60 seconds / minute * 60 minutes / hour * 3 hours = 3600*3/45 = 240 rotations

Rosita is making pizzas for a party. Each pizza requires 1 cups of sauce and 1 cups
of cheese. How many whole pizzas can Rosita make if she only has 8 cups of sauce
and 132 cups of cheese?



8 pizzas

Step-by-step explanation:

I cAn Do mAtH!!


She can only make 8 pizzas.

Step-by-step explanation:

She only needs 1 cup of sauce and 1 cup of cheese. She has 8 cups of sauce, so she can only make 8 pizzas. What can she do with all the cheese in her refrigerator? Hmmmmm...

What is the equation in slope-intercept form of the line that passes through the points (−12, −4) and (4, 8)?

A. y = −43 − 20
A. , y = −43 − 20

B. y = 43 − 12
B. , y = 43 − 12

C. y = −0.75 − 13
C. , y = −0.75 − 13

D. y = 0.75x +5
D. , y = 0.75x +5



Choice D. [tex]y = 0.75\, x + 5[/tex].

Step-by-step explanation:

The general equation for the slope-intercept form of a line in a cartesian plane is [tex]y = m\, x + b[/tex], where:

[tex]m[/tex] is the slope of the line, and[tex]b[/tex] is the [tex]y[/tex]-intercept of the line. (The [tex]y\![/tex]-intercept of a line in a cartesian plane is the [tex]y\!\![/tex]-coordinate of the point where the line intersects the [tex]\! y \![/tex]-axis.)

Start by finding the value of [tex]m[/tex]. The slope of a line is equal to its rise-over-run. For the two points in this question:

The "rise" is the [tex]y[/tex]-coordinate of the first point minus that of the second: [tex]\text{rise} = (-4) - 8 = -12[/tex].The "run" is the [tex]x[/tex]-coordinate of the first point minus that of the second. The order of these two points should stay the same: [tex]\text{run} = (-12) - 4 = -16[/tex].

Calculate the slope of this line:

[tex]\displaystyle m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{-12}{-16} = 0.75[/tex].

The equation of the line becomes:

[tex]y = \underbrace{0.75}_{m}\, x + b[/tex].

Substitute the coordinates of either of the two points to find [tex]b[/tex]. For example, for the first point [tex](12, -4)[/tex], substitute in the following:

[tex]x = -12[/tex], and[tex]y = -4[/tex].

The equation becomes:

[tex]-4 = 0.75 \times (-12) + b[/tex].

Solve for the value of [tex]b[/tex]:

[tex]b = 5[/tex].

Hence, the slope-intercept form of this line shall be:

[tex]y = 0.75\, x + 5[/tex].

Yousef typed a 36-word paragraph in 23 minute. What is his typing speed, in words per minute?



1.5652173913 words per minute.

Step-by-step explanation:

When you divide 36 words/ 23 minutes, you get 1.5652173913.

Subtract the equation
-7 - 12 =




Step-by-step explanation:



Step-by-step explanation:

Which series of transformations would result in a figure that is congruent to the triangle ABC?
rotation of 180' clockwise about the origin and then dilation with a scale factor of -1, centered at the origin
rotation of 180' clockwise about the origin and then dilation with a scale factor of 2, centered at the origin
reflection across the line y ox and then dilation with a scale factor of -3, centered at the origin
translation 3 units to right and then dilation with a scale factor of 3, centered at the origin


Answer: rotation of 180' clockwise about the origin and then dilation with a scale factor of -1, centered at the origin

Step-by-step explanation:

Two triangles are congruent if all the corresponding angles are equal, and if the corresponding side lengths are equal.

Now, this means that if we have a given triangle, and we apply a dilation/contraction, then the resultant triangle will have different side lengths than the initial triangle, then those two triangles can not be congruent.

So we can already discard all the options with dilations.

Now, remember how the dilation works.

If we have a point that is at a distance L of the origin, and we do a dilation of scale factor K centered at the origin, then the new distance of the point and the origin will be K*L.

Notice that if the scale factor is equal to 1, then there is no change.

In the options, we have one where the scale factor is -1.

The minus sign means that will cause the enlargement to appear on the other side of the center of enlargement. But the scale factor still is of absolute value 1, so it will not change anything.

Then the only option that not dilates the triangle is:

Rotation of 180' clockwise about the origin and then dilation with a scale factor of -1, centered at the origin

So this is the only transformation that results in a figure congruent to the triangle ABC

Other Questions
Maggies brother is 11 years younger than four times her age. The sum of their ages is 54. How old is Maggie? What were the 3 classes in France???????? help please!!For the following situation, identify the explanatory and response variable. Then discuss what reservations (think cautions) you might have about the results of the experiment. Asking What if type questions may help.Do the high, centered brake lights (which have been required on all U.S. cars since 1986) really reduce the frequency of rear-end collisions? Randomized comparative experiments with fleets of rental and business cars, done before the lights were required, showed that the third brake light reduced rear-end collisions by as much as 50%. After the lights were required the drop was only 5%. research and tell the story about a case in which bones helped identify a missing person or helped solve a crome [Government] can have no right except as this is derived fromthe individual right of each man to protect himself and his property.-John LockeQUESTION : What idea from the quote above was also used by Thomas Jefferson to justify American Independence?answer choices : A. Natural RightsB. Universal SuffrageC. Consent of the GovernedD. Checks and Balances Which landform is created by glaciers? A. sandbar B. moraine C. alluvial fan (will give brainliest to the best answer) [tex] \frac{1}{11} \times x = [/tex]NEED THIS NOW/URGENTi dont know how to simplify with fractions please need this right now! do any of you know how to do algebra 2 ? help ASAP. Which relation is a function? PLZ HURRY IT'S URGENT!!!What is the slope of the line?y=3options:013undefined In order to be an active reader, you need to? in a survey 8 out of 32 students preferred vanilla ice cream if 252 students are surveyed how many should theoretically prefer vanilla ice cream make a prediction? Pls help!!! will give brainly!!!!pls wright this sentence Spanish: I pay for the party with coins.Use the correct form of the verb pagar to state what form of payment you use Hannah starting playing video games as soon as she got home from school. She played video games for 20 minutes. Then, it took Hannah 1 hour and 35 minutes to finish her homework. When Hannah finished her homework, it was 4:40 P.M. What time did Hannah get home from school? What is the primary difference between a citizen and a non-citizen? A. Citizens can own vehicles; non-citizens cannot. B. Citizens have political rights; non-citizens do not. C. Citizens can go to school; non-citizens cannot. D. Citizens have lots of money; non-citizens do not. Before i thought the dividing functions Aliza needs to run at a rate faster than 8.8 feet per second in order to exceed her fastest time in a race. After running for 15 minutes, her coach determines that she is running at an average rate of 5.8 miles per hour. He concludes that she is not running fast enough to exceed her fastest time. Behind the Qur'an, what is the second most important source of authority for Muslims? a The 5 Pillars of Islam b The Sunnah c Ramadan d Mecca Read the sentence and choose the word that connects its meaning to the day circled on the image.La clase de ciencias es el ________ 12. mircoles lunes martes domingo What is the independent clause and what is the dependent clause in this sentence.When Sally came out of the bathroom with toilet paper on her shoe and everyone laughed, she knew she had committed a huge faux pas.