Jane needs open-topped boxes to store her excess inventory at year's end. She purchases large
rectangles of thick cardboard with a length of 12 inches and width of 8 inches to make the boxes.
She is interested in maximizing the volume of the boxes and wants to know what size squares to
cut out at each corner of the cardboard (which will allow the corners to be folded up to form the
box) to do this. What is the maximum volume?

Answers

Answer 1

Answer:

x = 1,6 in   ( the side of the corner squares )

V(max) = 67,58 in³

Step-by-step explanation:

The cardboard is:

L = 12 in          w  =  8 in

Let´s call "x" the side of the square from the corner:

Then the sides of the base of the open box are:

( L - 2*x )    and    ( w - 2*x )     and   x is the height

( 12  -  2*x )  and  ( w  - 2*x )

V(ob) = (L - 2*x ) * (  - 2*x ) * x

Wich is a function of x

V(x) = [( 12 - 2*x ) * ( 8 - 2*x ) ]*x

V(x) = ( 96 - 24*x - 16*x + 4*x²) * x

V(x) = 96*x - 40*x² + 4*x³

Tacking derivatives on both sides of the equation

V´(x) = 12*x² - 80*x + 96

V´(x) = 0     12*x² - 80*x + 96 = 0

Solving for x

x₁,₂ = 80 ± √ 6400 - 4608 / 24

x₁,₂ = 80 ± 42,33 / 24

x₁ = 5,10       We dismiss this solution since 2*x  becomes 2*5,10 = 10,20

ant this value is bigger than 8 inches

x₂ = 1,60 in

Therefore dimensions of the box

a = 12 - 2*x     ;  a =  12 - 3,20  ;  a = 8,8 in

b = 8 - 2*x   ; b = 8 - 3,20  ; b =  4,80  in

And the volume of the open box is:

V(max) = 8,8*4,8*1,6

V(max)  =  67,58 in³

How do we Know that is the maximun value for V?

We find  V´´(x)  = 24*x - 80     for  x = 1,6 is negative ( V´´(x) = - 41,6 therefore V (x) has  a local maximun for a value of x = 1,6


Related Questions

An angle measures 14° less than the measure of its complementary angle. What is the measure of each angle?

Answers

Answer:

38 and 52

Step-by-step explanation:

x = small angle

Complementary angles add to 90 degrees

x + x+14 = 90

2x = 76

x= 38

90-38 = 52

52-14 = 38

38 +52 = 90

kaley and tom both work at publix

Answers

Answer:

Sorry, is there more to the question?

Step-by-step explanation:

Put it DB and I'll gladly answer!

Is there more to this??

Units of Length
Write a conversion factor that you would use to convert
from feet to meters.
Customary System
Units
Metric System Units
new units of measure
original units of measure
1 inch
2.54 centimeters
numerator =
1 foot
0.3048 meters
denominator =
1 mile
1.61 kilometers
Intro
Done

Answers

Answer:

1 foot = 0.3048 meters

Step-by-step explanation:

1 foot = 12 inches

1 inch = 2.54 centimeters

1 foot = 12 * 2.54 = 30.48 centimeters

1 meter = 0.01 centimeters

1 foot = 0.3048 meters

Answer:

numerator =  

✔ 0.3048 meters

denominator =  

✔ 1 foot

Step-by-step explanation: I got it right

The equations y = tan^2 (x) and y = 1 are graphed below. According to the graphwhich of the following is part the solution to the trigonometric inequality tan^2 (x) < 1 over the interval 0 <= x <= 2pi radians?

Answers

Answer: a

Step-by-step explanation:

just took test edg2020

The part of the solution of the equation that lies in the solution region of the inequality is x = π/6, the correct option is A.

What is an Inequality?

An Inequality is a statement that is formed when two expressions are joined using an inequality operator.

The equation is

y = tan²x

y = 1

the inequalities are

tan²x < 1   ( 0≤ x ≤ 2π)

The graph of the system of equations and the inequality is plotted, and attached with the answer.

The answer can be seen as  x = π/6.

To know more about Inequality

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42.2 to the nearest whole number

Answers

Answer:

42

Step-by-step explanation:

If the decimal was 5 or higher, that means it would've been 43, but since it's lower than 4, it rounds down to 42.

Which algebraic expression is equivalent to the expression below? 4/5x - 7/5y


A. 1/5(4x - 7y)

B. 5(7x - 7y)

C. 5(4x - 7y)

D. 1/5(7x + 4y)

Answers

I think the answer is A sorry if I get it wrong :(

Agrocery store sells 4 avocados for $6.00. What is the unit cost of each avocado?
A) $2.40
B) $0.10
C $1.50
D
$0.60

Answers

$0.60
Just take 4 and try dividing by 6
$6.00 divided by 4 avocados = $1.50


If you can do your homework at a rate of 3 minutes for every problem, how many problems
could be done after 4.81 hours? Round to a whole number.

Answers

Answer:

96.2

Step-by-step explanation:

well, it rounds up to 96, so the answer would be 96, but the actual answer is 96.2

HELP ILL GIVE BRAINLIEST!!!!

Answers:
A. 0 and 4
B. -4,-2 and 0
C. 0,2, abs 4
D. -4 and 0

Answers

Its A i had the same one. Hope it helps.

Express the following
ratio in its simplest
terms: 4 weeks 6 days

Answers

Answer:

it would be 2/3............

Answer:

=4:6

4

6

=2

3

2:3

Step-by-step explanation:

Hope this helps

Pls mark me the brainliest

question: what is the relationship between measures of angle 1 and angle 2
pls help need answer fast​

Answers

Answer:

C.

Step-by-step explanation:

m1 is greater than m2 because the angle is larger

What is the greatest common factor of 24 and 48?

Answers

Answer:

Greatest Common Factor: 24

Step-by-step explanation:

The greatest common factor is found by multiplying all the factors which appear in both lists: So the GCF of 24 (2x2x2x3) and 48 (2x2x2x2x3) is 2x2x2x3 which is 24.

The greatest common factor of 24 and 48 is 24.

Here, we have,

To find the greatest common factor (GCF) of 24 and 48,

we can use the prime factorization method:

Step 1: Find the prime factorization of each number.

Prime factorization of 24:

24 = 2 * 2 * 2 * 3

Prime factorization of 48:

48 = 2 * 2 * 2 * 2 * 3

Step 2: Identify the common prime factors and their lowest exponents.

Common prime factors: 2 * 2 * 2 * 3

Step 3: Multiply the common prime factors with their lowest exponents.

GCF(24, 48) = 2 * 2 * 2 * 3 = 24

So, the greatest common factor of 24 and 48 is 24.

To learn more on GCF click:

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The land area of a country is about 3.5 million square miles and the population is about 200 million people. What is the average population density?

Answers

So the Population density of this place is 800 million people / 3.5 million miles squared

Which equals: 228.57 people / square mile

9514 1404 393

Answer:

  about 57.1 persons per square mile

Step-by-step explanation:

To find persons per square mile, divide persons by square miles:

  (200 million persons)/(3.5 million mi^2) = (200/3.5) persons/mi^2

  ≈ 57.1 persons/mi^2

The sum of three consecutive numbers is 84. What is the largest of these numbers?

26
27
28
29

Answers

Step-by-step explanation:

set 3 no is ,x _ I ,x,x+1

X-1 + x + x + I = 84

3 X = 84

X= 84/ 3

X= 2 8

largest no is 29

The largest number is 29.
Step-by-step explanation:
Let the numbers be = x , x+1 , x+2
As given, their sum is 84, we can write as:

=>
=>
=>
x = 27
So, x+1=
And x+2 =
Hence, the largest number is 29.

The volleyball team raised $350 from its fundraiser. It wants to buy jerseys and volleyballs. The jerseys cost $12 each, and the team needs 22 jerseys. Volleyballs cost $15 each. What is the maximum number of volleyballs the team can buy and still have money left? Write your answer and explain the steps you took. WILL GIVE BRAIKNLIEST IF RIGHT!!!!!!!!

Answers

Answer:

6

Step-by-step explanation:

Answer:

The team can buy 5 volleyballs at maximum with no money left over.

Step-by-step explanation:

First, you need to find the money that the jerseys used:

12 × 22 = $264

Next, you have to subtract that from the total amount:

350 - 264 = $86

Finally, you have to see how many volleyballs you can buy:

If volleyballs cost $15 each then, 86 ÷ 15 = 5.73333333(Infinite)

But you can not buy part of a volleyball so you would have to buy 5 volleyballs. Then you would only have .7333333(Infinite) money left over, but that wouldn't be good for anything.

Is this a good answer????

Answer correctly for Brainliest. Show work.

Answers

Answer:

B) y=x/6 is the linear equation

Step-by-step explanation:

a) y=x^6 is a quadratic equation

c) y=6/x is a rational equation

d) y=|x+6| is a absolute value equation

Answer:

B is linear.

Step-by-step explanation:

You can plug the equations into y= or your calculator and compare the graphs

Salim walked113miles to school and then212miles to work after school. How far did he walk in all? 316 miles 325 miles 345 miles 356 miles

Answers

Answer:

Total distance traveled = 1/3 + 1/2 = 2+3/6 = 5/6 miles

Hope this helps!

Step-by-step explanation:

A 2 1/2 pound of grass seed covers 1/8 of an acre. How much grass seed is needed for 1 acre

Answers

Answer:

20 pounds

Step-by-step explanation:

You just multiply 2 1/2 by 8 to figure out the whole acre

2.5x8=20

Answer:

20 pounds of grass seeds

Step-by-step explanation:

2.5: 1/8

?    : 1

multiply both sides by 8.

2.5*8=20

? = 20

Convert the binary number 11011010 to decimal.

Answers

Answer:

218

Step-by-step explanation:

im not sure im sorry in advance if im not

128+64+0+16+8+0+2+0=218

11011010 is 218 in decimal

Look at the picture please!!!!

Answers

Answer:

Angle 1: 131

Angle 6: 49

Answer:

m∠1 = 131º

m∠6 = 49º

Step-by-step explanation:

So we know that a straight line is equal to 180º and we already know the measure of angle 8 (49º) so to find the measure of angle 7, we just subtract 49 from 180.

180 - 49 = 131

So m∠7 = 131º

Now because of the alternate exterior angle rule, we can now find out that m∠7 is the same as m∠1

Now for the measurement of angle 6, we know that it's 49º because of the vertical angles rule.

A submarine started at the surface of the water and
was moving down at –15 kilometers per
minute
toward the ocean floor. The submarine traveled at
this rate for 52 minutes before coming to rest on the
ocean floor. What is the depth of the ocean floor?

Answers

Answer:

The depth of the Ocean Floor is -780 kilometers

Step-by-step explanation:

The answer is -780 because -15 multiplied by 52 equals -780.

The heights of women in the USA are normally distributed with a mean of 64 inches and a standard deviation of 3 inches.

Required:
a. A random sample of six women is selected. What is the probability that the sample mean is greater than 63 inches?
b. What is the probability that a randomly selected woman is taller than 66 inches?
c. What is the probability that the mean height of a random sample of 100 women is greater than 66 inches?

Answers

Answer:

(a) 0.2061

(b) 0.2514

(c) 0

Step-by-step explanation:

Let X denote the heights of women in the USA.

It is provided that X follows a normal distribution with a mean of 64 inches and a standard deviation of 3 inches.

(a)

Compute the probability that the sample mean is greater than 63 inches as follows:

[tex]P(\bar X>63)=P(\frac{\bar X-\mu}{\sigma/\sqrt{n}}>\frac{63-64}{3/\sqrt{6}})\\\\=P(Z>-0.82)\\\\=P(Z<0.82)\\\\=0.20611\\\\\approx 0.2061[/tex]

Thus, the probability that the sample mean is greater than 63 inches is 0.2061.

(b)

Compute the probability that a randomly selected woman is taller than 66 inches as follows:

[tex]P(X>66)=P(\frac{X-\mu}{\sigma}>\frac{66-64}{3})\\\\=P(Z>0.67)\\\\=1-P(Z<0.67)\\\\=1-0.74857\\\\=0.25143\\\\\approx 0.2514[/tex]

Thus, the probability that a randomly selected woman is taller than 66 inches is 0.2514.

(c)

Compute the probability that the mean height of a random sample of 100 women is greater than 66 inches as follows:

[tex]P(\bar X>66)=P(\frac{\bar X-\mu}{\sigma/\sqrt{n}}>\frac{66-64}{3/\sqrt{100}})\\\\=P(Z>6.67)\\\\\ =0[/tex]

Thus, the probability that the mean height of a random sample of 100 women is greater than 66 inches is 0.

Write an equation in point-slope form and slope-intercept form for the line.

passes through (6, -6), slope = 5

Answers

Answer:

y = 5- 36

Step-by-step explanation:

6 is the x

-6 is the y

y=mx+b

so m times x, m is the slope

5 times 6 is 30

-6=30+b

so -36

y=5x-36

SORRY IF IM WRONG I TRIED

The population f(x), in millions, of State A of a country after x years is represented by the function shown below: f(x) = 4(1.08)x The graph shows the population g(x), in millions, of State B of the country after x years: Graph of function g of x equals 2 multiplied by 1.08 to the power of x Which conclusion is correct about the population of State A and State B? The original population of State A was double of the original population of State B. The original population of State B was double of the original population of State A. The original population of State A was four times of the original population of State B. The original population of State A was equal to the original population of State B.

Answers

We want to compare the population equations for two given states.

The correct option is:

"The original population of State A was double of the original population of State B"

We know that the population of state A is given by:

f(x) = 4*(1.08)^x

While for state B, the population is given by:

g(x) = 2*(1.08)^x

We want to see which conclusion is correct about the population of state A and state B.

First, the general exponential growth equation is given by:

h(x) = A*(r)^x

Where A is the initial value and r is the rate of growth.

With this, we can see that f(x) and g(x) have the same rate, and f(x) has an initial value that is twice the one of g(x).

Then the correct option is "The original population of State A was double of the original population of State B"

Because the initial value represents the initial population.

If you want to learn more, you can read:

https://brainly.com/question/12490064

A deep ocean trench is 30,000 feet deep How deep is that in miles? Round your answer to two decimal places. (There are 5,280 feet in a mile.)

Answers

If a mile =5,208 ft that means you need to divide 30,000÷5,280 to get your answer!

Step-by-step explanation:

30,000÷5,280= 5.68mil

1m/5,280ft = xm/30,000
solve for x miles
x = 5.68 miles

ksdfjsbjgdfg f sndm f

Answers

Answer:

Nice

Step-by-step explanation:

Answer:

hahhhh funny one nice :-)

According to a disease control​ center, ​26% of people in a certain country have high blood pressure. If a person in this country is selected at​ random, determine the odds in favor of this person having high blood pressure.

Answers

Answer:

Squirrel has High BLood pressure

Step-by-step explanation:

Squirrels

2:3= 10:

12:15 = 4:

1:5= 5:

15:30 = 1 :

Answers

Answer:

I think this is the answer

Step-by-step explanation:

2:3=10:15

12:15=4:5

1:5=5:25

15:30=1:2

Find the area of the shaded region

Answers

Answer: To find the area of the shaded region :

Calculate the area of both shapes. The area of a rectangle is determined by multiplying its length times its width. The area of a circle is Pi (i.e., 3.14) times the square of the radius. Find the area of the shaded region by subtracting the area of the small shape from the area of the larger shape.

Step-by-step explanation: I cannot see the picture to answer your question but I hope this helps you find put your answer. :)

Answer:

13

Step-by-step explanation:

Start with the smaller sqaure without the x varible which is 7 x 3 = 21

Then the larger square without the varible, 8 x 4 = 32

Now we subtract 21 from 34 and get 13.

So the area for the shaded region should be 13

What is the surface area of a rectangular prism that measures 10 in x 6 in x 4 in?

Answers

Answer: 240 in

Step-by-step explanation: 10 x 6 x 4 = 240 in

I hope that helps

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