1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
8_murik_8 [283]
3 years ago
9

An open-top box is to be made from a 42-inch by 48-inch piece of plastic by removing a square from each corner of the plastic an

d folding up the flaps on each side. What should be the length of the side x of the square cut out of each corner to get a box with the maximum volume
Mathematics
1 answer:
Marina CMI [18]3 years ago
4 0

Answer:

x  = 7.45 inch for the maximum volume.

The area of the square = x² = 7.45² = 55.5025  inch²

Step-by-step explanation:

From the given information:

An open-top box is to be made from a 42-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side.

The objective is to determine the length of the side x of the square cut out of each corner to get a box with the maximum volume

The volume of the box = l×b×h

The volume of the box = (42 - 2x)   \times (48-2x) \times (x)

The volume of the box = (2016 - 84x - 96x +4x^2)x

The volume of the box = (2016 -180x+4x^2)x

The volume of the box = (2016x -180x^2+4x^3)

The volume of the box = 4x^3 - 180x^2 +2016x

For the maximum volume V' = 0

V' = 12x^2 - 360x + 2016

Using the quadratic formula; we have:

=  \dfrac{-b \pm \sqrt{b^2 -4ac}}{2a}

where;

a = 12 , b = -360 c = 2016

=  \dfrac{-(-360) \pm \sqrt{(-360)^2 -4(12)(2016)}}{2(12)}

=  \dfrac{360 \pm \sqrt{129600 -96768}}{24}

=  \dfrac{360 \pm \sqrt{32832}}{24}

=  \dfrac{360 \pm 181.196}{24}

=  \dfrac{360 + 181.196}{24} \ \ \  OR \ \ \   \dfrac{360 - 181.196}{24}

=  \dfrac{541.196}{24} \ \ \  OR \ \ \   \dfrac{178.804}{24}

=  22.55 \ \ \  OR \ \ \  7.45

For the maximum value , we check the points in the second derivative term

V'' = 24x - 360

V'' ( 22.55) = 24(22.55) - 360

V'' ( 22.55) = 541.2 - 360  

V'' ( 22.55) = 181.2   (minimum)

V'' ( 7.45) = 24(7.45) - 360

V'' ( 7.45) = 178.8 - 360  

V'' ( 7.45) = -181.2  < 0  (maximum)

Therefore, x  = 7.45 inch for the maximum volume.

The area of the square = x² = 7.45² = 55.5025  inch²

You might be interested in
Mr. Washington is buying a gallon of milk for $3.99 and some boxes of cereal for $4.39 each. If Mr. Washington has
wolverine [178]

Answer:

$20-$3.99(milk)=$17.01÷$4.39(cereal)=4 boxes of cereal

5 0
3 years ago
A box without a top is made from a rectangular piece of cardboard, with dimensions 27 centimeters by 18 centimeters, by cutting
GaryK [48]
By cutting each corner we have that the resulting dimensions are:
 27 - 2x
 18 - 2x
 Height = x
 Therefore, the volume of the box in terms of the variable x, is given by:
 V (x) = (x) * (27-2x) * (18-2x)
 Answer:
 
The volume of the box in terms of x is:
 
(27-x) (18-x) x
8 0
4 years ago
Read 2 more answers
Y=-1/3x^2-4x-5 in vertex form
grigory [225]

Answer:

Y=-\frac{1}{3}(x+6)^2+7   [Vertex form]

Step-by-step explanation:

Given function:

Y=-\frac{1}{3}x^2-4x-5

We need to find the vertex form which is.,

y=a(x-h)^2+k

where (h,k) represents the co-ordinates of vertex.

We apply completing square method to do so.

We have  

Y=-\frac{1}{3}x^2-4x-5

First of all we make sure that the leading co-efficient is =1.

In order to make the leading co-efficient is =1, we multiply each term with -3.

-3\times Y=-3\times\frac{1}{3}x^2-(-3)\times4x-(-3)\times 5

-3Y=x^2+12x+15

Isolating x^2 and x terms on one side.

Subtracting both sides by 15.

-3Y-15=x^2+12x-15-15

-3Y-15=x^2+12x

In order to make the right side a perfect square trinomial, we will take half of the co-efficient of x term, square it and add it both sides side.  

square of half of the co-efficient of x term = (\frac{1}{2}\times 12)^2=(6)^2=36

Adding 36 to both sides.

-3Y-15+36=x^2+12x+36

-3Y+21=x^2+12x+36

Since x^2+12x+36 is a perfect square of (x+6), so, we can write as:

-3Y+21=(x+6)^2

Subtracting 21 to both sides:

-3Y+21-21=(x+6)^2-21

-3Y=(x+6)^2-21

Dividing both sides by -3.

\frac{-3Y}{-3}=\frac{(x+6)^2}{-3}-\frac{21}{-3}

Y=-\frac{1}{3}(x+6)^2+7   [Vertex form]

8 0
3 years ago
Please help me :)), do not 'troll' with answers, it's graded...
wolverine [178]

The first one, decreasing then increasing

4 0
3 years ago
Someone should help me out please​
Maurinko [17]

Answer:

18 in

Step-by-step explanation:

First you would multiply 2·5=10 then for the area of the bottom shape you already have 4 for one side to get the other side you subtract 7-5 and get 2 so 4·2=8 so then add the areas of both shapes and get 18

4 0
3 years ago
Other questions:
  • Stephan buys a melon and divides it into 6 servings. he eats 1/3 of the melon over the weekend. how many piecesof melon does ste
    7·2 answers
  • For homework, Fred has to read 30 minutes daily. It takes him 2 minutes per page, and he spends 4 minutes viewing graphics in th
    10·1 answer
  • If the basketball hoop is 10 ft high and located 15 ft away, what initial velocity v should the basketball have?
    7·1 answer
  • Wires manufactured for use in a computer system are specified to have resistances between 0.14 and 0.16 ohms. The actual measure
    5·1 answer
  • (3x^5 + x^3 - 4) divided by (x - 2)​
    10·1 answer
  • Question 7(Multiple Choice Worth 1 points) (08.02 LC) Eva conducted a survey to find out the amount of time high schoolers spend
    13·1 answer
  • A sample is generated from a population of 20 items. Each of the 20 items are given a label, and a 20-sided die is rolled 3 time
    11·1 answer
  • In which quadrant is the solution to this system of equations? 2x + 5y = 1<br> 3x - 4y = 13
    14·1 answer
  • Anyone know the answer?
    7·1 answer
  • A point is shown on the coordinate plane below. Where is the point located? *
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!