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
kvv77 [185]
3 years ago
14

You work for a cereal company as a box designer. Your job is to find the dimensions of a box. The volume of the box is given by

the following:
v(x)=8x^3-108x^2+360


a. Use a graphing calculator to determine the length, width, and height of the box so it will hold the largest possible volume.


b. Factor the polynomial completely to find the x-intercepts of the graph. In context of this problem, what do the intercepts represent?


c. What is the domain and range for the box?
Mathematics
1 answer:
ivolga24 [154]3 years ago
7 0

Question:

You work for a cereal company as a box designer. Your job is to find the dimensions of a box. The volume of the box is given by the following:

v(x)=8·x³-108·x²+360·x

Answer:

a. To hold the largest possible volume, the length, height, and width are;

Length, l of the box = 10.583

Width, w of the box = 3.791

Height, h of the box = 8.835

b. The x intercepts are;

x = 0

x = 15/2

x = 6

The intercept represents the points of the function at which the volume of the box change from negative to negative and back to positive

c. The domain range is all real numbers, that is from -∞ to +∞

Step-by-step explanation:

Since the volume V(x) = 8·x³ - 108·x² + 360·x, we have fo maximum volume;

\frac{\mathrm{d} \left (8\cdot x^{3} - 108\cdot x^{2} + 360\cdot x  \right )}{\mathrm{d} x}= 0

Therefore. 24·x² - 216·x + 360 = 0

Solving the quadratic equation, we have

(2·x - (9 - √21))(2·x - (9 + √21))

Therefore, x = (9 - √21)/2 or x = (9 + √21)/2

P

Therefore, from the v(x) equation we have by factorizing;

v(x) = 8·x³ - 108·x² + 360·x = 4·x·(2·x - 15)·(x - 6)

Plugging the possible values of x for maximum v(x), we obtain

For x = (9 - √21)/2, v(x) = 354.468

For x = (9 + √21)/2, v(x) = -30.468

Hence the for maximum v(x), x = (9 - √21)/2

The sides are thus;

4·x·(2·x - 15)·(x - 6)

Where:

4·x = 8.835

\left |2\cdot x - 15  \right | = 10.583

\left | x - 6\right | = 3.791

The length, l of the box = 10.583

The width, w of the box = 3.791

The height, h of the box = 8.835

b. The factorization of the polynomial gives;

v(x) =  4·x·(2·x - 15)·(x - 6)

Therefore, the x intercepts are;

x = 0

x = 15/2

x = 6

The intercept represents the points of the function at which the volume of the box will be positive, hence the possible values of the dimensions

c. The domain of the function v(x) = 8·x³ - 108·x² + 360·x (which is a polynomial without variables or radicals as the denominator) is the set of all real numbers from -∞ to ∞.

You might be interested in
PLEASEEEEEEEEEEEEEE HELPPPPPPPPPP MEEEEEEE D: PLEASEEEEEEEEEEEEEEEE
gizmo_the_mogwai [7]

Answer:

what is it

Step-by-step explanation:

3 0
2 years ago
Read 2 more answers
Q.3 Write the factors of 28 in circle A and the factors of 32 in circle B. Write
marissa [1.9K]

Answer: The largest common factor of 28 and 32 is 4.

Step-by-step explanation:

The factors of 28 are 1, 2, 4, 7, 14, 28

The factors of 32 are 1, 2, 4, 8, 16, 32

5 0
2 years ago
if 1/4 of a gallon of paint is needed to paint 2/3 of a fence, how many gallons are needed to paint the entire fence? site:socra
trasher [3.6K]

\bf \begin{array}{ccll} \stackrel{paint}{gallons}&fence\\ \cline{1-2} \\\frac{1}{4}&\frac{2}{3}\\\\ x&1 \end{array}\implies \cfrac{~~\frac{1}{4}~~}{x}=\cfrac{~~\frac{2}{3}~~}{1}\implies \cfrac{~~\frac{1}{4}~~}{\frac{x}{1}}=\cfrac{2}{3}\implies \cfrac{1}{4}\cdot \cfrac{1}{x}=\cfrac{2}{3} \\\\\\ \cfrac{1}{4x}=\cfrac{2}{3}\implies 3=8x\implies \cfrac{3}{8}=x

7 0
3 years ago
Read 2 more answers
Integrate 1 - x / x(x2 + 1) d x by partial fractions.
solniwko [45]

Answer:

log x-\frac{log(x^{2}+1) }{2}-tan^{-1} x

Step-by-step explanation:

step 1:-   by using partial fractions

[tex]\frac{1-x}{x(x^{2}+1) } =\frac{A(x^{2}+1)+(Bx+C)(x }{x(x^{2}+1) }......(1)

<u>step 2:-</u>

solving on both sides

1-x=A(x^{2} +1)+(Bx+C)x......(2)

substitute x =0 value in equation (2)

1=A(1)+0

<u>A=1</u>

comparing x^2 co-efficient on both sides (in equation 2)

0 = A+B

0 = 1+B

B=-1

comparing x co-efficient on both sides (in equation 2)

<u>-</u>1  =  C

<u>step 3:-</u>

substitute A,B,C values in equation (1)

now  

\\\int\limits^ {} \, \frac{1-x}{x(x^{2}+1) } d x =\int\limits^ {} \frac{1}{x} d x +\int\limits^ {} \frac{-x}{x^{2}+1 }  d x -\int\limits \frac{1}{x^{2}+1 }  d x

by using integration formulas

i)  by using \int\limits \frac{1}{x}   d x =log x+c........(a)\\\int\limits \frac{f^{1}(x) }{f(x)} d x= log(f(x)+c\\.....(b)

\int\limits tan^{-1}x  dx =\frac{1}{1+x^{2} } +C.....(c)

<u>step 4:-</u>

by using above integration formulas (a,b,and c)

we get answer is

log x-\frac{log(x^{2}+1) }{2}-tan^{-1} x

6 0
3 years ago
Pls help me geometry is so confusing
Zolol [24]
Dab on them haters brobro
6 0
2 years ago
Other questions:
  • What is 20/40 as a decimal
    5·1 answer
  • Using the Breadth-First Search Algorithm, determine the minimum number of edges that it would require to reach
    8·1 answer
  • Pls i need help with 14 tysm
    13·1 answer
  • Solve for h <br> A= 2πrh + πr2
    13·1 answer
  • round 63,849 to the nearest ten, to the nearest hundred, to the nearest thousand,and the nearest ten thousand
    8·2 answers
  • 2 poin
    6·2 answers
  • Thank u guys for helping me out but please I need help
    7·2 answers
  • Write sin81 in terms of cosine
    8·1 answer
  • This is the other question!
    12·2 answers
  • As used in line 19, "capture" is closest in meaning to
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!