The elves at Santa’s Workshop are trying to wrap the biggest box they can with the last bit of wrapping paper they have. All of the boxes they have are rectangular prisms with a base whose length is three times the width. Assume that they can make the box have any height they wish. There is 240 square inches of wrapping paper left.

What dimensions will maximize the volume of the box given the amount of wrapping paper the elves have? Make sure to show all your work and write a sentence at the end to interpret your results.

Answer:

The length of the box is 2×√30 inches long, the width of the box is 2×√(10/3) inches long, while the height of the box is √30 inches long

Step-by-step explanation:

The given parameters are;

The length of the last rectangular prism box = 3 × The width of the box

The area of the available wrapping paper = 240 in.²

Let L represent the length of the rectangular prism box, let W represent the width of the rectangular prism box and let h, represent the height of the rectangular box

Therefore, we have;

L = 3 × W

The surface area of the box, A = 2 × L × W + 2 × L × h + 2 × W × h

Whereby, L = 3 × W, we have;

A = 2 × 3 × W × W + 2 × 3 × W × h + 2 × W × h = 8·h·W + 6·W²

A = 8·h·W + 6·W² = 240

8·h·W + 6·W² = 240

8·h·W = 240 - 6·W²

h = (240 - 6·W²)/(8·W)

The volume,  V = L × W × h

∴ V = 3 × W × W × h = 3 × W² × h

V = 3 × W² × h

∴ V = 3 × W² × (240 - 6·W²)/(8·W) = 3/8 × W × (240 - 6·W²) = 90·W - 9/4·W³

dV/dW = d(90·W - 9/4·W³)/dW = 90 - 27/4·W²

At the maximum point, dV/dW = 0, which gives;

dV/dW = 90 - 27/4·W² = 0 at maximum point

27/4·W² = 90

W² = 4/27 × 90 = 40/3

W = ±√(40/3) = ±2×√(10/3)

Differentiating again and substituting gives;

d²V/dW² = d(90 - 27/4·W²)/dW =  - 27/4·W

Given that when W = 2×√(10/3), d²V/dW² is negative, and when W = -2×√(10/3), d²V/dW² is positive, W = 2×√(10/3) is the local maximum point

Therefor, the dimensions of the box are;

W = 2×√(10/3)

L = 3 × W = 3 × 2×√(10/3)

L = 3 × 2×√(10/3) = 2×√30

L = 2×√30

L = 6 ×√(10/3)

h = (240 - 6·(2×√(10/3))²)/(8·(2×√(10/3))) = √30

h = √30

Therefore, the length of the box is 2×√30 inches long, the width of the box is 2×√(10/3) inches long, while the height of the box is √30 inches long.

2/5x + 1/2=3/5
A.4/15
B.125
C.1/4

The diagram shows a dilation. Which dilation does it represent?
What is the scale factor?

What do you learn in Algebra 2 that you did not learn in Algebra 1. Is Algebra 2 just a review?

Samson owes his friend
$42.75. If Samson just
got paid $167.25 and
pays back his friend.
How much money will
he have left over?

What is the square root of 1,567?

Find the Error A student graphs the relationship y = x on a coordinate plane. She says the slope is O because there is no coefficient. Find her
mistake and correct it.

PLZ HELP ME!!!!!
If a scale uses 2 centimeters to represent 5 feet, how many centimeters will be used to represent 17.5
feet?
A 7
B5
C 3.5
D7.5

What is the area of the figure?
3 yd
6 yd
9 yd
Syd

what is the slope of the line? SOMEONE HELP ME PLSSS

how to find square root of 8653

please solve-3(×-2)=2(×-7) and write a reason for each step ​

Plz help me with 6th grade math- I’m dumb

PLEASEE HELP.!! ILL GIVE BRAINLIEST.!! *EXTRA POINTS* DONT SKIP:((

I need help if you can help me plz I would really appreciate it ?

the domain _ is the set of inputs for a function

???????? Helppppp :(

Alana biked 15 miles in 1 hour. How many miles did she bike in 3 hours?​

Why is 25 a composite number? (1 point)
It is odd.
OIt has more than two factors.
O All of its factors are prime numbers.
It is not divisible by any number except one and itself.

solve for x: 2x- 2 =3(x- 2)​

Give 3 ratios that are equivalent to 10:8

apesapesapesapesapesapesapesapesapesapesapesapesapesapes

Which expression would be easier to simplify if you used the associative property to change the grouping?
A. ( 50 + 50 ) + 63
B. ( 43 + 60 ) + 40
C. ( 2/5 + 3/5 ) + 7/3
D. [ - 14 + ( - 26 ) ] + 67

Which table represents y as a function of x?
PLEASE HELP??
OR BRAINLIEST:((??

Which of the following ordered pair(s) is/are solutions to the following system of
equations?
3x - 5y = 15
5x – 10y = 25
(11,3)
(3. - 1)
(-7,- 6)
All of the above

What is the opposite of –0.8?

18inch/4secs how far would the car travel in 1 second?

HELP !
CLICK ON THE PHOTO FOR QUESTION. ALOT OF POINTS GIVEN AWAY!

Determine if the figures below are congruent? Explain why or why not. I need it for a test please answer fast!

Pre-calc. I need help NOW

A grass fertilizer is sold in 5-pound bags. Stuart calculates that he needs enough fertilizer to cover 4,350 square feet. If each pound of fertilizer covers 362.5 square feet, how many 5-pound bags does Stuart need to buy?