If an open box has a square base and a volume of 115 in.3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. (round your answers to two decimal places.)

Answers

Answer 1

Let h = height of the box,
x = side length of the base.

Volume of the box is [tex]V=x^{2} h = 115 [/tex].
So [tex]h = \frac{115}{ x^{2} } [/tex]

Surface area of a box is S = 2(Width • Length + Length • Height + Height • Width).
So surface area of the box is
[tex]S = 2( x^{2} + hx + hx) \\ = 2 x^{2} + 4hx \\ = 2 x^{2} + 4( \frac{115}{ x^{2} } )x[/tex]
[tex]= 2 x^{2} + \frac{460}{x} [/tex]
The surface are is supposed to be the minimum. So we'll need to find the first derivative of the surface area function and set it to zero.

[tex]S' = 4x- \frac{460}{ x^{2} } = 0[/tex]
[tex] 4x = \frac{460}{ x^{2} } \\ 4x^{3} = 460 \\ x^{3} = 115 \\ x = \sqrt[3]{115} = 4.86 [/tex]
Then [tex]h= \frac{115}{4.86^{2}} = 4.87 [/tex]
So the box is 4.86 in. wide and 4.87 in. high.

Answer 2

length = 6.13 Width = 6.13 Height = 3.06 Let's use the variable L to represent the length of a bottom edge of the box. With that, the height of the box will be 115 / w^2. Since the box is open, the box will have a bottom, and 4 sides. No top. So the expression for the surface area of the box will be w^2 + 4w(115/w^2) = w^2 + 460/w Since we're looking for a local minimum, that will be at a point where the first derivative of the function is 0. So the first derivative is: 2w - 460/w^2 = 2w^3/w^2 - 460/w^2 = (2w^3 - 460)/w^2 = 2(w^3 - 230)/w^2 The above derivative has a real zero at the cube root of 230 which is approximately 6.126926 So the length of the sides of the box are 6.13 inches and the height of the box is 115/(6.13^2) = 3.06 inches.

Related Questions

An Uber driver carries riders to and from a local airport and a neighboring suburb. Due to traffic and the time of day the driver works, the trip to the airport takes 60 minutes longer than the trip to the suburb. The car’s average speed in traffic is 30 miles per hour. Without traffic, the car travels 15 miles per hour. The equation can be used to determine how many miles apart the two locations are shown. What is the value of m, the distances between the two locations?

A) 0.5 miles
B) 22.5 miles
C) 30 miles
D) 90 miles

Answers

Answer: Choice B) 22.5 miles

--------------------------------------------
--------------------------------------------

To find that answer, we simply solve for m

m/(30-15) = m/(30+15) + 1
m/15 = m/45 + 1
45*m/15 = 45*(m/45 + 1) ... multiply both sides by the LCD to clear out fractions
3m = 45*(m/45) + 45*(1)
3m = m+45
3m-m = m+45-m
2m = 45
2m/2 = 45/2
m = 45/2 <--- answer as a fraction
m = 22.5 <--- answer in decimal form

Use a calculator or long division to convert from fraction to decimal form.

What is the value of k?

2k+9=7−3k

Answers

To find the value of k in the equation 2k+9=7−3k, we rearrange and simplify to isolate k, resulting in k = -2/5.

To solve for the value of k in the equation 2k+9=7−3k, we must isolate the variable on one side of the equation. Let's start by adding 3k to both sides to get all the k terms on one side:

2k + 3k + 9 = 7 − 3k + 3k
5k + 9 = 7

Next, subtract 9 from both sides to get:

5k = 7 − 9
5k = −2

Now, to find the value of k, divide both sides by 5:

k = −2/5

Therefore, the value of k is −2/5.

Write an algebraic expression that is equivalent to the expression cot(arctan x)

Answers

cosX/sinX [tan-¹X]
cos²x/sin²x = cot²x
recall cot²x = cosec²x - 1

The algebraic expression equivalent to cot(arctan x) is 1 / x, derived by recognizing that cot(θ) is the reciprocal of tan(θ), and tan(θ) = x for θ = tan-1(x)

To find an algebraic expression for cot(arctan x), let's follow a step-by-step transformation using trigonometric identities: cot(arctan x) is denoted as cot(tan-1(x))

First, let
θ = tan-1(x). By definition of the arctan function, we have the relation:tan(θ) = x.Next, we need to find cot(θ). Recall that:cot(θ) = 1 / tan(θ).Since tan(θ) = x, we substitute to get:cot(θ) = 1 / x .Thus, we have cot(tan-1(x)) = 1 / x.

Therefore, the algebraic expression equivalent to cot(tan-1(x)) is 1 / x.

Gerald bought a home for $300,000. His largest up–front cost is 10 percent, or $30,000. What is this cost called?

Answers

It is called 'down payment'.

Hope this helps! (Please rate) :)

Answer:

The answer is down payment.

Step-by-step explanation:

Gerald bought a home for $300,000. His largest up–front cost is 10 percent, or $30,000.

This cost is called down payment.

Down payment is an amount in cash that is given while purchasing a home. This lies between 5% to 20% of the purchase price.

This payment reduces the amount of money you will have to borrow from banks. Each loan requires some sort of down payment.

Slope is 3, and (1,4) is on the line

Answers

The first step is since you know what the slope is, you can just plug that into the equation. We now have y=3x+b, b being the y intercept.
3 is equal to 3/1, and you can use that to find the Y intercept. Start at the point that they game you and go up 3 and over 1. But this makes you further away from the Y axis, so you want to do the opposite. Go down 3 and over 1.\
This makes you end up on the Y intercept at (0,1)
Making the equation y=3x+1

Hello!

Formular: y = mx + c
Where, m = slope

Given that m = 3,
Substitude 3 into m

y = 3x + c
Then add the coordinates (1, 4)
Where 1 being x and 4 being y

4 = 3(1) + c
Solve for C next

c = 1
Substitude m and c into the formular

y = 3x + 1
And it's done!

Use this same method for the rest of the questions :)

If two siblings are boys, what is likelihood that the third sibling will also be a boy

Answers

the is a 50% chance of being a boy or a girl.....previous siblings do not matter. Every child is an independent event. So even if u had 2 siblings that were boys, the result of the next one being a boy is still gonna be 50%...just as the odds of it being a girl, is 50% as well.

hi hold on,
there are two possibilities
first possibility is boy and the second Is girl.
:)

ab is tangent to the circle ad=4 and bc=3 solve for x

Answers

Answer:

Step-by-step explanation:

alright, lets get started.

bc and cd will be same as both are radius of the circle.

[tex]bc=cd=3[/tex]

[tex]ac=ad+cd[/tex]

[tex]ac=4+3[/tex]

[tex]ac=7[/tex]

as we know ab is tangent to the circle so, triangle abc will be a right triangle.

Using Pythagorean theorem

[tex]a^2+b^2=c^2[/tex]

[tex]3^2+x^2=7^2[/tex]

[tex]9+x^2=49[/tex]

Subtracting 9 from both sides

[tex]x^2=49-9=40[/tex]

[tex]x=\sqrt{40} =2\sqrt{10}[/tex]

So, the answer is [tex]2\sqrt{10}[/tex] : answer

Hope it will help. :)

Brandon says to get that shade of green, 6 pints of yellow paint need to be added to the bucket. Bianca says that won't work. In two or more complete sentences, explain who is correct and why. In your explanation, include the calculations to support the correct answer.

Answers

There are 2 pints of blue and 2 pints of yellow in the bucket.
To be 80% yellow, the new color will be (100%-80%=) 20% blue.

2 pints is 20% of (2 pints/0.20=) 10 pints.
With 4 pints in the bucket, to make 10 pints you should add
(10 pints-4 pints=) 6 more pints of yellow paint.

ANSWER: You should add 6 pints of yellow to the 4 pints of green in the bucket to make the new color that is 80% yellow.

A neighborhood swim club charges a one-time fee of $25 to join and $85 each month. If n is the number of months of membership and t is the total amount spent, then the variables are related by the equation t = 85n + 25. Which variable is the independent variable?

Answers

The independent variable is n.

I think that the answer is 110 becaus
85+25= 110

The volume of a rectangular prism is 18 cubic ft. If the width is 6ft. and the height is 3ft. what is the length

Answers

18 = 6 * 3 * length

length = 1 foot

If my son was born june 2016 what year will he graduate high school

Answers

Since the child was born in 2016, he will be 6 years old in 2022. Twelve years after he will be 18 in 2034. This should be his school graduation year.

NEED ANSWER ASAP!!! THANKS!!
A fish swims at a speed of 12 miles per hour. A boy swims at a speed of 4.4 feet per second.

3 ft = 1 yd

5280 ft = 1 mi

How much faster does the fish swim than the boy in yards per minute?

Answers

The Boy: 4.4 x 60 = 264ft per minute. 264 / 3 = 88yrds Per minute

The Fish: 12 x 5280 = 63360ft hr. 5280 / 60 = 1056ft minute 1056 / 3 = 352yrds

352 - 88 = 44 yards quicker then the boy.

44 x 3 = 132 feet quicker then the boy.

First, you convert 12 miles per hour into yards per hour:
21120 yards per hour.

Then you convert 4.4 feet per second into yards per second:
1.47 yards per second.

Then convert 1.47 yards per second into yards per hour:
5292 yards per hour.

Subtract 5292 from 21120:
15828 yards per hour.

Then convert that into minutes:
263.8 yards per minute.

The fish is 263.8 yards per minute faster than the boy.

What will this problem look like? The radius, r, of a circle is one-half of the length of its diameter, d.

1. 1/2r= d

2. 2r=d

3. 1/2 + r = d

4. r= 1/2d

5. r= d/2

There is more than one answer

Answers

2. 2r=d
4. r= 1/2d
5. r= d/2

r = 1/2d

r = d/2.......d/2 is the same as 1/2d

2r = d

Need help on 9 and 10

Answers

9.) The relation can not be a function only if the domain is the same. If you plot the points on a graph and use the Vertical Line Test(VLT) you would see that none of the points would be on the same line. It doesn't matter if the range is the same

10.)On the graph your y-intercept would be 2 so that is where you plot your first point. In order to find the slope you need to go up three spaces and one space to the left this is also known as "rise over run". *Remember you always go up spaces for "rise over run" but if the slope is positive then you go to the right and if its negative you go to the left.

I hope this helps love! :)

Seventy-five times an integer, minus 36, equals 21 times the square of the integer. Which equation could be used to solve for the unknown integer?

Answers

The equation: [tex]$ 21x^2 - 75x + 36 = 0 $[/tex] represents "75 times an integer minus 36 equals 21 times the square of the integer."

To solve this problem, let's represent the unknown integer with the variable [tex]$ x $[/tex]. Then we can set up the equation based on the given information.

The problem states: "Seventy-five times an integer, minus 36, equals 21 times the square of the integer."

Mathematically, this can be represented as:

[tex]\[ 75x - 36 = 21x^2 \][/tex]

So, the equation that could be used to solve for the unknown integer is:

[tex]\[ 21x^2 - 75x + 36 = 0 \][/tex]

A tile factory earns money by a flat fee for delivery and sales price per tile sold. For one customer the tile factory shipped 10,000 units of retail stores at the rate of $0.25 Per tile. The total tile factory earned from that sale was 3000 which of the following equation describes the revenue of the tile from this sale in terms of tiles
Sold?
A) y=0.25x+10,000
B) y=0.25x+3,000
C) y=0.25x+500
D) y=0.25x+0.25

Answers

10,000 tiles * 0.25 = 2500

3000 -2500 = 500

so the delivery fee is 500 and the tiles cost 0.25 each

so the equation would be y = 0.25x +500

Answer is C

The answer is c hope this helps

Please helppp

(750/512)^1/3
what is equivilant to this??

Answers

[tex]\bf a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \qquad \sqrt[{ m}]{a^{ n}}\implies a^{\frac{{ n}}{{ m}}}\\\\ -------------------------------[/tex]

[tex]\bf \left( \cfrac{750}{512} \right)^{\frac{1}{3}}\implies \cfrac{750^{\frac{1}{3}}}{512^{\frac{1}{3}}}\implies \cfrac{\sqrt[3]{750^1}}{\sqrt[3]{512^1}}\quad \begin{cases} 750=2\cdot 3\cdot 5\cdot 5\cdot 5\\ \qquad 2\cdot 3\cdot 5^3\\ \qquad 6\cdot 5^3\\ 512=8\cdot 8\cdot 8\\ \qquad 8^3 \end{cases} \\\\\\ \cfrac{\sqrt[3]{750}}{\sqrt[3]{512}}\implies \cfrac{\sqrt[3]{6\cdot 5^3}}{\sqrt[3]{8^3}}\implies \cfrac{5\sqrt[3]{6}}{8}[/tex]

Answer:

[tex]\frac{5\sqrt[3]{6}}{8}[/tex]

Step-by-step explanation:

We have been given an exponential number. We are supposed to simplify our given number.

[tex](\frac{750}{512})^{\frac{1}{3}[/tex]

Using fractional exponent rule [tex]x^\frac{m}{n}=\sqrt[n]{x^m}[/tex], w ecan write our given number as:

[tex](\frac{750}{512})^{\frac{1}{3}}=\sqrt[3]{(\frac{750}{512})^1}[/tex]

[tex](\frac{750}{512})^{\frac{1}{3}}=\sqrt[3]{\frac{750}{512}}[/tex]

We can rewrite 512 as [tex]8^3[/tex] and 750 as 125*6.

[tex](\frac{750}{512})^{\frac{1}{3}}=\sqrt[3]{\frac{125*6}{8^3}}[/tex]

We can rewrite 125 as [tex]5^3[/tex]

[tex](\frac{750}{512})^{\frac{1}{3}}=\sqrt[3]{\frac{5^3*6}{8^3}}[/tex]

Using radical rule [tex]\sqrt[n]{a^n}=a[/tex], we will get:

[tex](\frac{750}{512})^{\frac{1}{3}}=\frac{5\sqrt[3]{6}}{8}[/tex]

Therefore, [tex]\frac{5\sqrt[3]{6}}{8}[/tex] is equivalent to our given number.

please show me how to solve ... -12x - 17 = -89

Answers

Hey there again!

First , let's add 17 on each of your sides (we have to get the "X" by itself , right?

Like: -12x - 17 + 17
= -89 + 17

-89 + 17 = -72

So, divide by -12 on each of your sides since both numbers goes into 12

Like: 12x / -12 = -1

= -72 / -12 = 6

X = 6

Good luck on your assignment and enjoy your day!

~MeIsKaitlyn:)

Riki rides her bike 26 miles every week. What is the total numbers of miles she will ride in one dozen. What is the total number of miles she will ride in 7 weeks

Answers

in one dozen week she will ride for 2184 mi
for 7 weeks is 1274 weeks

While driving in the United States, Franklin sees that the height of a tunnel is marked as 10’6”. He knows his truck is 3.3m tall. Can he drive through the tunnel?

Answers

to my knowledge (im not the smartest person lol) I would say no he cannot

no he cannot :D
hope this helpXD

What is the 6th value in the sequence with the explicit formula an=−2n−14?

A −26

B −12

C −10

D 6

Answers

plug in n= 6

a6 = -2(6) - 14 = -26

Its A

The answer is A: -26

Please help asap!!
A recipe calls for 2 1/4 cups of flour to make a batch of cookies. How many batches of cookies can be made if you have 15 3/4 cups of flour?

Answers

You can make 7 batches. If you need any help showing work I'd be happy to let you know in a comment :0 hope this helps!

you could make 7 batches

A standard coffee mug has a capacity of 16 fluid ounces if Annie needs to fill 26 mugs with coffee how many total quarts of coffee does she need

Answers

She will need 13 quarts, because 16 x 26 is 416 fluid ounces equals to 13 quarts.
Hope this helped
Mark me brainliest!

Find the surface area of a cube with 4.1 in. edges.

Answers

4.1*4.1*6=100.86 inches^2

HELP WITH CALCULUS!

Create an original rational function that has at least one asymptote (vertical, horizontal, and/or slant) and possibly a removable discontinuity. List these features of your function: asymptote(s) (vertical, horizontal, slant), removable discontinuity(ies), x-intercept(s), y-intercept, and end behavior. Provide any other details that would enable another student to graph and determine the equation for your function. Do not state your function.

Answers

This function has a hole at x = -3 (point: (-3, -1/6) which is removable)
This function has a vertical asymptote x = 3
This function has a horizontal asymptote y = 0 (degree of numerator < degree of denominator)
It has no x - intercept
Its' y - intercept is at (0, -1/3)
As x →-∞, f(x) →0 and as x →∞, f(x) →0

The function is [tex]f(x)= \frac{x+3}{(x+3)(x-3)} [/tex]

How to find equation of axis of symmetry when given 2 points?

Answers

axis of symmetry of 3 points

How can you find the x-intercept of the graph of a linear equation using the equation? How is using the graph of a linear equation to find the intercepts like using the equation?

Answers

The first one is change of y over the change of x.

The second one is when the y-intercept is 0 and the x-intercept is a #

Malachi took cans to the recycling plant and put them into the can-crusher one at a time. after he had already crushed some of the cans, he started recording the number of cans he crushed per minute. this table shows how many cans malachi put in the can-crusher at different times. minutes 3 5 7 9 cans crushed 49 65 81 97 what is the equation that represents malachi's situation? enter your answer in the boxes

Answers

2/3
it is 2/3 or not I don't know

Answer:

your answer would be y-65=8(x-5)

When you sign up for a movie service, you get three free months of movies. Then you are charged $15 per month to rent movies. If you pay $90, how many months of use do you get?

Answers

nine months because you divide 90 by 15 and get six and then you add three for the free months for a total of nine

You get three free months of the movie service. Then you get 6 additional months of the service because 90/15=6. You receive a total of 9 months for paying $90.

Find the average rate of change for f(x) = x2 + 9x + 18 from x = −8 to x = −2.
A)−1
B)−3
C)1
D)3

Answers

In order to find the average rate of change, we must take the values at x=-8 and x= -2, and determine what the slope would be if a line was drawn between the two points.
f(-8) = -8² +9(-8) + 18
f(-8) = 10

f(-2) = -2² + 9(-2) + 18
f(-2) =4 -18 +18
f(-2) = 4

f(-8) =10 and f(-2)=4

if a line was drawn between the two points (-8,10) and (-2,4) the slope (rise/run) would be -6/6, since the line would go down 6 units from 10 to 4, and over 6 units from x=-8 to x=-2.

The average rate of change, therefore, is -6/6, or
A)-1

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