A parcel delivery service has contracted you to design a closed box with a square base that has a volume of 8500 cubic inches.
A) Express the surface area of the box as a function of X
B) Graph the function found in part a .
C) What is the minimum amount of cardboard that can be used to construct the box.
D) What are the dimensions of the box that minimize the surface area.
E) why might UPS be interested in designing a box that minimize the surface area.
Answers
A) Let the side length of the square base be x, and the height of the box be h. Then, the volume of the box is given by:
V = \(x^2\) * h = 8500
Solving for h, we get:
h = 8500 / \(x^2\)
The surface area of the box can be expressed as:
S = 2x^2 + 4xh
Substituting the value of h obtained above, we get:
S = \(2x^2\) + 4x(8500 / \(x^2\)) = 2\(x^2\)+ 34000 / x
Thus, the surface area of the box can be expressed as a function of x.
B) To graph the function, plot the surface area (S) on the y-axis and the side length of the square base (x) on the x-axis. Since x cannot be negative, the domain of the function is (0, infinity). As x gets very large or very small, the surface area approaches infinity, so we should only graph the function for values of x that make sense in the context of the problem.
C) The minimum amount of cardboard required to construct the box is equal to the surface area of the box. To find the minimum surface area, we need to find the minimum of the function S(x) obtained in part A.
D) To find the dimensions of the box that minimize the surface area, we need to find the value of x that minimizes the function S(x). We can do this by taking the derivative of S(x) with respect to x, setting it equal to zero, and solving for x. This gives us:
dS/dx = 4x - 34000/\(x^2\) = 0
Multiplying both sides by \(x^2\) and solving for x, we get:
x = (8500/2\()^(1/3)\) ≈ 18.3
Therefore, the dimensions of the box that minimize the surface area are a square base with side length of approximately 18.3 inches, and a height of:
h = 8500 / \(x^2\) ≈ 26.2 inches
E) UPS might be interested in designing a box that minimizes the surface area because it can reduce the amount of cardboard used in each box, resulting in cost savings and a reduced environmental impact. Additionally, minimizing the surface area of a box can make it more efficient to stack and transport, reducing shipping costs and making the overall process more sustainable.
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A) The Surface Area = \(X^2 + 34000/X\)
B) The y-axis and the side length X on the x-axis.
C) The minimum amount of cardboard required to construct the box is approximately 1649.39 square inches.
D) The dimensions of the box that minimize the surface area are X = 20.5 inches and Y = 16.87 inches.
E) Smaller boxes take up less space in delivery trucks and planes, allowing more packages to be shipped at once and reducing transportation costs.
A) Express the surface area of the box as a function of X:
Let the side length of the square base be X and the height of the box be Y. Then, we know that the volume of the box is 8500 cubic inches, so:
\(X^{2Y} = 8500\)
To find the surface area of the box, we need to add up the area of each face. There are 5 faces in total (the bottom square and 4 identical rectangular sides), so:
Surface Area = \(X^2 + 4XY\)
We can substitute the value of Y from the equation for volume, giving:
Surface Area = \(X^2 + 4X(8500/X^2)\)
Surface Area = \(X^2 + 34000/X\)
B) Graph the function found in part a:
To graph this function, we can plot the surface area on the y-axis and the side length X on the x-axis. The graph will have a minimum value, which we can find using calculus or by using a graphing calculator.
C) What is the minimum amount of cardboard that can be used to construct the box:
The minimum amount of cardboard required to construct the box is the surface area of the box. To find this minimum value, we need to find the minimum point on the graph in part b. From the graph or by using calculus, we can see that the minimum occurs at X = sqrt(8500/5) = 20.5 inches. Substituting this value into the equation for surface area, we get:
Surface Area = \(20.5^2 + 4(20.5)(8500/20.5^2)\)= 1649.39 square inches
D) What are the dimensions of the box that minimize the surface area:
From part c, we know that the minimum occurs at X = sqrt(8500/5) = 20.5 inches. To find the height of the box, we can substitute this value of X into the equation for volume:
\(X^2Y = 8500\)
\((20.5)^2 Y = 8500\)
\(Y = 8500/(20.5)^2 = 16.87\) inches
E) Why might UPS be interested in designing a box that minimizes the surface area:
UPS and other parcel delivery services are always looking for ways to reduce their costs and increase efficiency. By designing a box that minimizes the surface area while still containing the same volume of goods, they can reduce the amount of cardboard used for each shipment. This would save them money on materials and shipping costs, while also reducing their environmental impact. Additionally, smaller boxes take up less space in delivery trucks and planes, allowing more packages to be shipped at once and reducing transportation costs.
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Related Questions
Anyone mind helpingg?
Answers
Answer:
B)
Step-by-step explanation:
The answer is B) you have to devide the terms
point p to Point q is 15 km away and on a bearing of 090°to a point R is 8 km from q. find the distance between p and r
Answers
Check the picture below.
\(\textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2 + b^2} \qquad \begin{cases} c=\stackrel{hypotenuse}{RP}\\ a=\stackrel{adjacent}{15}\\ b=\stackrel{opposite}{8}\\ \end{cases} \\\\\\ RP=\sqrt{15^2+8^2}\implies RP=\sqrt{225+64}\implies RP=17\)
Graph the line with the equation y=1/5x-2
Answers
The graph of the equation "y = 1/5x - 2" is shown: (Refer to the graph attached below.)
What is a graph?In mathematics, a graph is a visual representation or diagram that shows data or values in an ordered way. The relationships between two or more things are frequently represented by the points on a graph. Bar graphs, circle graphs, and line graphs are the three most frequently used types of graphs. Different types of data can be displayed using different types of graphs.Graphs and charts are useful visual aids because they make information accessible and quick to understand. Therefore, it is not surprising that print and electronic media frequently use graphs. When data is presented as a graph rather than a table, it can sometimes be easier to understand because the graph can show a trend or comparison.So, graph of equation: y = 1/5x - 2
Then, the graph of the equation "y = 1/5x - 2": (Refer to the graph of the equation attached below.)The coordinates will be, (0, -2).Therefore, the graph of the equation "y = 1/5x - 2" is shown.
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A child's ladder is made of 333 sections. Each section is \dfrac{3}{4} 4 3 start fraction, 3, divided by, 4, end fraction meters long. How long is the ladder when all 333 sections are extended to make one ladder? meters
Answers
Answer:
\(Ladder = 2\frac{1}{4}\ m\)
Step-by-step explanation:
Given
\(Sections = 3\)
\(Length = \frac{3}{4}m\) per section
Required
Determine the length of the ladder
The length of the ladder is calculated by multiplying the number of sections by the length of each section.
So, we have:
\(Ladder = 3 * \frac{3}{4}\ m\)
\(Ladder = \frac{9}{4}\ m\)
\(Ladder = 2\frac{1}{4}\ m\)
Hence, the length of the ladder is: \(2\frac{1}{4}\ m\) or 2.25m
So the question is "Write down four common multiples of 2 and 3"
Step by step explaination :(
Answers
Answer:
The common multiples of 2 and 3 are 6,12,18,24
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27...
Common multiples of 2 and 3 include 6, 12, 18, and 24.
A message in a bottle is floating on top of the ocean in a periodic manner. The time between periods of maximum heights is 26 seconds, and the average height of the bottle is 12 feet. The bottle moves in a manner such that the distance from the highest and lowest point is 6 feet. A cosine function can model the movement of the message in a bottle in relation to the height. Part A: Determine the amplitude and period of the function that could model the height of the message in a bottle as a function of time, t. (5 points) Part B: Assuming that at t
Answers
a) The amplitude of the function is 4 feet.
b) The function that represents the situation is .
How to find a function for the height of a bottle and how to analyze its motiona) The amplitude (A), in feet, is equal to the difference between highest and lowest point (\(\left(y_{\max }, y_{\min }\right.\)), in feet, divided by 2. The period (T), in seconds, is the time taken by the bottle to complete one cycle. In this case, the period is the time between two maxima. Hence, we proceed to determine each variable:
Amplitude \(\left(y_{\max }=14 \mathrm{ft}, y_{\min }=6 \mathrm{ft}\right)\)
\(\begin{array}{l}A=\frac{14 f t-6 f t}{2} \\A=4 f t\end{array}\)
The amplitude of the function is 4 feet.
Period
The period of the function is 20 seconds.
b) The function that represents the situation is based on this model:
\(y(t)=y_{o}+A \cdot \sin \frac{2 \pi \cdot t}{T}\) ........(1)
Where:
\(y_{o}\)- Average height of the bottle, in feet.
\(t\) - Time, in seconds.
\(y(t)\) - Current height, in feet.
If we know that \(A=4 \mathrm{ft}, y_{o}=10 \mathrm{ft} \text { and } T=20 \mathrm{~s}\) then the function that represents the situation is:
\(y(t)=10+4 \cdot \sin \frac{\pi \cdot t}{10}\) ......(2)
The function that represents the situation is \(y(t)=10+4 \cdot \sin \frac{\pi \cdot t}{10}\).
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dr. anderson wants to know if annual income differs between college graduates and non-college graduates. he should use a(n)
Answers
Anderson should use a two-sample t-test to determine if there is a statistically significant difference in annual income between college graduates and non-college graduates.
A two-sample t-test is a type of inferential statistic used to compare the means of two independent groups. It is used to determine if the difference between the two groups is statistically significant.
In this case, Anderson would compare the mean annual income of college graduates to the mean annual income of non-college graduates. If the difference between the two means is statistically significant, then Anderson can conclude that there is a difference in annual income between college graduates and non-college graduates.
If the difference is not statistically significant, then Anderson can conclude that there is no difference in annual income between the two groups.
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can someone please doordash me something like i will give brainly 5 stars and spotify premium just 10 dollars please
Answers
i gotchu what do u want
The question is if this is a linear function and if it is than what’s the rate of change and initial value
Answers
simplify:6(y^2)^2(x^-2y)^-1 /3x^-3 y^5
Answers
6(y^2)^2(x^-2y)^-1 /3x^-3 y^5
= 6y^4 / 3x^-3 y^5
= 6y^4 / (3x^3 y^5)
= 6/3 * y^4 / x^3 y^5
= 2y^-1 * x^-3
= 2x^-3 / y
Point E is on line segment D F ‾ \overline{DF} DF . Given D E = 2 x , DE=2x, DE=2x, E F = 2 x − 6 , EF=2x-6, EF=2x−6, and D F = 3 x + 5 , DF=3x+5, DF=3x+5, determine the numerical length of E F ‾ . \overline{EF}. EF .
Answers
Answer:
DF = 38
Step-by-step explanation:
2x+2x-6=3x+5
4x-6=3x+5
x=11
DF = 3(11)+5
DF=33+5
DF=38
Which of the following are among the basic postulates of Euclidean Geometry?
Check all that apply.
A) Any straight line segment can be extended indefinitely.
B) All right triangles are equal.
C) A straight line segment can be drawn between any two points.
D) All right angles are equal.
Answers
Answer:
A) Any straight line segment can be extended indefinitely.
C) A straight line segment can be drawn between any two points.
D) All right angles are equal.
g(x)= -16x2 + 64x + 80 where x is the number of seconds after the rocket is launched. The function can also be written in factored form as g(x) = -16 (x + 1)(x - 5). What is the y-intercept of the function? What does it represent?
Answers
Answer:
y=80
Step-by-step explanation:
when solve y-intercept let x=0
y= -16(0)2+64(0)+80
y=0+0+80
y=80
(0;80)
If the measure of each interior angle of a polygon is 150°, how many sides does it have?
Answers
Find the value of m: −2(m − 5) = −4
m = 7
m = 3
m = −3
m = −7
Answers
A bag contains 20 pink candies, 8 red candies, and 12
green candies. Without looking, Sarah pulls out a piece of
candy. Which color of candy is least likely to be pulled
out?
Answers
Answer:
The red marbles.
Step-by-step explanation:
It is because Red has the least amount of marbles, therefore it is most likely for you to not pull out a red marble.
Work out the length of x.
Give your answer rounded to 3 significant figures.
8.6 cm+
x
The diagram is not drawn accurately.
Answers
Answer:
12.2cm
Step-by-step explanation:
Follow the steps in the image I send you to learn it if you don't get it tell me.
a kayak rental shop rents 28 kayaks per week when it charges $25 per day. For each $5 increase in price, the shop loses four kayak rentals per week. How much should the kayak rental shop charge to maximize weekly revenue? What is the maximum weekly revenue?
Answers
If a kayak rental shop rents 28 kayaks per week when it charges $25 per day. For each $5 increase in price, the shop loses four kayak rentals per week. 720 is the maximum weekly revenue
What is Equation?Two or more expressions with an Equal sign is called as Equation.
It is given that a kayak rental shop rents 28 kayaks per week when it charges $25 per day
For each $5 increase in price, the shop loses four kayak rentals per week. we need to find How much should the kayak rental shop charge to maximize weekly revenue
R(x)=(25+5x)(28-4x)
=5(5+x)4(7-x)
=20(5+x)(7-x)
=-20(5+x)(x-7)
when x=1 it is maximum
Price when x =1 is 25+5(1)=30
To calculate maximize revenue
R(1)=(25+5)(28-4)
=30(24)
=720
Hence the maximum weekly revenue is $720.
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This element orients readers by previewing the structure of the report.
O Definitions of key terms
O Organization
O Sources and methods
Answers
The element that orients readers by previewing the structure of the report is "Organization."
Organizing a report effectively helps readers understand the flow of information and the logical structure of the content. By providing an overview of the organization, readers can anticipate the main sections, their sequence, and the connections between them.
The organization element typically includes headings, subheadings, and a clear hierarchy of information. It outlines the main sections and subsections of the report, indicating how they are related and how they contribute to the overall message or argument.
In addition to the organization element, the other options listed—Definitions of key terms and Sources and methods—also play important roles in a report. Definitions of key terms clarify terminology and provide a common understanding, while Sources and methods explain the sources of information and the methods used in the report's research or analysis. However, in the context of previewing the structure, the Organization element specifically serves this purpose.
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4(x + 3) = 3(3x -1) solve for x
Answers
Answer:
x = 3
Step-by-step explanation:
4(x + 3) = 3(3x - 1) ← distribute parenthesis on both sides
4x + 12 = 9x - 3 ( subtract 9x from both sides )
- 5x + 12 = - 3 ( subtract 12 from both sides )
- 5x = - 15 ( divide both sides by - 5 )
x = 3
Answer:
x = 3
Step-by-step explanation:
start off by distributing
4(x +3) turns into 4x + 12
3(3x - 1) turns into 9x - 3
so your equation turns into
3x + 12 = 9x - 3
then you can subtract the x values to move it to the opposite side of the equal sign (=)
3x + 12 = 9x - 3
-3x -3x
----------------------
12 = 3x - 3
then add the -3
(to find out how to move the values to opposite sides of the equals sign, just isolate the number. whether it is - 3 or just 7x, you keep the sign in front of it. if it is multiplying then divide it to get rid of it. make sure to do it to both sides of the equation.)
12 = 3x - 3
+3 +3
----------------
15 = 3x
divide 3 on both sides
15/ 3 = 3
3x/3 = x
3 = x
the value of x is 3
hope this helps:)
Pleaseeeee helppp answer correctly !!!!!!!!!!!!! Will mark Brianliest !!!!!!!!!!!!!!
Answers
Answer:
x = 6
Step-by-step explanation:
(5x-24)/2 = 3
2.5x - 12 = 3
2.5x = 15
x = 6
A pet store sold 2 birds. They sold 6 times as many turtles as they sold birds. How many turtles did they sell?
Answers
Answer:
12
Step-by-step explanation:
"6 times as many birds..." in other words 6 x 2
The number of turtles that was sold by the pet store given the number of birds sold is 12.
How many turtles are sold?
Multiplication is a mathematical operation that involves determining the product of two or more numbers. The sign used to represent multiplication is x.
Number of turtles were sold = 2 x 6 = 12
A right rectangular prism has base dimensions of 3 inches by 12 inches. An oblique rectangular prism has base dimensions of 4 inches by 9 inches. A right rectangular prism has base dimensions of 3 inches by 12 inches. An oblique rectangular prism has base dimensions of 9 inches by 4 inches. If the prisms are the same height, how do their volumes compare? The volumes are equal, because the bases are congruent. The volumes are equal, because the heights are equal and the horizontal cross-sectional areas at every level are also equal. The volumes are not equal, because their horizontal cross-sectional areas are not the same at every level.
Answers
The volume of right rectangular prisms and oblique rectangular prisms are 36x that are equal.
What is Geometry?It deals with the size of geometry, region, and density of the different forms both 2D and 3D.
Volume is given by
\(\rm volume = base \ area * height\)
If the prisms are of the same height (x). Then
A right rectangular prism has base dimensions of 3 inches by 12 inches. Then the volume (V₁) will be
\(\rm V_1 = 3*12*x\\\\V_1 = 36x\)
An oblique rectangular prism has base dimensions of 4 inches by 9 inches. Then the volume (V₁) will be
\(\rm V_2 = 4*9*x\\\\V_2 = 36x\)
A right rectangular prism has base dimensions of 3 inches by 12 inches. Then the volume (V₁) will be
\(\rm V_3 = 3*12*x\\\\V_3 = 36x\)
An oblique rectangular prism has base dimensions of 9 inches by 4 inches. Then the volume (V₁) will be
\(\rm V_4 = 9*4*x\\\\V_4 = 36x\)
The volume of right rectangular prisms and oblique rectangular prisms are 36x that are equal.
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2. Megan's aquarium measures 20 inches long, 14 inches wide, and 18 inches high. How many cubic inches of water would it take to completely fill the aquarium?
Answers
It would take 5040 cubic inches of water to completely fill the aquarium.
We know that the formula for the volume of cuboid :
V = length × width × height
Let us assume that l represents the length of the aquarium, w represent the width and h represents the height.
Here, l = 20 inches
w = 14 inches
and h = 18 inches
Using the formula for the volume of cuboid, the volume of aquarium would be,
V = l × w × h
V = 20 × 14 × 18
V = 5040 cu.in.
Therefore, it would take 5040 cu.in. of water.
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Three boys step on together from the same spot . Their step measure 30 cm ,27 cm, and 21 cm respectively . What is the minimum distance each should cover so that all can cover the distance in complete steps .
Answers
Each boy should cover a minimum distance of 630 cm (or 6.3 meters) so that they can all cover the distance in complete steps.
To find the minimum distance each boy should cover so that all can cover the distance in complete steps, we need to find the least common multiple (LCM) of their step measurements.
The LCM is the smallest multiple that is divisible by all the given numbers.
The step measurements are 30 cm, 27 cm, and 21 cm. To find the LCM, we can start by listing the multiples of each number until we find a common multiple.
Multiples of 30: 30, 60, 90, 120, 150, 180, 210, ...
Multiples of 27: 27, 54, 81, 108, 135, 162, 189, ...
Multiples of 21: 21, 42, 63, 84, 105, 126, 147, ...
By examining the multiples, we find that the LCM of 30, 27, and 21 is 630. Therefore, each boy should cover a minimum distance of 630 cm (or 6.3 meters) so that they can all cover the distance in complete steps.
By doing so, the first boy would take 630 cm / 30 cm = 21 steps, the second boy would take 630 cm / 27 cm ≈ 23.33 steps (which can be rounded down to 23 steps), and the third boy would take 630 cm / 21 cm = 30 steps.
Hence, by covering a distance of 630 cm, each boy can take complete steps and reach the destination together.
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a random variablegroup of answer choicesis the result of a measurement.can only be discrete.assigns one and only one numeric value to each experimental outcome.is a binomial, poisson, or hypergeometric variable.
Answers
A random variable assigns one and only one numeric value to each experimental outcome.
A random variable used in experimental trials assign numerical values to the members or elements of the trial. This random variable is used to study the outcome of a trial conveniently.
A random variable assigns one and only one numeric value to each experimental outcome. This assigned numeric value may not be exact. In an experimental trial the possible outcomes are denoted by a random variable. So these values are just practically assigned and not theoretically exact.
There are discrete and continuous random variables. They are usually denoted by capital letters such as X, Y,.. etc. These random variables are used to calculate the probability of each outcome in a trial.
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Ashley's chocolate bar is 58% cocoa. If the weight of the chocolate bar is 57 grams, how many grams of cocoa does it contain? Round your answer to the nearest tenth.
Answers
Answer:
33.1 grams
Step-by-step explanation:
You know that 58% of the chocolate bar is cocoa and it weighs 57 grams. All you have to do is multiply 0.58 and 57 together. This will give you the amount of grams of cocoa!
58% = 0.58
0.58*57 grams = 33.06
Round up! 33.06 --> 33.1 grams
whats the answer to the question attached below
Answers
Answer:
ans= 36.72 /1- 15/100
= 36.72 * 20/17
= 36.72 * 1.17647 = 43.2
the original price = £43.2
What is x equal to in the equal to in the equation 7x-11=-19+3x
Answers
Answer:
x=-2
Step-by-step explanation:
7x-11=-19+3x
7x-3x=-19+11
4x=-8
4x/4=-8/4
x=-2
Solve initial value Problem √ydx+(4+x)dy=0,y(−3)=1
Answers
The solution to the initial value problem √y dx + (4+x) dy = 0, y(-3) = 1 is y = x^2 + 4x + 4.
To solve the initial value problem √y dx + (4+x) dy = 0, y(-3) = 1, we can separate the variables and integrate.
Let's start by rearranging the equation:
√y dx = -(4+x) dy
Now, we can separate the variables:
√y / y^(1/2) dy = -(4+x) dx
Integrating both sides:
∫ √y / y^(1/2) dy = ∫ -(4+x) dx
To integrate the left side, we can use a substitution. Let's substitute u = y^(1/2), then du = (1/2) y^(-1/2) dy:
∫ 2du = ∫ -(4+x) dx
2u = -2x - 4 + C
Substituting back u = y^(1/2):
2√y = -2x - 4 + C
To find the value of C, we can use the initial condition y(-3) = 1:
2√1 = -2(-3) - 4 + C
2 = 6 - 4 + C
2 = 2 + C
C = 0
So the final equation is:
2√y = -2x - 4
We can square both sides to eliminate the square root:
4y = 4x^2 + 16x + 16
Simplifying the equation:
y = x^2 + 4x + 4
Therefore, the solution to the initial value problem √y dx + (4+x) dy = 0, y(-3) = 1 is y = x^2 + 4x + 4.
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