A large container in the shape of a rectangular solid must have a volume of 480 m3. The bottom of the container costs $5/m2 to construct whereas the top and sides cost $3/ m2 to construct. Use Lagrange multipliers to find the dimensions of the container of this size that has the minimum cost
Answers
The dimension of the container of this size that has the minimum cost is \(x = \sqrt[3]{360}\), \(y = \sqrt[3]{360}\) and \(z =\frac{4}{3} \sqrt[3]{360}\)
The given parameters are:
Volume = 480m^3Cost: $5 per square meter for the bottom, and $3 per square meter for the sides.Assume the dimensions of the container are: x, y and z.
The volume (V) would be:
\(V = xyz\)
Substitute 480 for V
\(xyz =480\)
And the objective cost function would be:
\(C =8xy + 6xz + 6yz\)
Differentiate the cost function using Lagrange multipliers
\(8x + 6z = \lambda xz\)
\(8y + 6z = \lambda yz\)
\(6x + 6y = \lambda xy\)
Divide equation (1) by (2)
\(\frac{8x + 6z}{8y + 6z} = \frac{\lambda xz}{\lambda yz}\)
\(\frac{8x + 6z}{8y + 6z} = \frac{x}{y}\)
Factor out 2
\(\frac{4x + 3z}{4y + 3z} = \frac{x}{y}\)
Cross multiply
\(4xy + 3yz = 4xy + 3xz\)
Evaluate the like terms
\(3yz = 3xz\)
Divide both sides by 3z
\(y = x\)
Divide the first equation by the third
\(\frac{8y + 6z}{6x + 6y} = \frac{\lambda yz}{\lambda xy}\)
\(\frac{8y + 6z}{6x + 6y} = \frac{z}{x}\)
Factor out 2
\(\frac{4y + 3z}{3x + 3y} = \frac{z}{x}\)
Cross multiply
\(4xy+3xz = 3xz + 3yz\)
Cancel out the common terms
\(4xy = 3yz\)
Divide both sides by y
\(4x = 3z\)
Make z the subject
\(z =\frac{4}{3}x\\\)
So, we have:
\(y = x\) and \(z =\frac{4}{3}x\\\)
Recall that:
\(xyz =480\)
Substitute \(y = x\) and \(z =\frac{4}{3}x\\\)
\(x \times x \times \frac 43x = 480\)
So, we have:
\(\frac 43x^3 = 480\)
Multiply both sides by 3/4
\(x^3 = 360\)
Take the cube roots of both sides
\(x = \sqrt[3]{360}\)
Recall that:
\(y = x\)
So, we have:
\(y = \sqrt[3]{360}\)
Also, we have:
\(z =\frac{4}{3}x\\\)
So, we have:
\(z =\frac{4}{3} \sqrt[3]{360}\)
Hence, the dimension of the container of this size that has the minimum cost is \(x = \sqrt[3]{360}\), \(y = \sqrt[3]{360}\) and \(z =\frac{4}{3} \sqrt[3]{360}\)
Read more about Lagrange multipliers at:
https://brainly.com/question/4609414
Related Questions
An amusement park has 8 water slides and 26 other attractions.
What is the probability that a randomly selected attraction at this amusement park will be
a water slide?
Write your answer as a fraction or whole number.
P(water slide)
Answers
Answer:
The answer is 4/17
Step-by-step explanation:
A group of students sold 48 candles in 4 hours Part B At that rate, how many candles can be sold in 9 hours.
Answers
Answer:
108 candles would be sold in 9 hours
Step-by-step explanation:
First add 48 + 48 = 96
then 48 divided by 4 = 12
12 + 96 = 108
PLS help asap!!!!!
Which equation represents the line that is perpendicular to y = 6 and passes through (-8,-2)?
Answers
Answer:
The answer is a since non of the other answer would go throught that point
Hope This Helps!!!
Solve the following system of linear equations using the elimination method. 3m-n=-10
2m +n=-5
m = -2, n =4
m = -3, n=1
m=-3, n = -1
m = -4, n = -2
Answers
Answer:
m = -3, n = 1
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightEquality Properties
Algebra I
Solving systems of equations using substitution/eliminationStep-by-step explanation:
Step 1: Define Systems
3m - n = -10
2m + n = -5
Step 2: Solve for m
Elimination
Combine equations: 5m = -15Divide 5 on both sides: m = -3Step 3: Solve for n
Define equation: 3m - n = -10Substitute in m: 3(-3) - n = -10Multiply: -9 - n = -10Add 9 to both sides: -n = -1Divide -1 on both sides: n = 1Factor this & please show all work.. I will give brainliest:)
Answers
Answer:
\((7n - 4) (7n^{2} + 8)\)
Step-by-step explanation:
\(49n^{3} - 28n^{2} + 56n - 32\)
\(= (49n^{2} - 28n^{2}) + (56n - 32)\)
\(= 7n^{2} (7n - 4) + 8 (7n - 4)\)
\(= (7n - 4) (7n^{2} + 8)\)
The student council wants to determine the best date for the end-of-year dance for the 350 students. They survey every 5th student who gets off the bus one morning. Eighteen students voted for May 2, 39 students voted for May 9, and 13 students voted for May 16. Which date is it likely that 105 of the 350 students would choose for the dance?
Answers
There is a probability that 105 of the 350 students would select May 9 for the dance.
What is the sample about?To solve the above we need to find the proportion of students in the sample who voted for each date and this can be done by:
May 2: 18/70
= 0.257
May 9: 39/70
= 0.557
May 16: 13/70
= 0.186
if the whole population of 350 students voted, then
May 2: 0.257 x 350
= 90
May 9: 0.557 x 350
= 195
May 16: 0.186 x 350
= 65
From the above calculation, we can see that if 105 students out of 350) were to choose a date, the most likely date that they will select is May 9, since it is the one that has the highest proportion of votes in the sample.
Learn more about sample from
https://brainly.com/question/24466382
#SPJ1
you can determine if the inverse of a polynomial function is a function by using the ____ line test on the inverse.
Answers
You can determine if the inverse of a polynomial function is a function by using the Horizontal Line Test on the inverse.
The Horizontal Line Test is a graphical method that can be used to determine whether a given function is one-to-one, meaning that for each output value, there is at most one input value that maps to it.
If the inverse of a polynomial function is a function, it must pass the Horizontal Line Test, meaning that no horizontal line intersects the graph of the inverse more than once.
In other words, if the inverse of a polynomial function passes the Horizontal Line Test, it is guaranteed to have an inverse, which will be a function itself. If the Horizontal Line Test fails, then the inverse is not a function and does not have an inverse.
To know more about Inverse function:
https://brainly.com/question/2541698
#SPJ4
Radioactive radium has a half-life of approximately 1599 years. What percent of a given amount remains after 900 years? (Round your answer to two decimal places.)
Answers
64.77% of a given amount remains after 900 years.
To determine the percentage of radioactive radium remaining after 900 years, we'll use the half-life formula:
Final Amount = Initial Amount * \(1/2^{(time elapsed / half-life)}\)
In this case, the half-life is 1599 years and the time elapsed is 900 years. We want to find the percentage remaining, so let's assume the initial amount is 100%.
Final Amount = 100 * \((1/2)^{(900 / 1599)}\)
Now, we'll calculate the exponent:
Exponent = 900 / 1599 ≈ 0.5635
Then, we'll compute the final amount:
Final Amount ≈ 100 * \((1/2)^{0.5635}\) ≈ 100 * 0.6477 ≈ 64.77%
After 900 years, approximately 64.77% of the initial radioactive radium remains. This percentage was found using the half-life formula, which accounts for the exponential decay of radioactive substances over time.
The formula shows how the remaining amount of a substance decreases as time progresses based on its half-life, which is the time it takes for half of the substance to decay. In this scenario, radium's half-life of 1599 years and the 900-year time frame were used to determine that roughly 64.77% of the initial radium remains.
Know more about Radioactive radium here:
https://brainly.com/question/31436084
#SPJ11
Here is a shape ABCD.
Calculate the perimeter of the shape.
Give your answer correct to 3
significant figures.
B
С
D
9 cm
O
9 cm
140°
21 cm
24°
А
The shape is made from a triangle and a
sector of a circle, centre O and radius 9 cm.
AD = 21 cm
Angle AOD = 140°
Angle OAD = 24°
Total marks: 5
Answers
9514 1404 393
Answer:
59.8 cm
Step-by-step explanation:
The length of arc ABC is ...
s = rθ
s = (9 cm)(220/180π) = 11π cm ≈ 34.558 cm
The length of segment CD is 9 cm less than the length of segment OD. OD can be found using the law of sines.
OD/sin(A) = AD/sin(O)
OD = AD·sin(A)/sin(O) = (21 cm)sin(24°)/sin(140°) ≈ 13.288 cm
Then the length of CD is ...
CD = OD -9 cm = (13.288 cm) -(9 cm) = 4.288 cm
The perimeter is the sum of the segment and arc lengths:
P = ABC + CD +AD
P = 34.558 cm + 4.288 cm + 21 cm = 59.846 cm
The perimeter of the shape is about 59.8 cm.
What is the first step in evaluating the expression shown below?
12 ÷ (74 – 36) + 8 – 2
a
Subtract 74 – 36.
b
Divide 12 ÷ 74.
c
Add 36 + 8.
d
Subtract 8 – 2.
Answers
Answer:
A
Step-by-step explanation:
because it is is parenthesis
The amplitude of the graphs of the sine and cosine functions is.
Answers
The amplitude of the graphs of the sine and cosine functions is the maximum value of the function.
In trigonometry, the amplitude of a sine or cosine function determines the maximum distance between the graph of the function and its central axis (usually the x-axis). It represents the magnitude of oscillation or the maximum displacement from the equilibrium position. The amplitude is always positive and can be identified by looking at the highest and lowest points of the graph. For both the sine and cosine functions, the amplitude is equal to the absolute value of the coefficient of the trigonometric term. For example, in the function y = Asin(x), the amplitude is A, and in the function y = Bcos(x), the amplitude is B. The larger the amplitude, the more stretched or compressed the graph becomes vertically.
Learn more about sine or cosine function here
brainly.com/question/30353030
#SPJ11
What is the amplitude of the graphs of the sine and cosine functions?
PLEASE HELP ILL GIVE BRAINLIEST
SHOW YOUR WORK
Answers
what are the ordered pairs of the solutions for this system of equations?
f(x)=x^(2)-2x+3; f(x)=-2x+12
Answers
The ordered pairs for the system of equations f(x) = x^2 -2x + 3 and f(x) = -2x + 12 are (3, 6) and (-3, 18)
What is a quadratic equation?A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term. The first condition for an equation to be a quadratic equation is the coefficient of x2 is a non-zero term(a ≠ 0)
f(x) = x^2 -2x +3 and f(x) = -2x + 12
which means
x^2 -2x +3 = -2x + 12
x^2 -2x +3 + 2x - 12 = 0
x^2 -9 = 0
by factorizing we have
(x-3)(x+3) = 0
x = 3 or -3
when x = 3
f(x) = -2x + 12
f(3) = -2(3) + 12 which is 6
when x = -3
f(-3) = -2(-3) + 12 which is 18
ordered pairs are (3, 6) and (-3, 18)
In conclusion, (3, 6) and (-3, 18) are the ordered pairs
Learn more about Quadratic equation: https://brainly.com/24334139
#SPJ1
How can you decompose the composite figure to determine its area? as a circle, three rectangles, and a triangle as a circle, a trapezoid, and four triangles as a semicircle, three rectangles, and a square as a semicircle, a trapezoid, and two rectangles.
Answers
Answer:
The best way to decompose the composite figure to determine its area is as a semicircle, a trapezoid, and two rectangles. This way, we can use the following formulas to find the area of each part:
Area of a semicircle = 21πr2, where r is the radius of the circle.
Area of a trapezoid = 21(b1+b2)h, where b1 and b2 are the bases and h is the height of the trapezoid.
Area of a rectangle = l×w, where l is the length and w is the width of the rectangle.
Then, we can add up the areas of each part to find the total area of the composite figure. The other options are not as convenient because they either involve more parts or more complicated shapes. For example, option A would require finding the area of a triangle, which involves using trigonometry or the Pythagorean theorem. Option B would require finding the area of four triangles, which is more tedious than finding the area of two rectangles. Option C would require finding the area of a square, which is redundant because a square is a special case of a rectangle.
Step-by-step explanation:
The composite figure can be decomposed into a semicircle, a trapezoid, and two rectangles.
Option D is the correct answer.
What is a trapezium?It is a quadrilateral that has one pair of parallel sides and a height.
The area is calculated as: 1/2 x sum of the parallel sides x height.
Examples:
Area of a trapezium that has the parallel sides as 3 cm and 4 cm and a heght o 5 cm.
Area = 1/2 x (3 + 4) x 5
Area = 1/2 x 7 x 5
Area = 35/2 = 17.5 cm^2
We have,
The given figure can be decomposed into the following figure.
- Semicircle
- Trapezoid
- Two rectangles
Thus,
A semicircle, a trapezoid, and two rectangles.
Learn more about trapezium here:
https://brainly.com/question/22607187
#SPJ5
Choose the correct description of the graph of the inequality X - 3 greater than or equal to 25
A. Open circle on 8, shading to the left
B. Closed circle on 8, shading to the left.
C. Open circle on 8, shading to the right.
D. Closed circle on 8, shading to the right.
I’m pretty sure it’s D
Answers
Answer:
D. Closed circle on 8, shading to the right.
√1-x² Let 1 = ²₁-** √3-2√x+y² dzdydx. By converting I into an equivalent triple integral in cylindrical coordinates, we obtain: 1 = ²³² rdzdrde None of these O This option 23-2r² 1 = √²² ²²-²³ rdzdrde. O This option 1 = ²₁² ²¹² rdzdrdo
Answers
The correct option is: 1 = ∫∫∫r dz dr dθ
By converting the given expression into an equivalent triple integral in cylindrical coordinates, we obtain:
1 = ∫∫∫r dz dr dθ
This option represents the correct conversion of the integral into cylindrical coordinates. The integral is taken over the region defined by the limits of integration for z, r, and θ, which are not provided in the given question.
Know more aboiut coordinates here:
https://brainly.com/question/32836021
#SPJ11
What is the y-intercept of the line shown?
-1
0
1
Answers
Answer:
-1
Step-by-step explanation:
The y-intercept is where the line crosses the y-axis.
Answer:
\(\huge\boxed{\sf y-intercept = -1}\)
Step-by-step explanation:
Y-intercept is the point of the line of the equation where:x = 0the line of the equation passes through the y-axis.Here,
y-intercept = -1Since, this is the point where x = 0 and where the line passes through the y-axis.
\(\rule[225]{225}{2}\)
Graphing inequality
Answers
Answer:
1) open dot on 1 and make a line going towards -10
2) closed dot on -6 doing towards 10
Step-by-step explanation:
When it has the line under it means that its included which is why u use a closed dot :)
hope this helps
HELPPPP MEEEEEEEEEEEEEEEEEE
Answers
Answer:
c. 41
Step-by-step explanation:
What is the length of AC? And Length of AB?
Answers
Answer:
The length of AC is 5 units. The length of AB is 3 units.
Step-by-step explanation:
(4,8), (x,12); slope=2
Answers
Answer:
x=6
Equation: y=2x
Step-by-step explanation:
we can use slope intercept form y=mx+b
M is slope and b is y intercept
We know slope is 2
y=2x+b
To find b we fill in the coordinates we know
8=2(4)+b
8=8+b
-8 -8
The y intercept is 0 now to find x we fill in what know with those coordinates
12=2x
/2. /2
6=x
Hopes this helps please mark brainliest
A rancher owns 8 full sections and 3 half sections. He sells off 5 quarter sections. How many acres are left
Answers
The rancher has 5,280 acres left after selling off 5 quarter sections.
There are different ways to approach this problem, but one possible method is to convert all the sections and subsections to acres and then subtract the number of acres sold.
One section is equal to 640 acres (1 mile x 1 mile)
One-half section is equal to 320 acres (1/2 mile x 1 mile)
The one-quarter section is equal to 160 acres (1/2 mile x 1/2 mile)
The rancher owns:
8 full sections x 640 acres/section = 5,120 acres
3 half sections x 320 acres/half section = 960 acres
Total = 6,080 acres
The rancher sells off:
5 quarter sections x 160 acres/quarter section = 800 acres
The number of acres left is:
6,080 acres - 800 acres = 5,280 acres
Therefore, the rancher has 5,280 acres left after selling off 5 quarter sections.
Answer: There are 5,280 acres left.
To learn about the selling price here:
https://brainly.com/question/31211894
#SPJ11
Can somebody help me solve this The diameter of a cylinder is 1 yd. The height is 12 yd. What is the volume of the cylinder type you answer with yards
Answers
Answer:
9.42
Step-by-step explanation:
Volume of cylinder= πr^2h
3.14 × 0.5 × 0.5 × 129.42 cube.ydAnswer:
37.68 yd
Step-by-step explanation:
= \(\pi\)r²h
= (3.14)(1)²(12)
= 3.14 x 1 x 12
= 3.14 x 12
= 37.68 yd
Solve b over negative 4.6 is greater than or equal to negative 5.3 for b.
Answers
the solution for the given Mathematical operation for b will gives us, b ≥ -25.76.
What are mathematical operations?The term "operation" in mathematics refers to the process of computing a value utilizing operands and a math operator. For the specified operands or integers, the math operator's symbol has predetermined rules that must be followed. In mathematics, there are five basic operations: addition, subtraction, multiplication, division, and modular forms.
Given, b over negative 4.6 is greater than or equal to negative 5.3
So, in numerical form
=> b/ 4.6 ≥ -5.6
Thus simplification for b
=> b ≥ -5.6* 4.6
=> b ≥ -25.76
Therefore, the solution for the given Mathematical operation for b will gives us, b ≥ -25.76.
Learn more about Mathematical operations here:
https://brainly.com/question/20628271
#SPJ1
Evaluate the following -3- (-6)
Answers
Answer:
\( - 3 - ( - 6) \\ = - 3 + 6 \\ = 3\)
Step-by-step explanation:
3 is correct answer
hope it helped you:)
Answer:
The answer is 3
Step-by-step explanation:
when you have two minus signs next to each other like minus negative 6, it becomes a plus so the new equation would be -3+6. And that equals 3.
when using a graduated pipet, at which point do you measure the volume?
Answers
The volume is measured at the bottom of the meniscus when the liquid level is between the two etched marks on the pipet.
1. Place the graduated pipet on a flat surface and make sure it is level
2. Draw the liquid up until the meniscus is level with the etched line that corresponds to the desired volume
3. Make sure the liquid is not touching the etched line above it
4. Observe the bottom of the meniscus and take the reading at the point where the liquid level is between the two etched lines
The volume is measured at the bottom of the meniscus when the liquid level is between the two etched marks on the pipet.
Learn more about volume here
https://brainly.com/question/16134180
#SPJ4
The altitude h, in feet, of the balloon x hours after starting its ascent from the hill can be modeled by the function h(x)=-16x²+64x+80. Does the parabola open up or down?
Answers
Based on the altitude function h(x)=-16x²+64x+80, the parabola opens down.
What is the graph of a quadratic function?In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the given quadratic function, we can logically deduce that the graph would be a downward parabola because the coefficient of x² is negative and the value of "a" is less than zero (0).
Since the leading coefficient (value of a) in the given quadratic function y = -16x² + 64x + 80 is negative 16, we can logically deduce that the parabola would open downward and the x-intercept (roots) represent the roots or zeros.
In conclusion, the vertex is given by the ordered pair (2, 144) and it has x-intercepts at (-1, 0) and (5, 0) as shown in the image attached below.
Read more on quadratic functions here: brainly.com/question/29499209
#SPJ1
Need help ASAP pls solve!!!
Answers
Answer:
18 (to the nearest tenth)
Step-by-step explanation:
Use theorem and solve basically
Answer:
Pythagorean theorem: \(a^{2} + b^{2} = c^{2}\)
\(10^2+b^2= 15^2\\100+b^2 = 225\\b^2 = 225-100\\b^2 = 125\\b = \sqrt{25} \\b = 5\)
Answer is 5.
Help pls I will give Brianlist
Answers
Answer:
A
Step-by-step explanation:
Hope this helps :)
Answer:
A
Step-by-step explanation:
How many roots do the functions have in common?
f(x)=x² - 4x - 5
Choose 1 answer:
f and g share the same root(s).
f and g share one root in common but each have another root that
is not shared.
f and g share no roots in common.
Answers
Answer:
Step-by-step explanation:
To determine how many roots the function f(x) = x² - 4x - 5 has in common with another function, we need the equation of the other function. Without that information, we cannot determine the number of common roots. Please provide the equation of the other function, and I will be able to assist you further.
Hope this answer your question
Please rate the answer and
mark me ask Brainliest it helps a lot