Answer:
The maximum volume of the box is:
[tex]V =\frac{5}{3}\sqrt{\frac{5}{3}}[/tex]
Step-by-step explanation:
Given
[tex]Surface\ Area = 10m^2[/tex]
Required
The maximum volume of the box
Let
[tex]a \to base\ dimension[/tex]
[tex]b \to height[/tex]
The surface area of the box is:
[tex]Surface\ Area = 2(a*a + a*b + a*b)[/tex]
[tex]Surface\ Area = 2(a^2 + ab + ab)[/tex]
[tex]Surface\ Area = 2(a^2 + 2ab)[/tex]
So, we have:
[tex]2(a^2 + 2ab) = 10[/tex]
Divide both sides by 2
[tex]a^2 + 2ab = 5[/tex]
Make b the subject
[tex]2ab = 5 -a^2[/tex]
[tex]b = \frac{5 -a^2}{2a}[/tex]
The volume of the box is:
[tex]V = a*a*b[/tex]
[tex]V = a^2b[/tex]
Substitute: [tex]b = \frac{5 -a^2}{2a}[/tex]
[tex]V = a^2*\frac{5 - a^2}{2a}[/tex]
[tex]V = a*\frac{5 - a^2}{2}[/tex]
[tex]V = \frac{5a - a^3}{2}[/tex]
Spit
[tex]V = \frac{5a}{2} - \frac{a^3}{2}[/tex]
Differentiate V with respect to a
[tex]V' = \frac{5}{2} -3 * \frac{a^2}{2}[/tex]
[tex]V' = \frac{5}{2} -\frac{3a^2}{2}[/tex]
Set [tex]V' =0[/tex] to calculate a
[tex]0 = \frac{5}{2} -\frac{3a^2}{2}[/tex]
Collect like terms
[tex]\frac{3a^2}{2} = \frac{5}{2}[/tex]
Multiply both sides by 2
[tex]3a^2= 5[/tex]
Solve for a
[tex]a^2= \frac{5}{3}[/tex]
[tex]a= \sqrt{\frac{5}{3}}[/tex]
Recall that:
[tex]b = \frac{5 -a^2}{2a}[/tex]
[tex]b = \frac{5 -(\sqrt{\frac{5}{3}})^2}{2*\sqrt{\frac{5}{3}}}[/tex]
[tex]b = \frac{5 -\frac{5}{3}}{2*\sqrt{\frac{5}{3}}}[/tex]
[tex]b = \frac{\frac{15 - 5}{3}}{2*\sqrt{\frac{5}{3}}}[/tex]
[tex]b = \frac{\frac{10}{3}}{2*\sqrt{\frac{5}{3}}}[/tex]
[tex]b = \frac{\frac{5}{3}}{\sqrt{\frac{5}{3}}}[/tex]
Apply law of indices
[tex]b = (\frac{5}{3})^{1 - \frac{1}{2}}[/tex]
[tex]b = (\frac{5}{3})^{\frac{1}{2}}[/tex]
[tex]b = \sqrt{\frac{5}{3}}[/tex]
So:
[tex]V = a^2b[/tex]
[tex]V =\sqrt{(\frac{5}{3})^2} * \sqrt{\frac{5}{3}}[/tex]
[tex]V =\frac{5}{3} * \sqrt{\frac{5}{3}}[/tex]
[tex]V =\frac{5}{3}\sqrt{\frac{5}{3}}[/tex]
The maximum volume of the box which has a 10 m² surface area is given below.
[tex]\rm V_{max} = \dfrac{5}{3} *\sqrt{\dfrac{5}{2}}[/tex]
What is differentiation?The rate of change of a function with respect to the variable is called differentiation. It can be increasing or decreasing.
We want to construct a box with a square base and we currently only have 10 m² of material to use in the construction of the box.
The surface area = 10 m²
Let a be the base length and b be the height of the box.
Surface area = 2(a² + 2ab)
2(a² + 2ab) = 10
a² + 2ab = 5
Then the value of b will be
[tex]\rm b = \dfrac{5-a^2}{2a}[/tex]
The volume of the box is given as
V = a²b
Then we have
[tex]\rm V = \dfrac{5-a^2 }{2a}* a^2\\\\V = \dfrac{5a - a^3}{2}\\\\V = \dfrac{5a}{2} - \dfrac{a^3}{2}[/tex]
Differentiate the equation with respect to a, and put it equal to zero for the volume to be maximum.
[tex]\begin{aligned} \dfrac{dV}{da} &= \dfrac{d}{da} ( \dfrac{5a}{2} - \dfrac{a^3}{2} ) \\\\\dfrac{dV}{da} &= 0 \\\\\dfrac{5}{2} - \dfrac{3a^2 }{2} &= 0\\\\a &= \sqrt{\dfrac{5}{2}} \end{aligned}[/tex]
Then the value of b will be
[tex]b = \dfrac{5-\sqrt{\dfrac{5}{2}} }{2*\sqrt{\dfrac{5}{2}} }\\\\\\b = \sqrt{\dfrac{5}{2}}[/tex]
Then the volume will be
[tex]\rm V = (\sqrt{\dfrac{5}{2}} )^2*\sqrt{\dfrac{5}{2}} \\\\V = \dfrac{5}{3} *\sqrt{\dfrac{5}{2}}[/tex]
More about the differentiation link is given below.
https://brainly.com/question/24062595
What is the area of the triangle centimeters squared 23 CM 16 CM 14 CM
Answer:
Step-by-step explanation:
82
Answer:
Step-by-step explanation:
.5(half) x 14(base) x 16(height)=112cm
=
Evaluating an algebraic expression: Whole nu
Evaluate the expression when b=38 and c=8,
B/2+3c^2
Simplify your answer as much as possible.
9514 1404 393
Answer:
211
Step-by-step explanation:
Put the numbers where the variables are and do the arithmetic.
b/2 +3c^2 = 38/2 +3(8^2) = 19 +3·64 = 211
Answer:
211
Step-by-step explanation:
given : b = 38 and c = 8
Evaluate :
= b/2 + 3c²
= 38 / 2 + 3 ( 8 )²
= 19 + 3 × 64
= 19 + 192
= 211
Many service companies collect data via a follow-up survey of their customers. For example, to ascertain customer sentiment, Delta Air Lines sends an e-mail to customers immediately following a flight. Among other questions, Delta asks:
How likely are you to recommend Delta Air Lines to others?
The possible responses are:
1. Definitely will
2. Probably Will
3. May or May Not
4. Probably Will Not
5. Definitely Will Not
Required:
a. Are the data collected by Delta in this example quantitative or categorical?
b. What measurement scale is used?
Answer:
The answer is "Categorical and Ordinal scale"
Step-by-step explanation:
Categorical factors affect data types that can be grouped. This analysis is the examination of data in which the answer variables are grouped in a series of reciprocal classifications, like the age group or even the unordered groups of eye colors.
An ordinal chart is a form of the category with such a specific order or scale. It is the 2nd assessment level that provides the ordering and order of data, without really measure the extent of variability. The first of the four measuring scales is the ordinal scale of listening.
The Fighting Hens' record has improved every year. The 2016 team won 3
more games than the 2015 team, and the 2017 team won 2 more games
than the 2016 team. If the 2017 team won 20 games, how many did the 2015
team win?
Answer:
15
Step-by-step explanation:
just calculate backwards
20 -2 -3
Answer:
The answer would be 15.
Step-by-step explanation:
I agree with the previous answer, in these cases you just calculate backward.
You start out with 20, then it states 2017 team won 2 more games, so you subtract 2. So now we know the 2016 team won 18 games.
Next we subtract 3 to get how many wins the 2015 team got. 18 - 3 is 15.
The answer would be 15.
Customers at a restaurant can build their own burrito by choosing one item from each category shown in the table.
Answer:
E
Step-by-step explanation:
Beans: 2 possibilities
Meat: 3 possibilities
Vegetables: 4 posibilities
Toppings: 4 possibilities
Then mulltiply the numbers:
2 x 3 x 4 x 4 which equals 96
A farmer finds that if she plants 55 trees per acre, each tree will yield 25 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. How many trees should she plant per acre to maximize her harvest
Answer:
31 trees per acre will maximize the harvest
Step-by-step explanation:
Given
[tex]Plant \to 55[/tex] trees/acre
[tex]Yield \to 25[/tex] bushels
[tex]x \to trees[/tex]
Required
Number of trees to maximize harvest
From the question, we understand that:
Yield will decrease by 4 i.e. 25 - 4x
For every additional tree planted, i.e. 55 + x
So, the function is:
[tex]F(x) = (25- 4x)*(55+x)[/tex]
Open bracket
[tex]F(x) = 25 * 55 -4x * 55 + 25 * x -4x*x[/tex]
[tex]F(x) = 1375 - 220x + 25x -4x^2[/tex]
[tex]F(x) = 1375 -195x -4x^2[/tex]
Rewrite as:
[tex]F(x) = -4x^2 -195x +1375[/tex]
The maximum of a quadratic function is calculated as:
[tex]Max = -\frac{b}{2a}[/tex]
In the above equation:
[tex]a = -4; b =-195; c = 1375[/tex]
So:
[tex]x = -\frac{-195}{2 * -4}[/tex]
[tex]x = -\frac{195}{8}[/tex]
[tex]x = -24.375[/tex]
Recall that the number of trees to be planted is: 55 + x
So, we have:
[tex]Trees = 55+x[/tex]
[tex]Trees = 55-24.375[/tex]
[tex]Trees = 30.625[/tex]
Approximate
[tex]Trees = 31[/tex]
simplify the square root of 11.
Answer:
3.316
Step-by-step explanation:
6x+3y=24
3x-4y=-21
What is the solution of system of equations
Answer:
i don't know if it is correct or not, sorry if i were mistaken
The value of x and y for the given system of equations is 1 and 6 respectively.
What is system of equations?
A system of equations is a collection of two or more equations with a same set of unknown variables.
What is solution to the system of equation?A solution to a system of equations is a set of values for the variable that satisfy all the equations simultaneously.
What is substitution method?The substitution method is a method to solve simultaneous linear equations. In which the value of one variable from one equation is substituted in the other equation.
According to the given question
we have a system of equations
[tex]6x + 3y = 24..(i)[/tex]
and, [tex]3x -4y = -21..(ii)[/tex]
From equation (i)
[tex]6x + 3y = 24[/tex]
⇒ [tex]2x + y = 8[/tex]
⇒ [tex]y = 8-2x[/tex]
Substitute the value of y in equation (ii)
⇒[tex]3x - 4(8-2x)=-21[/tex]
⇒[tex]3x-32+8x =-21[/tex]
⇒[tex]11x = 32-21[/tex]
⇒[tex]11x = 11[/tex]
⇒ [tex]x =1[/tex]
Substitute, x = 1 in either the equations (i) and (ii)
⇒[tex]3(1)-4y =-21[/tex]
⇒[tex]3 -4y=-21[/tex]
⇒[tex]-4y = -21-3[/tex]
⇒-4y = -24
⇒[tex]y =6[/tex]
Hence, the value of x and y for the given system of equations is 1 and 6 respectively.
Learn more about the substitution method here:
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Which one is it !? How do you do it ? It’s timedddd
heres a key for then
SIN= O/H
COS= A/H
TAN= O/A
CSC= H/O
SEC= H/A
COT= A/O
let me know if u have anymore questions
Answer:
3/5 is ur answer
Step-by-step explanation:
ok so its SIN
SIN is o/h (o= opposite and h= hypotenuse)
so we can look at the picture and it says find the SIN of a
ok so the O would be 3 and the H would be 5
so ur answer would be 3/5
I need help anyone?
Answer:
0
Step-by-step explanation:
When you substitute x=1 for f(1),
you get f(1)=0
and the absolute value of 0 is 0.
hope I answered it correct.
please give brainliest, thanks!
what is the solution to the system of linear equations
Answer:
(0, -5)
General Formulas and Concepts:
Algebra I
Reading a coordinate planeCoordinates (x, y)Solving systems of equations by graphingStep-by-step explanation:
According to the systems of equations graph, we see both lines intersect at (0, -5). Where the 2 lines intersect is the solution to the systems.
∴ our answer is (0, -5)
Determine if the two triangles are congruent. If they are, State how you know. NO LINKS!!!!! Show work please. Part 3c
Answer:
7. Not enough information to determine congruency
8.congreunt by AAS
Step-by-step explanation:
7. We know one side and the vertical angles between the triangles. That is not enough to determine the triangles are congruent
8. We know one angle and one side are congruent. The angle between are vertical angles and they are congruent. We know two angles and one side, so they are congruent by AAS
Which values are solutions to the inequality below? Check all that apply.
x < 64
O A. 9
11
B. 66
C. 64
D. 6
E. 8
F. no solutions
The following frequency distribution shows the yearly tuitions (in $1,000s) of a sample of private colleges.
Yearly Tuition Frequency
12-16 5
17-21 4
22-26 3
27-31 2
For the above data, compute the mean yearly tuition.
Answer:
19.714
Step-by-step explanation:
Given :
Yearly Tuition Frequency
12-16 5
17-21 4
22-26 3
27-31 2
We calculate the midpoint (x) = (lower + upper) / 2
The mean yearly tuition :
The midpoint (x) :
X ___ F ___ Fx
14 __ 5
19 __ 4
24 __ 3
29 __ 2
The mean = Σfx / Σf
Σfx / Σf = (14*5 + 19*4 + 24*3 + 29*2) / (5+4+3+2)
Mean = 276 / 14
Mean yearly tuition = 19.714
5. In the diagram shown below, BC is drawn tangent
to circle OA, and BD is a chord in that circle. If
mBD = 144º, what is mZCBD?
8
(1) 36
(2) 72
(3) 1440
(4) 180°
please help!!!
9514 1404 393
Answer:
(2) 72°
Step-by-step explanation:
In this geometry, the angle at the tangent is half the measure of the intercepted arc.
∠CBD = (arc BD)/2 = 144°/2
∠CBD = 72°
__
Additional comment
Consider a point X anywhere on long arc BD. The inscribed angle at X will have half the measure of short arc BD, so will be 144°/2 = 72°. This is true regardless of the position of X on long arc BD. Now, consider that X might be arbitrarily close to point B. The angle at X is still 72°.
As X approaches B, the chord XB approaches a tangent to the circle at B. Effectively, this tangent geometry is a limiting case of inscribed angle geometry.
10 - 5 2/3 = ? i need this for math
[tex]\huge\color{purple}\boxed{\colorbox{black}{♡Solution♡}}[/tex]
[tex]4 \frac{1}{3} [/tex] ✅Step-by-step explanation:
[tex]10 - 5 \frac{2}{3} \\ = 10 - \frac{17}{3} \\ = \frac{10 \times 3}{1 \times 3} - \frac{17}{3} \\ = \frac{30 - 17}{3} \\ = \frac{13}{3} \\ = 4 \frac{1}{3} [/tex]
[tex]\large\mathfrak{{\pmb{\underline{\orange{Happy\:learning }}{\orange{!}}}}}[/tex]
A rectangular field is two times as long as it is wide. If the perimeter of the field is 390 feet, what are the dimensions of the field?
A) Write an equation you can use to answer the given question. Let w be the width of the field. Do not solve the equation yet. The equation is ______________ (Make sure you use the correct variable.)
B) Use your equation to find the dimensions of the field. The width of the field is _________ feet. The length of the field is _________ feet.
Answer:
2(2w) + 2(w) = 390
6w = 390
w = 65
2w or l = 130 feet
A sphere with a volume of 1200 cm³ is dilated by a factor of 3/5.
The volume of the dilated image is ___ cubic centimeters.
Answer:
259.2 cm³
Step-by-step explanation:
The volume of the dilated image is
(3/5)³ × 1200= 27/125 × 1200
= 259.2 cm³
A privately owned lake contains two types of game fish, bass and trout. The owner provides two types of food, A and B, for these fish. Bass require 2 units of food A and 4 units of food B,
and trout require 5 units of food A and 2 units of food B. If the owner has 400 units of each food, find the maximum number of fish the lake can support.
fish
Need Help?
Read
Watch it
Answer:
133 fishes
Step-by-step explanation:
Units of food A = 400 units
Units of food B = 400 units
Fish Bass required 2 units of A and 4 units of B.
Fish Trout requires 5 units of A and 2 units of B.
i. For food A,
total units of food A required = 2 + 5
= 7 units
number of bass and trout that would consume food A = 2 x [tex]\frac{400}{7}[/tex]
= 114.3
number of bass and trout that would consume food A = 114
ii. For food B,
total units of food B required = 4 + 2
= 6 units
number of bass and trout that would consume food B = 2 x [tex]\frac{400}{6}[/tex]
= 133.3
number of bass and trout that would consume food B = 133
Thus, the maximum number of fish that the lake can support is 133.
PLZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ HELP ME OUT THIS I DO ANYTHING JUST HELP AND SHOW YOUR WORK
Task #1 Creating A Table Task
Create a table of x and y values that represents a proportional relationship.
a) Explain how you know the relationship is proportional.
b) What equation models the values in the table?
2) Create a table of x and y values that represents a linear, non-proportional relationship.
a) Explain how you know the relationship is non-proportional.
b) What equation models the values in the table?
Answer:
sis
Step-by-step explanation:
What is the fraction to this picture?
for his new year's resolution olvin decides to meal prep instead of spending so much money on takeout everyday: a cereal bar for breakfast, a burger for lunch had 200 calories. assuming the burger and chicken sandwich are equal in calories and olvin eats 8,400 calories a week how many calories are in one chicken sandwich
Answer:
She had the whole week off from work, spent a chunk of it vacationing in ... beef burger (assuming 85% lean, which is fairly standard, would mean 60 calories per ... Out to dinner: 1/2 of a large Asian Chicken Salad (from a small deli/restaurant; ~350).
Step-by-step explanation:
She had the whole week off from work, spent a chunk of it vacationing in ... beef burger (assuming 85% lean, which is fairly standard, would mean 60 calories per ... Out to dinner: 1/2 of a large Asian Chicken Salad (from a small deli/restaurant; ~350).
Translate to an algebraic expression.
The difference of five times a number and two
I can't find the surface area lol
n=3.14
Answer:
216 square feet
Step-by-step explanation:
The formula for the surface area of a rectangular prism is:
2 (lw + hl + hw)
w= width
h= height
l= length
Use the formula with the given dimensions:
2 (6 • 2 + 12 • 6 + 12 • 2)
= 2 (12 + 72 + 24)
= 2 (108)
= 216
Surface area is measured in square feet
(feet in this case)
Hope this helps
buggy’s bugs buggles buuuugles
Can y’all help me on question 24?!
Answer:
C
Step-by-step explanation:
:) glad I could help! Give Brainiest answer if you can! It doesn't cost you and it would make my day! :D
Answer:
b
helpful answer to your question
Without graphing, determine the range of the function f(x) = -5|x-7|+2 over
the interval (-5,10).
A. [2 -62
B. [2, 17]
C. (17, 62]
D. (-7, 2]
To prove ABC= ADC, To prove which triangle postulate could you use?
SSS
SAS
ASA
AAS
Answer:
ASA
Step-by-step explanation:
..................
To prove ABC = ADC, We use ASA triangle postulate.
What is congruency?We know two similar planer figures are congruent when we have sides or angles or both that are the same as the corresponding sides or angles or both.
The similarity of triangles on the other hand is when the corresponding angles are equal but the corresponding sides are not equal but they have the same common ratio.
From the given diagram a quadrilateral ABCD is given,
Line segment AC divides quadrilateral ABCD into two triangles namely
ABC and ADC, and both have a common side AC.
m∠BAC ≅ m∠DAC and m∠BCA ≅ m∠DAC.
Therefore, Triangle ABC and ADC are congruent by ASA, with two angles and a side included between them.
learn more about congruency here :
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A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. What is the probability that fewer than 11 of them will vote
Answer:
0.0479 = 4.79% probability that fewer than 11 of them will vote
Step-by-step explanation:
For each voter, there are only two possible outcomes. Either they will vote, or they will not. The probability of a voter voting is independent of any other voter, which means that the binomial probability distribution is used to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
70% of all eligible voters will vote in the next presidential election.
This means that [tex]p = 0.7[/tex]
20 eligible voters were randomly selected from the population of all eligible voters.
This means that [tex]n = 20[/tex]
What is the probability that fewer than 11 of them will vote?
This is:
[tex]P(X < 11) = P(X = 10) + P(X = 9) + P(X = 8) + P(X = 7) + P(X = 6) + P(X = 5) + P(X = 4) + P(X = 3) + P(X = 2) + P(X = 1) + P(X = 0)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 10) = C_{20,10}.(0.7)^{10}.(0.3)^{10} = 0.0308[/tex]
[tex]P(X = 9) = C_{20,9}.(0.7)^{9}.(0.3)^{11} = 0.0120[/tex]
[tex]P(X = 8) = C_{20,8}.(0.7)^{8}.(0.3)^{12} = 0.0039[/tex]
[tex]P(X = 7) = C_{20,7}.(0.7)^{7}.(0.3)^{13} = 0.0010[/tex]
[tex]P(X = 6) = C_{20,10}.(0.7)^{6}.(0.3)^{12} = 0.0002[/tex]
[tex]P(X = 5) = C_{20,5}.(0.7)^{5}.(0.3)^{15} \approx 0[/tex]
The probability of 5 or less voting is very close to 0, so they will not affect the outcome. Then
[tex]P(X < 11) = P(X = 10) + P(X = 9) + P(X = 8) + P(X = 7) + P(X = 6) + P(X = 5) + P(X = 4) + P(X = 3) + P(X = 2) + P(X = 1) + P(X = 0) = 0.0308 + 0.0120 + 0.0039 + 0.0010 + 0.0002 = 0.0479[/tex]
0.0479 = 4.79% probability that fewer than 11 of them will vote
please help
no links please
Answer:
3rd choice
Step-by-step explanation:
multiply by 9/5: F=9/5 C
then add 32: F = 9/5C +32