Let (x, y, z) be a point on the plane in the first octant. The box formed by this point has volume xyz, and you want to maximize this subject to the equation of the plane.
Use the method of Lagrange multipliers: the Lagrangian is
L(x, y, z) = xyz - λ (x + 2y + 3z - 6)
Find its critical points:
∂L/∂x = yz - λ = 0
∂L/∂y = xz - 2λ = 0
∂L/∂z = xy - 3λ = 0
∂L/∂λ = -(x + 2y + 3z - 6) = 0
Solving the first three equations for λ gives
λ = yz = xz/2 = xy/3
Solve these equations for y and z :
• yz = xz/2 => y = x/2 => 2y = x
• yz = xy/3 => z = x/3 => 3z = x
Substitute these solutions into the last equation and solve for x, then again for y and z :
x + 2y + 3z - 6 = 3x - 6 = 0 => 3x = 6 => x = 2, y = 1, z = 2/3
At this critical point, the maximum volume is
xyz = 2*1*2/3 = 4/3
Fill in the blank with the correct fraction. Make sure to express your answer in its simplest, most reduced form.
Answer:
3/4
Step-by-step explanation:
as 1 hour is 60 minutes, to express in fraction form, we have to divide.
so
45/60 can be simplified to 3/ 4 when 15 is taken out as common of both. so the answer is 3/4
Answer:
3/4
Step-by-step explanation:
1 hour= 60 minutes
45/60
simplify the fraction by dividing the numerator and denominator by 5
45÷5=9 and 60÷5=12
new fraction =9/12
simplify 9/12 by dividing the numerator and denominator by 3
9÷3=3 and 12÷3=4
Ans.=3/4
45 minutes is 3/4 of an hour
ow many three-letter "words" can be made from 8 letters "FGHIJKLM" if repetition of letters
Answer:
336
Step-by-step explanation:
A total of 512 possible words can be made.
What is a expression? What is a mathematical equation? What is Equation Modelling?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have a 8 letters word "FGHIJKLM"
Assume that there are [y] possible combinations. Then, the total number of [x] digit words that can be made are -
y = 8ˣ
y = 8ˣ
y = 8 x 8 x 8
y = 512
Therefore, a total of 512 possible words can be made.
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Really struggling and stressing out. Anyone mind helping?
Answer:
it should be 314.16, so I would choose 314
There are 20 rows of seats on a concert hall: 25 seats are in the 1st row, 28 seats on the 2nd row, 31 seats on the third row, and so on. How many seats are in the last row?
Answer:
It is a sequence topic.
The general term of the sequence is 25+3(n-1) where n is the number of the term (here it is the row no.)
So the 20th term is the term with n=20. Substitute n=20 into the above general term, we will get 25+3(20-1) .
Please go on by yourself to work out the answer.
The equation of the graphed line is x+2y=5. What is the x-intercept of the graph?
A coordinate plane with a line passing through (negative 5, 5) and (5, 0).
2
2.5
5
5.5
Answer:
5 is the answer. hhhjjh y fl jj g gshwhwjqjajajw
Answer:
c
Step-by-step explanation:
it's 5
Which table represents a linear function?
Answer:
Option 3 (C)
Step-by-step explanation:
It is the only one that changes the same amount every time ( times 2 )
Determine the equation of the circle graphed below.
Answer:
(x - 6)^2 + (y + 4)^2 = 4
Step-by-step explanation:
formula for a circle
( x - h )^2 + ( y - k )^2 = r^2
(h,k) is the center
r is the radius
(6,-4) is center
2 is radius
Substitute in numbers and watch the negatives.
Find the surface area of the square pyramid 8mm 6mm
Answer:
136 mm²
Step-by-step explanation:
[tex]A=a^{2} +2a\sqrt{\frac{a^{2} }{4} } +h^{2}[/tex]
[tex]A=6^{2} +2(6)\sqrt{\frac{6^{2} }{4} } +8^{2}[/tex]
[tex]A=36 +12\sqrt{\frac{36 }{4} } +64[/tex]
[tex]A=36 +12\sqrt{9 } +64[/tex]
[tex]A=36 +12(3)+64[/tex]
[tex]A=36 +36+64[/tex]
A = 136
how to find the area of a circle with the diameter 21
Answer:
A ≈ 346.36
Step-by-step explanation:
To find the answer, you must use the formula, A=πr²
Since only the diameter was provided, you need to divide it by 2 to find the radius. 21 divided by 2 equals 10.5, so that will be your radius.
Another formula that can be used while using only the diameter would be A= 1/4 π d²
Please can someone answer quickly
4x-2 3x+14 how do I find x?
Answer:
x = 16
Step-by-step explanation:
4x - 2 = 3x + 14
4x - 2 + 2 = 3x + 14 + 2
4x = 3x + 16
4x - 3x = 3x - 3x + 16
x = 16
if arc QR=3x+38 degrees , arc PS=5x-10 degrees ,and
Answer:
Arc QR = 111.5°
Step-by-step explanation:
Given:
Arc QR = (3x + 38)°
Arc PS = (5x - 10)°
m<QMP = 68°
Required:
Measure of arc QR
Solution:
Recall: Angles of intersecting chords theorem states that the angle formed equals half of the sum of the intercepted arcs
This by implication means:
m<1 = ½(arc QR + arc PS)
We are given arc QR, and arc PS.
m<1 = 180° - m<QMP (angles on a straight line)
m<1 = 180° - 68°
m<1 = 112°
Plug in the values into the equation and solve for x
112° = ½(3x + 38 + 5x - 10)
Multiply both sides by 2
2*112 = 3x + 38 + 5x - 10
224 = 3x + 38 + 5x - 10
Add like terms
224 = 8x + 28
224 - 28 = 8x + 28 - 28
196 = 8x
196/8 = 8x/8
24.5 = x
x = 24.5
✅Let's find arc QR
Arc QR = (3x + 38)°
Plug in the value of x
Arc QR = 3(24.5) + 38 = 73.5 + 38
Arc QR = 111.5°
Suppose that for a certain company C(x)=35x+300,000 represents the total cost function, and R(x)=75x represents the total revenue function. Find the total-profit function and break-even point.
Which of the following functions represents the total-profit function?
A) 40x-300,000
B) 40x+300,000
C) 110x+300,000
Answer:
40x - 300,000
7,500
Step-by-step explanation:
profit = revenue - cost
p(x) = 75x - (35x + 300,000)
p(x) = 40x - 300,000
Break even is when profit is zero
40x - 300,000 = 0
40x = 300,000
x = 7,500
The functions that represents the total-profit function would be p(x) = 40x - 300,000. so the correct option is A.
What is a function?Function is a type of relation, or rule, that maps one input to specific single output.
Given that for a certain company C(x) =35x+300,000 represents; the total cost function, and R(x)=75x represents the total revenue function.
WE can calculate the profit by the following formula;
Profit = revenue - cost
p(x) = 75x - (35x + 300,000)
p(x) = 40x - 300,000
Now Break even is when profit is zero,
40x - 300,000 = 0
40x = 300,000
x = 7,500
Hence, the break-even point is 7500.
Therefore, the functions that represents the total-profit function would be p(x) = 40x - 300,000. so the correct option is A.
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D(p) is the price, in dollars per unit, that consumers are willing to pay for p units of an item, and S(p) is the price, in dollars per unit, that producers are willing to accept for p units. Find the equilibrium point. D(p)=3500−20p, S(p)=2500+5p
What are the coordinates of the equilibrium point?
Answer:
(40,2700)
Step-by-step explanation:
Given
[tex]D(p)=3500-20p[/tex]
[tex]S(p)=2500+5p[/tex]
Required
The equilibrium point
To do this, we have:
[tex]D(p) = S(p)[/tex]
So, we have:
[tex]2500+5p = 3500-20p[/tex]
Collect like terms
[tex]20p+5p = 3500-2500[/tex]
[tex]25p = 1000[/tex]
Divide both sides by 25
[tex]p = 40[/tex]
Substitute [tex]p = 40[/tex] in [tex]D(p)=3500-20p[/tex]
[tex]D(40) = 3500 - 20 * 40[/tex]
[tex]D(40) = 2700[/tex]
Hence, the coordinates of the equilibrium is: (40,2700)
Find the inner product for (4,8) (6,-3) and state whether the vectors are perpendicular.
Answer:
[tex] = \binom{4}{8} . \binom{6}{ - 3} \\ = (4 \times 6) + (8 \times - 3) \\ 24 + ( - 24) \\ = 0 \\ since \: inner \: product \: is \: zero : \\ the \: vectors \: are \: perpendicular[/tex]
Answer:
0
The vectors are perpendicular
Step-by-step explanation:
(4,8) (6,-3)
The inner product is also known as the dot product
4*6 + 8*-3 = 24-24 = 0
The vectors are perpendicular since the dot product is zero
show how to solve 1/4 (4 + x) = 4/3
pls solve for 2 brainliest
Plzz help find the value of angle MHA and MHT
Answer:
Step-by-step explanation:
2x+11+6x-7=180 (being linear pair)
8x+4=180
8x=180-4
8x=176
x=176/8
x=22
MHA=2x+11
=2*22+11
=44+11
55
MHT=6x-7
=6*22-7
=132-7
=125
=
Answer:
2X+11+6X-7 =180
8X+4 =180
8X. =180-4
8X. =176
X. =22
MHA=2X+11
=44+11
=55
MHT=6X-7
=132-7
=125
5. In the figure Find X (In the picture) (giving points to best answer/brainlest)
Answer:
x = 93
Step-by-step explanation:
*for angle labels, see picture below
Using angle addition postulate, we can say that angle EAC is the sum of angle CAB and angle BAE:
m<EAC = m<CAB + m<BAE
m<EAC = x + 39
Since line l and m are parallel, you can say that angle ACD is an alternate interior angle to angle EAC. This means they are equal.
Now, we can write an equation to solve for x:
m<EAC = m<ACD
x + 39 = 132
x = 93
Answer:
x=93°
Step-by-step explanation:
If l ıı m
l=180° ; 132°+?(y) =180° (Co-interior angle)
Find the value of y
132°+y=180°(Transpose)
y=180°-132°
y=48°
...
Find x
39°+x+48°=180°(Transpose)
x=180°-48°-39°
x=93°
combine like terms to simplify the expression of -5.55-8.55c+4.35c
Answer:
-5.55-4.2c
Step-by-step explanation:
-5.55-8.55c+4.35c
-8.55c and 4.35c are like terms so you have to combine them.
-8.55c+4.35c=-4.2c
-5.55 is NOT a like term so it is on its own.
final answer -> -5.55-4.2c
*** don't forget -5.55 is negitive ***
hope this helped negative:)
One of the legs of a right triangle measures 15 cm and the other leg measures 6 cm.
Find the measure of the hypotenuse. If necessary, round to the nearest tenth.
Answer:
16.2 cm
Step-by-step explanation:
use the pythagoran theorem
a² + b² = c²
15² + 6² = c²
225 + 36 = c²
261 = c²
Take the square root of both sides
16.1554944214 = c
Rounded
16.2 cm
Sylvia is twice as old as her brother. Find their ages now if in seven years her brother will be what her age was last year.
Answer:
Sylvia=16
brother=8
Step-by-step explanation:
8+7=15
16-1=15
Hurry which one ITS NOT 270
A.84
C.128
D.540
Answer:
84 cm^2
Step-by-step explanation:
The area of a triangle is
A = 1/2 bh where b is the length of the base and h is the height
A = 1/2 ( 5+9) * 12
A = 1/2 (14) * 12
A =84
given sin x =-4/5 and x is in quadrent 3, what is the value of tan x/2
Answer:
We can write sin x in terms of tan x/2 using the formula:
⇒ sin x = (2 tan (x/2)) / (1 + tan2(x/2))
Therefore, using the above formula, we can find the values of tan x/2 by putting the value of sin x.
⇒ -4/5 = (2 tan (x/2)) / (1 + tan2(x/2))
Now, if we replace tan (x/2) by y, we get a quadratic equation:
⇒ 0.8y2 + 2y + 0.8 = 0
⇒ 2y2 + 5y + 2 = 0
By using the quadratic formula, we get y = -0.5, -2
Hence, the value of tan (x/2) = -0.5, -2
Now, we have two solutions of tan (x/2).
Now, let's check for the ideal solution using the formula tan x = (2 tan (x/2)) / (1 - tan2(x/2)).
For tan (x/2) = -0.5:
⇒ tan x = 2(-0.5) / 1 - (-0.5)2 = -4/3
It is also given that x lies in the third quadrant. We know that tan is positive in the third quadrant, and here we get tan x = -4/3 which is negative.
Hence, we can say that tan (x/2) = -0.5 is not a correct solution. Hence it is rejected.
Now let's check for tan (x/2) = -2.
⇒ tan x = 2(-2) / 1 - (-2)2 = 4/3
Here, we get tan x = 4/3 which is positive.
Hence, we can say that tan (x/2) = -2 is a correct solution.
Exponential Growth
You are a financial advisor You have a client who is 25 years old and has $10,000 they want to invest into a retirement account. After investing this money, the client will not be adding any money to the account. The money will just let it grow until the client reaches age 65 As you are helping them decide the best investment for their money, you discover the following options.
I also need account C and account D
help me please
Answer:
compounded daily as this added up would be the best
Step-by-step explanation:
In a political science class, the teacher gives a midterm exam and a final. The association between midterm and final scores is linear. The summary statistics are shown below. Midterm Mean=75, Midterm Standard Deviation=10 Final Mean=75, Final Standard Deviation=10 r=0.8 According to the regression equation, for a student who gets 85 on the midterm (one standard deviation above average) what is the predicted final exam grade?
Answer:
The predicted final exam grade is 83
Step-by-step explanation:
Given
[tex]\bar x = 75[/tex] -- midterm mean
[tex]\sigma_x = 10[/tex] --- midterm standard deviation
[tex]\mu =75[/tex] --- final mean
[tex]\sigma = 10[/tex] --- final standard deviation
[tex]r = 0.8[/tex]
Required
The predicted final grade of student who score 85 in midterm
The prediction is represented as:
[tex]y = \alpha + \beta x[/tex]
Where:
[tex]\beta = r * \frac{\sigma}{\sigma_x}[/tex]
[tex]\beta = 0.8 * \frac{10}{10}[/tex]
[tex]\beta = 0.8 * 1[/tex]
[tex]\beta = 0.8[/tex]
and
[tex]\alpha = \mu - r * \bar x[/tex]
[tex]\alpha = 75- 0.8 * 75[/tex]
[tex]\alpha = 75- 60[/tex]
[tex]\alpha = 15[/tex]
So:
[tex]y = \alpha + \beta x[/tex]
[tex]y = 15 + 0.8x[/tex]
For a student who gets 85, the prediction is:
[tex]y = 15 + 0.8*85[/tex]
[tex]y = 15 + 68[/tex]
[tex]y = 83[/tex]
The symbols for elements with accepted names Group of answer choices consist of a single capital letter consist of a capital letter and a small letter consist of either a single capital letter or a capital letter and a small letter no answer is correct
Answer:
consist of either a single capital letter or a capital letter and a small letter.
Step-by-step explanation:
A chemical reaction can be defined as a reaction in which two or more atoms of a chemical element react to form a chemical compound.
In Chemistry, a chemical element can be defined as any pure substance that is typically made up of only atoms and cannot be broken down into simpler substances through chemical processes. Thus, the atomic number (the number of protons in the nuclei of an atom) of a particular chemical element distinguishes from other chemical elements.
Generally, all chemical elements are denoted or represented by a symbol, which may either be single capital letter or a capital letter and a small letter.
This ultimately implies that, the symbols for elements with accepted names consist of either a single capital letter or a capital letter and a small letter. For example, the symbol for sodium is Na, copper is Cu, carbon is C, oxygen is O, iron is Fe, nitrogen is N, magnesium is Mg, potassium is K, argon is Ag, hydrogen is H, helium is He, phosphorus is P, etc.
Can anyone help me solve this
[tex]\longrightarrow{\green{ D. \:3 {a}^{4} \sqrt{2a} }}[/tex]
[tex]\large\mathfrak{{\pmb{\underline{\red{Step-by-step\:explanation}}{\red{:}}}}}[/tex]
[tex] \sqrt{18 {a}^{9} } \\ \\ ➝ \: \sqrt{2 \times 3 \times 3 \times {a}^{9} } \\ \\ ➝ \: \sqrt{2 \times ({3})^{2} \times {( {a}^{4}) }^{2} a } \\ \\ [∵( { {a}^{4} )}^{2} a = {a}^{4 \times 2 + 1} = {a}^{9}] \\ \\ ➝ \: 3 \times {a}^{4} \sqrt{2a} \\ \\ ➝ \: 3 {a}^{4} \sqrt{2a} [/tex]
[tex]\pink{\large\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese \: \: Mystique35}}}}}[/tex]
Answer:
The answer is [tex]3a^{4}\sqrt{2a}[/tex]
Step-by-step explanation:
To simplify the radical, start by factoring 9 out of 18 for step 1. Next, for step 2, rewrite 9 as [tex]3^{2}[/tex]. Then, factor out [tex]a^{8}[/tex] for step 3. For step 4, rewrite [tex]a^{8}[/tex] as [tex](a^{4})^{2}[/tex]. Then, for step 5, move the 2 in the radical. Rewrite [tex]3^{2}(a^{4})^{2}[/tex]as [tex](3a^{4})^{2}[/tex] for step 6. Then, add parentheses to the radical for step 7. Finally, for step 8 pull the terms out from under the radical, and the answer is [tex]3a^{4}\sqrt{2a}[/tex].
Step 1: [tex]\sqrt{9(2)a^{9} }[/tex]
Step 2: [tex]\sqrt{3^{2}*2a^{9}[/tex]
Step 3: [tex]\sqrt{3^{2}*2(a^{8}a) }[/tex]
Step 4: [tex]\sqrt{3^{2}*2((a^{4})^{2}) }[/tex]
Step 5: [tex]\sqrt{3^{2}(a^{4})^{2}*2a }[/tex]
Step 6: [tex]\sqrt{(3a^{4} )^{2}*2a }[/tex]
Step 7: [tex]\sqrt{(3a^{4})^{2}*(2a) }[/tex]
Step 8: [tex]3a^{4}\sqrt{2a}[/tex]
Consider this right triangle. What is the value of sin(B)?
Answer:
[tex] \frac{3}{5} [/tex]
Step-by-step explanation:
using SOHCAHTOA
[tex]sin = \frac{opposite}{hypoteneus} [/tex]
Therefore
[tex]sin = \frac{3}{5} [/tex]
The value of sin B is 3/5.
What is Trigonometry?Trigonometry is one of the branches of mathematics which deals with the relationship between the sides of a triangle (right triangle) with its angles. There are 6 trigonometric functions for which the relation between sides and angles are defined.
We have,
P = 3
B = 4
H = 5
Using Trigonometry
sin B = P /H
sin B = 3/5
Thus, the value of sin B is 3/5.
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A laptop company claims up to 8.1 hours of wireless web usage for its newest laptop battery life. However, reviews on this laptop shows many complaints about low battery life. A survey on battery life reported by customers shows that it follows a normal distribution with mean 7.5 hours and standard deviation 27 minutes. (a) What is the probability that the battery life is at least 8.1 hours
Answer:
0.0918 = 9.18% probability that the battery life is at least 8.1 hours.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean 7.5 hours
This means that [tex]\mu = 7.5[/tex]
Standard deviation 27 minutes.
An hour has 60 minutes, which means that [tex]\sigma = \frac{27}{60} = 0.45[/tex]
What is the probability that the battery life is at least 8.1 hours?
This is 1 subtracted by the p-value of Z when X = 8.1. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{8.1 - 7.5}{0.45}[/tex]
[tex]Z = 1.33[/tex]
[tex]Z = 1.33[/tex] has a p-value of 0.9082.
1 - 0.9082 = 0.0918
0.0918 = 9.18% probability that the battery life is at least 8.1 hours.
Find the Value of x from the angles
[tex]\sf \bf {\boxed {\mathbb {TO\:FIND:}}}[/tex]
The value of [tex]x°[/tex].
[tex]\sf \bf {\boxed {\mathbb {SOLUTION :}}}[/tex]
[tex]\longrightarrow{\blue{ x°\:=\:35° }}[/tex]
[tex]\sf \bf {\boxed {\mathbb {STEP-BY-STEP\:EXPLANATION:}}}[/tex]
We know that,
[tex]\sf\pink{Sum\:of\:angles\:of\:a\:triangle\:=\:180°}[/tex]
➪ [tex]x°[/tex] + 125° + 20° = 180°
➪ [tex]x°[/tex] + 145° = 180°
➪ [tex]x°[/tex] = 180° - 145°
➪ [tex]x°[/tex] = 35°
Therefore, the value of [tex]x°[/tex] is 35°.
Now, the three angles of the triangle are 35°, 125° and 20°.
[tex]\sf \bf {\boxed {\mathbb {TO\:VERIFY:}}}[/tex]
✒ 35° + 125° + 20° = 180°
✒ 180° = 180°
✒ L. H. S. = R. H. S.
Hence verified.
[tex]\huge{\textbf{\textsf{{\orange{My}}{\blue{st}}{\pink{iq}}{\purple{ue}}{\red{35}}{\green{ヅ}}}}}[/tex]
