Solution :
Let x, y, z be the dimensions of the rectangle.
Volume of the rectangle (V) = xyz
Given that the vertex should lie on ellipse [tex]$4x^2 + y^2+4z^2=4$[/tex] .......(i)
So here the volume xyz must be maximum with constraints above.
We solve this using Lagranches method with variable λ
Lagranches function is
[tex]$F : xyz + \lambda(4x^2+y^2+4z^2-4)=0$[/tex] .....(ii)
To find λ, [tex]$\frac{dF}{dx}=0; \frac{dF}{dy}=0;\frac{dF}{dz}=0$[/tex]
[tex]$\frac{dF}{dx}=0 \Rightarrow yz+\lambda(16x)=0 \Rightarrow \lambda = -\frac{yz}{16x}$[/tex] .............(iii)
[tex]$\frac{dF}{dy}=0 \Rightarrow xz+\lambda(2y)=0 \Rightarrow \lambda = -\frac{xz}{2y}$[/tex] ..................(iv)
[tex]$\frac{dF}{dz}=0 \Rightarrow xy+\lambda(16z)=0 \Rightarrow \lambda = -\frac{xy}{16z}$[/tex] ...............(v)
Equating (iii) and (iv)
[tex]$-\frac{yz}{16x}=-\frac{xz}{2y} \Rightarrow y^2=8x^2 \Rightarrow y = \sqrt8 x$[/tex] ...............(vi)
Equating (iii) and (v)
[tex]$-\frac{yz}{16x}=-\frac{xy}{16z} \Rightarrow z^2=x^2 \Rightarrow z = x$[/tex] ....................(vii)
Substitute (vi) and (vii) in (i),
From (i),
[tex]$4x^2 + (\sqrt8 x)^2+4x^2 = 4$[/tex]
[tex]$\Rightarrow 4x^2 +8x^2+4x^2 = 4 \Rightarrow x = \frac{1}{4}$[/tex]
From (vi),
[tex]$y = \sqrt8 x$[/tex]
[tex]$\Rightarrow y= \sqrt8 \times \frac{1}{4} \Rightarrow y =\frac{\sqrt8}{4}$[/tex]
[tex]$\Rightarrow y =\frac{\sqrt{2\times4}}{4} \Rightarrow y = \frac{1}{\sqrt2}$[/tex]
From (vii),
z = x
[tex]$z =\frac{1}{4}$[/tex]
Therefore, for maximum volume the dimensions of a rectangle box are
[tex]$x =\frac{1}{4} ; y = \frac{1}{\sqrt2} ; z =\frac{1}{4}$[/tex]
What is the domain of the relation (8, -2), (4, -2), (-3, 2), (-5, -3)?
A. {8, 4, 3, -5}
B. {-8, -4, 3,5}
C. {-5, -3, 4, 8}
D. {-3, -2, 2}
Answer:
C
Step-by-step explanation:
Domain (and range) don't necessarily have to be in smallest to largest order unless your teacher requires it, so don't worry about the order.
Let's remind ourselves that the domain is the input value, or the x value, and y is the output value, or the y value. We know that in an ordered pair, the first # is x and the second # is y (because x comes before y in the alphabet). Lets look at our ordered pairs and see what the x values are.
{(8, -2), (4, -2), (-3, 2), (-5, -3)}
The x values are 8, 4, -3, and -5, therefore:
Domain: {8,4,-3,-5}
Range would equal all the y values. Remind yourself that if an x value is repeated to only write it down ONCE.
Remember, the order does not matter, so the answer would be C.
Hope this helped!
If someone can just answer the first question that would be so helpful plzzz
Answer:
I'm guessing that when you graph it, the graph changes direction when it's not on the x-axis. the "real zeros" of a graph are the ones that go through the x-axis and then imaginary ones are the ones that don't.
Step-by-step explanation:
The new owners of Bluegrass Natural Foods, Inc., have hired you to help them diagnose and cure problems that the company has had in maintaining adequate liquidity. As a first step, you perform a liquidity analysis. You then do an analysis of the company’s short-term activity ratios. 3
Your calculations and appropriate industry norms are listed.
Current ratio bluegrass 4.5 industry norms 4.0
Quick ratio bluegrass 2 industry norms 3.1
Inventory turnover bluegrass 6 industry norms 10.4
Average collection period bluegrass 73 days industry norms 52 days
Average payment period bluegrass 31 days industry norms 40 days
a. What recommendations relative to the amount and the handling of inventory could you make
to the new owners?
b. What recommendations relative to the amount and the handling of accounts receivable could
you make to the new owners?
c. What recommendations relative to the amount and the handling of accounts payable could you
make to the new owners?
Answer:
I don't understand make is more simplify
The graph y=f(x) is graphed below. What is the end behavior of f(x)?
Answer:
as x → -∞, y → ∞ and as x → ∞, y → -∞
Step-by-step explanation:
Just check the first quadrant of the given graph.
In the first quadrant, when x gets smaller, the value of y is increasing to positive infinity.
In other words, as x → -∞, y → ∞
Now, observe the behavior of the graph in the 4th quadrant.
In the 4th quadrant, when x gets larger, the value of y is decreasing to negative infinity.
In other words, as x → ∞, y → -∞
Therefore, the behavior of f(x):
as x → -∞, y → ∞ and as x → ∞, y → -∞
Therefore, option (C) is true.
The end behavior of a graph is defined as what is going on at the ends of each graph when the function approaches positive or negative infinity.
End behaviour of given graph is, x⇒ -∞ then y⇒∞ and x ⇒∞ then y⇒ -∞
Since, here a graph is attached.
From graph it is observed that, when x tends to negative side, graph is going to upward direction. and when x tends to positive side then graph is going to downward direction.
So, when x tends negative infinity then y tends to positive infinity.
when x tends to positive infinity then y tends to negative infinity.
Therefore, end behaviour of given graph are,
x⇒ -∞ then y⇒∞ and x ⇒∞ then y⇒ -∞
Learn more:
https://brainly.com/question/14962400
What is the predicted value of fuel efficiency (in miles per gallon) for a car with a 2.5 cubic liters engine? Round your answer to two decimals, if needed.
Answer:
25.92
Step-by-step explanation:
Fuel efficiency is the measure or effort that is required to convert the chemical energy in the fuel to the kinetic energy which the cars needs to move. It is meant by how good or effectively the fuel is utilized by the vehicle. It is the ability of the engine to extract more energy from the fuel.
In the context, the size of the engine is 2.5 cubic liters. Then the predicted value of the fuel efficiency will be 25.92 miles per gallon.
PLEASE HELP I WILL GIVE 100 points
Using the drawing below to answer the questions that follow
Part A what is the scale factor used when going from the larger rectangle to smaller one?
Part B
What is the length of missing side?
PLEASE HELP
Answer: the scale factor is 1/2
And the missing side is 5.5
Step-by-step explanation:
The larger triangle is twice the smaller triangle
The scale factor is 1/2 and the length of the missings side is 5.5 after using the concept of the scale factor.
What is a scale factor?The ratio among comparable dimensions of an object and a model with that object is known as an exponent in algebra. The replica will be larger if the scale factor is a whole number. The duplicate will be lowered if the step size is a fraction.
It is given that:
The two rectangles are shown in the picture:
Part A what is the scale factor used when going from the larger rectangle to the smaller one?
The scale factor = 8/16 = 1/2
The length of the missings side = 1/2(11) = 5.5
Thus, the scale factor is 1/2 and the length of the missings side is 5.5 after using the concept of the scale factor.
Learn more about the scale factor here:
brainly.com/question/22312172
#SPJ2
Which is smaller, one-ninth of 54 or one-fourth of 36?
Answer:
one-ninth of 54
Step-by-step explanation
54*1/9=6
36*1/4=9
Answer:
1/9 of 54 is smaller than 1/4 of 36
Step-by-step explanation:
1/9 of 54 is 54/9 and that is 6
1/4 of 36 is 36/4 and that is 9.
Please mark brainliest
Which of the following is equivalent to 12950+750-750
Answer:
12950
Step-by-step explanation:
Solve 11 < 3x+8 < 20.
Answer:
1<x<4
Step-by-step explanation:
this helps me out a lot in my math class
Answer:
x=4 and thats your answer thanks for helping you but if using the z-score it's 0
Step-by-step explanation:
1<x<4
Find the equation of the purple line.
Answer:
Y-3/5x+2
Step-by-step explanation:
determine whether the decimal is terminating repeating or irrational 0.4286
Answer:
irrational decimal
Step-by-step explanation:
It can't be repeated, can't go on forever, so it is an irrational decimal.
What is the value of x?
(6x – 1)º
(5x + 5)º
(4x + 8)º
(3x)º
(5x + 3)º
X =
Step-by-step explanation:
(1 point)
No, there isn't a more efficient way to solve this system.
Yes, a more efficient way is to multiply the first equation by 4, add to eliminate y, then solve for x.
Yes, a more efficient way is to multiply the first equation by −4, add to eliminate y, then solve for x.
Yes, a more efficient way is to multiply the first equation by −4, add to eliminate x, then solve for y.
Nineteen less than four times a number is greater than forty-seven.
Please i will give away all my points to the person that answers this in number form thanks
Answer: pretty sure it's the number you multiply by 4 is 17
Step-by-step explanation:
when you multiply 4 and 16 you'll get 64 then you subtract 19 and get 45 which is less than 47 so by reasoning if you were to multiply 4 and 17 you'll get 68 then you subtract 19 from 68 and you should get 49 ( you didn't include how much greater than 47 the number should be but the safest bet would be to use 17)
1998 hurricane Bonnie approached the United States at a speed of 8 miles per hour . The function y= 8x disproves how many miles y hurricane Bonnie traveled x hours . Use the graph to estimate how far the hurricane traveled in 2.5 hours
Answer:
y=20
Step-by-step explanation:
8x
8(2.5)
8×2=16
8×0.5=4
16+4=20
y=20 in this equation
(11 + 42 - 5) / (11 - 4) (11+42-5) 48/(11-4) 48/7
Answer:
= -24576/7
or
-3510.857142
both are the correct answer
Most living things need oxygen to survive you breathe in oxygen and it is carried to this Organ. here the red blood cells pick up the oxygen and carry the oxygen to all the body cells that need it what is this organ
Answer:
The lungs
Step-by-step explanation:
becuase biology
Hy can you figure out
(100 x 50) - (1/8 x 50) ÷ by 2?
Answer:
4996.875 or 39975/8
Step-by-step explanation:
brainly pls
how to find rations in triangles
Answer:
When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangles.
Step-by-step explanation:
Would appreciate some help first to answer gets brainliest
If $2000 is deposited into an account paying simple interest of 6% per year on a monthly basis, in how many months will this account increase by $90?
A. 22
B. 15
C. 9
D. 4
the reciprocal of 3 - 1/5 is
Answer:
[tex]\frac{5}{14}[/tex]
Step-by-step explanation:
First subtract the two terms by setting the first term as a fraction.
3 * [tex]\frac{5}{5}[/tex] = [tex]\frac{15}{5}[/tex]
Then subtract the terms.
[tex]\frac{15}{5}[/tex][tex]-\frac{1}{5}[/tex] = [tex]\frac{14}{5}[/tex]
Finally, flip the numerator and the denominator.
A package states that there are 60 calories in 12 crackers and 75 calories in 15 crackers. Since the relationship is
proportional, how many calories are there in 180 crackers?
x х
12
15
y
60
75
36 calories
108 calories
720 calories
900 calories
PLEASE HELP QUICK!!
HELP PLEASE!!!! WILL GIVE 100 POINTS
Answer:
w to r to a
Step-by-step explanation:
e to c to p
Answer:
w to r to a
Step-by-step explanation:
A flagpole casts a shadow that is 24 ft long at the same time you cast a shadow that is
4.5 ft long. If you are 6 ft tall, how tall is the flagpole?
Answer:
32 ft tall
Step-by-step explanation:
24 divided by 4.5 is 5.3 repeating. 6 times 5.3 is around 32
please help ALGRBRA II 10 points
Solve the equation.
8 (2-6v) +19=-61
Answer:
v = 2
Explanation:
8(2 − 6v) + 19 = −61
[ Simplify both sides of the equation ]
8(2 − 6v) + 19 = −61
(8)(2) + (8)(−6v) + 19 = −61 (Distribute)
16 + −48v + 19 = −61
(−48v) + (16 + 19) = −61 (Combine Like Terms)
−48v + 35 = −61
−48v + 35 = −61
[ Subtract 35 from both sides ]
−48v + 35 − 35 = −61 − 35
−48v = −96
[ Divide both sides by -48 ]
−48v / -48 = −96 / -48
v = 2
- PNW
Ariel bought 3 raffle tickets for $5. How many tickets could Ariel
buy for $15?
Answer:
9 raffle tickets
Step-by-step explanation:
Because 3 raffle tickets are $5
and to find out the number of tickets you can buy with $15 you have to
divide $15 by $5 which is 3
then multiply the 3 times 3 to get 9 raffle tickets
Have a great day, I hope this is helpful to you.
Answer:
9
Step-by-step explanation:
3=5 6=10 9=15
What is the slope of line that passes through the points (-3, 5) and (0, -1).
Answer:
[tex]m=-2[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Slope Formula: [tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Step-by-step explanation:
Step 1: Define
Point (-3, 5)
Point (0, -1)
Step 2: Find slope m
Simply plug in the 2 coordinates into the slope formula to find slope m.
Substitute [SF]: [tex]m=\frac{-1-5}{0+3}[/tex]Subtract/Add: [tex]m=\frac{-6}{3}[/tex]Divide: [tex]m=-2[/tex]Which of the following inequalities is correct?
Answer:
B
Step-by-step explanation:
Solve the equation.
x² + 6x+15=0
Answer:
x = -3 + i√6 and x = -3 - i√6
Step-by-step explanation:
Let's apply the "completing the square" method to find the roots of this equation.
Take half of the coefficient 6 of x, square it and add this result to x^2 + 6x. Then subtract the same quantity:
x^2 + 6x + 15 becomes
x^2 + 6x + 3^2 - 3^2 + 15 = 0
Rewriting the first three terms as the square of a binomial, we ge:
(x + 3)^2 - 9 + 15 = 0, which simplifies to:
(x + 3)^2 + 6 = 0, or (x + 3)^2 = -6
Taking the square root of both sides:
x + 3 = ±i√6
Then the two roots are complex:
x = -3 + i√6 and x = -3 - i√6
