g A pipe of negligible diameter is to be carried horizontally around a corner from a hallway 8 ft wide into a hallway 4 ft wide. What is the longest length that the pipe can be
Answers
The length of the longest pipe that can be carried horizontally is given by the minimum value of the function of the length of the pipe.
The longest length that the pipe can be is approximately 16.648 feet.
Reasons:
The length of the pipe, L
Width of the first hallway = 8 ft.
The width of the second hallway = 4 ft.
We have;
(L - a)·sin(θ) = 8
a·cos(θ) = 4
Therefore;
L - a = 8·csc(θ)
a = 4·sec(θ)
L(θ) = 8·csc(θ) + 4·sec(θ)
At the minimum length, we have;
L'(θ) = 4·sec(θ)·tan(θ) - 8·csc(θ)·cot(θ) = 0
4·sec(θ)·tan(θ) = 8·csc(θ)·cot(θ)
Therefore;
[tex]\displaystyle \frac{sec(\theta) \cdot tan(\theta)}{csc(\theta) \cdot cot(\theta)} = \frac{sin(\theta) \cdot tan(\theta) \cdot tan(\theta) }{cos(\theta) } =tan^3(\theta) = \frac{8}{4}[/tex]
[tex]\displaystyle tan(\theta) =\sqrt[3]{\frac{8}{4} } ={\sqrt[3]{2} }[/tex]
θ = arctan([tex]\sqrt[3]{2}[/tex]) ≈ 51.56°
a = 4×sec(arctan([tex]\sqrt[3]{2}[/tex]))
a ≈ 6.434
L = 8·csc(θ) + a = 8 × csc(arctan([tex]\sqrt[3]{2}[/tex])) + 4×sec(arctan([tex]\sqrt[3]{2}[/tex])) ≈ 16.648
The longest length that the pipe can be, L ≈ 16.648 ft.
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The longest length of the pipe is an illustration of trigonometry functions
The longest length of the pipe is 16.64 ft
The widths of the hallways are given as:
[tex]\mathbf{w_1 = 8}[/tex]
[tex]\mathbf{w_2 = 4}[/tex]
See attachment for the diagram that represents the scenario
From the diagram, we have:
[tex]\mathbf{sin(\theta) = \frac 4x}[/tex]
[tex]\mathbf{cos(\theta) = \frac 8y}[/tex]
Make x and y the subject
[tex]\mathbf{x = \frac 4{sin(\theta)}}[/tex]
[tex]\mathbf{y = \frac 8{cos(\theta)}}[/tex]
Where, the total length (L) of the pipe is:
[tex]\mathbf{L = x + y}[/tex]
So, we have:
[tex]\mathbf{L = \frac 4{sin(\theta)} + \frac 8{cos(\theta)}}[/tex]
Differentiate
[tex]\mathbf{L' = -\frac{4cos(\theta)}{sin^2(\theta)} + \frac{8sin(\theta)}{cos^2(\theta)}}[/tex]
Set to 0
[tex]\mathbf{-\frac{4cos(\theta)}{sin^2(\theta)} + \frac{8sin(\theta)}{cos^2(\theta)} = 0}[/tex]
Rewrite as:
[tex]\mathbf{\frac{4cos(\theta)}{sin^2(\theta)} = \frac{8sin(\theta)}{cos^2(\theta)}}[/tex]
Divide through by 4
[tex]\mathbf{\frac{cos(\theta)}{sin^2(\theta)} = \frac{2sin(\theta)}{cos^2(\theta)}}[/tex]
Cross multiply
[tex]\mathbf{2sin^3(\theta) = cos^3(\theta)}[/tex]
Divide both sides by [tex]\mathbf{2cos^3(\theta)}[/tex]
[tex]\mathbf{tan^3(\theta) = \frac 12}[/tex]
Take cube roots of both sides
[tex]\mathbf{tan(\theta) = \sqrt[3]{\frac 12}}[/tex]
[tex]\mathbf{tan(\theta) = 0.7937}[/tex]
Take arc tan of both sides
[tex]\mathbf{\theta = tan^{-1}(0.7937)}[/tex]
[tex]\mathbf{\theta = 38.4}[/tex]
Substitute [tex]\mathbf{\theta = 38.4}[/tex] in [tex]\mathbf{L = \frac 4{sin(\theta)} + \frac 8{cos(\theta)}}[/tex]
[tex]\mathbf{L = \frac 4{sin(38.4)} + \frac 8{cos(38.4)}}[/tex]
Evaluate
[tex]\mathbf{L = 6.43+ 10.21}}[/tex]
[tex]\mathbf{L = 16.64}}[/tex]
Hence, the longest length of the pipe is 16.64 ft
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[tex] {7}^{3} [/tex]
Answers
Answer:
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Step-by-step explanation:
[tex]7^{3} = 7 * 7 * 7[/tex]
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-Chetan K
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Answers
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Step-by-step explanation:
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Answers
Answer:
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Step-by-step explanation:
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Hello i need some help here the answer
A.x−3y=6 5x−y=10
B.3x−4y=24 2x+3y=−6
C.2x−9y=18 7x−2y=14
D.x−8y=16 5x+2y=20
Answers
9514 1404 393
Answer:
A. x −3y = 6; 5x −y = 10
Step-by-step explanation:
The first table gives the x- and y-intercepts, so you can write the equation for the line in intercept form as ...
x/a +y/b = 1 . . . . . . x-intercept = a; y-intercept = b
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The only matching equation is in answer choice A.
__
You can check that the second given equation in choice A matches the second table:
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Answers
Answer:
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Step-by-step explanation:
Step 1: Set up the expression.
[tex]\frac{612}{1-0.15}[/tex]Step 2: Simplify.
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Step-by-step explanation:
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Answers
Step-by-step explanation:
x = number of hours driven
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Answers
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Step-by-step explanation
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What is a linear equation in one variable ?A linear equation is an algebraic equation of highest degree 1 the number of variable is 1 also.
According to the given question Jane & Jim spent the same amount of money today.
Jim spent 5.25 dollars on food and rode the train 3 times, and Jane spent 12.45 dollars on food and rode the roller coaster twice.
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-6x + 30 < -18. Help pls
Answers
Hello, hope you are having a splendid day.
-6x+30<-18
First, move all the numbers to the right, using the opposite operation:
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Now, if we divide both sides by a negative variable, we change the inequlaity sign.
Hope it helps.
Ask me if you have any queries.
[tex]MagicalNature[/tex]
Good luck.
Answer:
[tex]\boxed{\boxed{\sf x>8}}[/tex]
Step-by-step explanation:
[tex]\sf -6x\:+\:30\:<\:-18[/tex]
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Answers
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Answers
Step-by-step explanation:
1.2 × 2.8
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Answer:
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I am hoping that this answer has satisfied your query and it will be able to help you in your endeavor, and if you would like, feel free to ask another question.
scientific form of 47500000000
Answers
[tex]▪▪▪▪▪▪▪▪▪▪▪▪▪ {\huge\mathfrak{Answer}}▪▪▪▪▪▪▪▪▪▪▪▪▪▪[/tex]
Scientific notation for above value is ~
[tex]4.75 \times 10 {}^{10} [/tex][tex]Hello[/tex] [tex]There![/tex]
First, let's make sure our coefficient is between 1 and 10:
4.75
All numbers in scientific notation look like this:
___x10^- (some number)
4.75x10^
Now, the number is large, so the exponent is positive.
So we move the decimal place some number of units. We move the decimal place 10 units, so the exponent is 10:
4.75x10^10
Hope this helps!
~Just a felicitous girlie
#CarryOnLearning
[tex]SilentNature[/tex]
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Answers
Answer:
83
Step-by-step explanation:
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Answers
Answer:
$0.97
Step-by-step explanation:
7328*100/108^(4*29)
a bunch of math later, with rounding...
0.97
*note-may be wrong
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Answers
Answer:
0.24
Step-by-step explanation:
There is nothing, after the hundredths (4), to round it, So 0.24 is already rounded to the nearest hundredths.
The number 0.024 after rounded off to the nearest hundred its become 0.02.
What is rounding off?Rounding numbers refers to changing a number's digits such that it approximates a value. The provided number is more simply represented by this value. For instance, 700,000 rather than 698,869 could be used to express a town's population.
Given that;
The number is,
⇒ 0.024
Now, After rounding off to the nearest hundred, we get;
⇒ 0.024 ≈ 0.02
( Because 4 is less than 5. )
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Consider the following equation.
x+4y=14
Determine the missing coordinate in the ordered pair (−2,?) so that it will satisfy the given equation.
Answers
Explanation:
We plug in x = -2 and solve for y
x+4y = 14
-2+4y = 14
4y = 14+2
4y = 16
y = 16/4
y = 4
Therefore, the point (-2, 4) is on the line x+4y = 14.
There are 10 girls in the class. This is three less thank half the number of boys . How many boys are in the class?
Answers
Answer:
there are 6 boys
Step-by-step explanation:
what i ls the value of
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Answers
Step-by-step explanation:
π can be taken as approximation of 22/7 ≈ 3.14
4. If the paint costs $30 per gallon and you need to spend $60 on supplies, and you can
save $45 per month, in how many months will you have enough money for the project?
Answers
Step-by-step explanation:
if I understand this right, then you need paint and additional supplies.
but I don't see any information indicating how many gallons of paint I will need.
so, I can go only with the 1 gallon specified.
so,
1×30 + 60 = $90
we can save $45 per month.
then we need
90 / 45 = 2
months to have enough money.
please update the numbers to how many gallons you really need, and then take that result and divide it by the 45 savings per month.