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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Related Questions
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
The x-value with y=0 is a=6; the y-value with x=0 is b=-2. So, the equation of the line for the first table is ...
x/6 +y/-2 = 1
Multiplying by 6 gives ...
x -3y = 6
The only matching equation is in answer choice A.
__
You can check that the second given equation in choice A matches the second table:
for (x, y) = (-3, -25), you have ...
5(-3) -(-25) = -15 +25 = 10 . . . . as required
) Which value is equivalent
[tex] {7}^{3} [/tex]
Answers
Answer:
343
Step-by-step explanation:
[tex]7^{3} = 7 * 7 * 7[/tex]
[tex]7^3 = 343[/tex]
-Chetan K
A desk is on sale for 15% off. The sale price is $612.
What is the regular price?
Answers
Answer:
$720
Step-by-step explanation:
Step 1: Set up the expression.
[tex]\frac{612}{1-0.15}[/tex]Step 2: Simplify.
[tex]\frac{612}{0.85}[/tex]Step 3: Convert 0.85 into a fraction.
[tex]612/ \frac{85}{100}[/tex] [tex]612/ \frac{17}{20}[/tex]Step 4: Multiply 612 by reciprocal of 17/20.
[tex]612 * \frac{20}{17}[/tex] [tex]36 * 20[/tex] [tex]720[/tex]Therefore, the regular price was $720.
(1 point)
Write the complex number z= -7-10i in polar form: z = r(cos (theta)+ i sin (theta)) where r= and 0 =
The angle should satisfy 0<=theta<=2pi
Answers
Answer:
When a complex number z is written in its polar form,z = r(cos theta + i sin theta), the nonnegative number r is called the magnitude, or modulus, of z.
In mathematics, magnitude is the size of a mathematical object, a property by which the object can be compared as larger as or smaller than other objects of the same kind. More formally, an object's magnitude is an ordering (or ranking) of the class of objects to which it belongs.
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.
what i ls the value of
[tex]\pi[/tex]
Answers
Step-by-step explanation:
π can be taken as approximation of 22/7 ≈ 3.14
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]
What is 024 rounded off to the nearest 100?
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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The high temperatures for the first five days in December were as follows 28, 3, -2, -4 and 12. What is the average temperature.
Answers
Answer:
28 +3 +2+4+12=49
49÷5=9.8
Find the product using formula ...
formula = ( a + b ) ( a - b )
a) 1.2 × 2.8
Answers
Step-by-step explanation:
1.2 × 2.8
(2 - 0.8 )(2 + 0.8)
2² - (0.8)²
4 - 0.64
3.36
[tex]1.2 \times 2.8 \\ = (2 - 0.8)(2 + 0.8) \\ = {(2)}^{2} - {(0.8)}^{2} \\ = 4 - 0.64 \\ = 3.36[/tex]
Answer:3.36
Hope it helps.
Do comment if you have any query
29 years ago, James put some money into a savings account with 8% interest compounded quarterly (4 times a year). He just remembered it and saw that there was now $7328 in it. Find the original amount he invested.
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
Simplify the expression.5(z+4)+5(2−z)
Answers
Answer:
30
Step-by-step explanation:
Hope this helps!!! :))
Find the missing number in the following series: __ 85 81 83 79 81 77
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Answer:
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Step-by-step explanation:
The pattern is -4 then +2
Please help! Will give 50 points!!
Answers
Answer:
- 5 | - 1 | 3 | 7Step-by-step explanation:
Given function:
y = 2x + 3Complete table by finding y values:
x = - 4 ⇒ y = 2*(-4) + 3 = - 8 + 3 = - 5x = - 2 ⇒ y = 2*(-2) + 3 = - 4 + 3 = - 1x = 0 ⇒ y = 2*0 + 3 = 0 + 3 = 3x = 2 ⇒ y = 2*2 + 3 = 4 + 3 = 7Consider 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.
Latonya swims 50 meters in 1/2 minute. What is the slope
Answers
Answer: 100
Step-by-step explanation: 50 x 1/2=100
Pls answer the rest for me I’ll give you brainilist
Answers
Answer:
29 x 8 = 191
62 x 19 = 1,178
8.7 x 12 = 104.4
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Jane & Jim spent the same amount of money today. If Jim spent
$5.25 on food and rode the train 3 times, and Jane spent $12.45 on
food and rode the roller coaster twice, then how much does a
token cost?
Answers
Answer:4$
Step-by-step explanation
The cost of one token is 7.2 dollars.
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.
Assuming cost of a token to be T.
∴ 5.25 + 3T = 12.45 + 2T.
3T - 2T = 12.45 - 5.25
T = 7.2.
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4. If the paint costs $30 per gallon and you need to spend $60 on supplies, and you can
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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.
what is the definition of fixed percent?
Answers
Answer:
the quotient, fixed as a percentage
Step-by-step explanation:
Answer:
Fixed Percentage means, the quotient, expressed as a percentage, obtained by dividing the unexpended and not withdrawn amount of the Allocated Amount, at the time a Net Profits Interest is assigned by the total of (i) the unexpended and not withdrawn amount of the Allocated Amount and (ii) the unexpended and not.
I drove to the beach at a rate of 40 miles per hour. If I had driven at a rate of 30 miles per hour instead, then I would have arrived 25 minutes later. How many miles did I drive?
Answers
Step-by-step explanation:
x = number of hours driven
25 minutes = 5/12 hour
remember,
1 hour = 60 Minuten
5 minutes = 60/12 minutes = 1/12 hour
25 minutes = 5×5 minutes = 5/12 hour
x hours × 40 miles/hour = (x + 5/12) hours × 30 miles/hour
the dimensions "hour" eliminate each other from the top and the bottom of the fractions leaving only miles. and the miles must be the same either way.
40x = (x + 5/12)×30 = 30x + 150/12
10x = 150/12
x = 15/12 = 5/4 hours
40 miles/hour × 5/4 hours = 40×5/4 miles = 10×5 = 50 miles.
I drove 50 miles.
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Answers
Answer:
there are 6 boys
Step-by-step explanation:
Each square below represents one whole …. What percent is represented by the shaded area?
Answers
Answer:
3/4
Step-by-step explanation:
because there are 4 parts and only three are shaded so 3/4
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Answer soon please 10 points Easy math
Answers
Answer:
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Step-by-step explanation:
yeah, this one was fun.
Allison is buying two birthday cards for $5.78. Sales tax on the two cards is $0.38. Including sales tax, what is the cost of each card?
1. $2.89
2. $3.27
3. $3.80
4. $3.08
Answers
Answer:
1.$2.89
Step-by-step explanation:
1/3 off $ 0.38 $ 0.25
2 for 1 at $ 5.78 $ 2.89
What is the interior degree measure of angle A?
Answers
Answer:
in the pictureStep-by-step explanation:
opposite corners first, then adjacent corners, and so on
-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:
-6x<-18-30
-6x<-48
Divide both sides by -6:
x>8
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]
Subtract 30 from both sides:
[tex]\sf -6x+30-30<-18-30[/tex][tex]\sf -6x<-48[/tex]Multiply both sides by -1 :
[tex]\sf \left(-6x\right)\left(-1\right)>\left(-48\right)\left(-1\right)[/tex][tex]\sf 6x>48[/tex]Divide both sides by 6:
[tex]\sf \cfrac{6x}{6}>\cfrac{48}{6}[/tex][tex]\sf x>8[/tex]___________________________________
the question is in the image!
Answers
Answer:
g ≤ 6
Step-by-step explanation:
The number of goldfish that can live in the tank is less than or equal to 6.
How many ? yards = 6 miles
Answers
Answer:
6 miles = 10560 yards
Step-by-step explanation:
To calculate a mile value to the corresponding value in yard, just multiply the quantity in mile by 1760 (the conversion factor). Here is the formula:
Value in yards = value in mile × 1760
Suppose you want to convert 6 mile into yards. Using the conversion formula above, you will get:
Value in yard = 6 × 1760 = 10560 yards
Write an equation for the hangers below and solve (will give brainless if correct!)
Answers
Answer:
1) 5x+2=17
2) 5(2+x)=17
Step-by-step explanation:
1) subtract 2 from both of the sides, then you get 5x=15. Divide 5 on both sides, then you get X=3
2) Use distributive property and multiply 5 and 2 and you get 10, then do the same thing with the x. You should get 10+5x=17. Subtract 10 from both sides. Then you are left with 5x=17, divide 5 on both sides you should get x=7/5
Hope this helps!
A father wants to buy a total of five milk drinks for his son and spend $7.95. Eggnog costs $1.55 and a peanut punch costs $1.65. Determine the number of each milk drink bought
Answers
Answer:
2
Step-by-step explanation:
$ 1.55 + $1.65 = $3.11
the budget is $7.95
$7.95 ÷ $3.11 = 2.5 but since you can't get half a milk it's 2