a rancher has 80 feet of fencing with which to enclose two adjacent rectangular corrals (see figure). what dimensions should be used so that the enclosed area will be a maximum?

On solving the given question, by the Help of LaPlace Inverse form we got - [tex]y(t) = \frac{-7}{2} + \frac{1}{2}e^{-8t}[/tex]

In terms of its usefulness in resolving physical issues, the Laplace transform is an important transform that is probably second only to the Fourier transform. The Laplace transform is especially helpful in solving linear ordinary differential equations, such as those that arise in the analysis of electronic circuits.

Take Laplace for DE we get -

[tex]L(y)" + 8L(y)' =0\\s^2Y (s) - sy(0) - y'(0) 8(sY(s) - y(0))=0\\Y(s)(s^2 + 8s) + 3s 4 + 24 =0\\Y(s)(s^2 + 8s) = -3s - 28\\Y(s) = \frac{-3s - 28}{(s^2 + 8s) } \\Right Side = \\\frac{-7}{2s} + \frac{1}{2(s+8)} \\ Now taking inverse LaPlace - \\y(t) = \frac{-7}{2} + \frac{1}{2}e^{-8t}[/tex]

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a. The solution of the equation x + 2 = √3x + 10 is x = -3 or x =2.

b. Extraneous solution is x= -3.

c. Extraneous solution does not satisfy original equation.

As given in the question,

Given equation :

x + 2 = √3x + 10

a. Squaring both the sides

( x + 2 )² = ( √3x + 10 )²

⇒ x² + 4x + 4 = 3x + 10

⇒ x² + x - 6 =0

⇒ x² + 3x -2x -6 =0

⇒ ( x+ 3) ( x -2) =0

⇒ x= -3 or x = 2

b. Extraneous solution:

x = -3

-3 + 2 = √3(-3)+ 10

⇒ -1  ≠ 1 extraneous solution.

x= 2

2+ 2 =√3(2) +10

⇒4 = 4 not extraneous.

c. Extraneous solution does not satisfy original equation

Therefore, the solution of the equation is x = -3 or x=2and its extraneous solution is x =-3. Extraneous solution does not satisfy original equation.

The complete question is :

Use the equation below to answer these questions.

x + 2 = √3x + 10

a) What is the solution to the equation?

b) What is the extraneous solution? Why?

c) In general, what is an extraneous solution?

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To maintain the same child-to-adult ratio, the number of adult needed for 108 children are 18.

As the stated question-

At the preschool, there's many 36 children and 6 adults.

Thus, this is solved by using unitary method.

36 children -----> 6 adults.

1 children -----> 6/36 adults.

For 108 children, multiply both sides by 108.

108 children -----> 6 x 108/36 adults.

108 children -----> 18 adults.

Thus, to maintain the same child-to-adult ratio, the number of adult needed for 108 children are 18.

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Answer:

one solution

Step-by-step explanation:

the solution to a system of equations given graphically is at the point of intersection of the 2 lines, that is (1, 2 )

Since there is only 1 point of intersection then there is only one solution.

5.56 fluid ounce of tea remain in the pitcher

In mathematics, a variable is an alphabet that represent an unknown number. We let required amount as variable and then try to calculate its value

According to the question,

Total ounces of iced tea a pitcher contains = 17.6 fluid ounces

Amount of tea Erin pours to drink = 12.06 fluid ounces

Let "x"  fluid ounce of tea remain in the pitcher

So , total was 17.6

=> x + 12.06 = 17.6

Solving for x

Subtracting by 12.06 both the sides

=> x + 12.06 - 12.06 = 17.6 -12.06

=> x = 5.56 fluid ounce

Hence , 5.56 fluid ounce of tea remain in the pitcher

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Answer:

It should be 1/5

Since the fractional versions of the ratios are equal, the ratios are equivalent.

Answer:

.5x + 10 > 20

Step-by-step explanation:

x represents each delivery he makes:

.5x + 10 > 20

This shows that for the inequality to be true, it must total over $20

Hope this helped!

Answer:

C no solution.

Step-by-step explanation:

The lines are parallel, they will never intersect.

Answer:

On the 4th day, the sum of penalties and charges will be the same in both the school library and the town library.

Step-by-step explanation:

School library

Penalty-25

Every day charge-5

City library

Penalty-5

Every day charge-10

Let the day when total fee at both libraries will be same be x

Total charge in school library-25+5x

Total charge in city library-5+10x
They both amount to be equal

The measure of the radius of the sphere is given as follows:

r = 359 cm.

The volume of a cylinder of radius r and height h is given by the equation presented as follows:

V = πr²h

Which is the base area given by πr² multiplied by the height h.

In the context of this problem, the parameters are given as follows:

The volume of the circumscribed cylinder is 2154.04π cm^3, hence the equation to solve for r is given as follows:

πr²h = 2154.04π

2πr³ = 2154.04π

2r³ = 2154.04

r = cube root of (2154.04/2)

r = 359.

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Answer: Yes, it will overload the truck.

Step-by-step explanation:

To begin, form an algebraic equation.

200(r) + 425(f)

Second, substitute your numbers for the variables.

200(18) + 425(11) = ?

Third, multiply your numbers of fridges by the weight of each.

3600 + 4675 = 8275 is the total

Finally, determine whether or not if it is greater than the weight limit the truck has.

7200 < 8275

Therefore, the total weight is greater than the weight limit.

The work that has been done on player 2 by player 1 is a negative work. Option A.

We define the work done as a situation in which the force that is applied moves a distance in the direction of the force. In this case we ought to know that a negative work is said to be done when the object is found to slow down at te application of the force. The meaning of this is that the force acts in such a way as to slow down motion which makes it a negative force.

For the fact that football player one pushes against football player two, causing player two to slow down from moving any farther down the field, a negative work is done.

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The dimensions of the can are radius of can = 4cm and

height of can = 78.6cm

What is the volume of cylinder?

A cylinder can be seen as a collection of multiple congruent disks, stacked one above the other. In order to calculate the space occupied by a cylinder, we calculate the space occupied by each disk and then add them up.

Thus, the volume of the cylinder can be given by the product of the area of base and height.

Volume of cylinder = πr²h

According to the given question:

We have V cylinder = V = 400cc

We will first find the areas involved

Let x = radius of base then area of the base =  π*x² and this is the area of the top too

For lateral area we need to get h the height of the cylinder as function of x

V = πx²h   ⇒  h= v/πx²    ⇒   h = 400/πx²

Now the total area of the cylinder is:

A(x) = Area of the base + area of the top + lateral area

A(x) = 2*π*x² + 2πx h ⇒  A(x) =  2*π*x² + 2πx (V/πx² )

A(x) =  2*π*x² + 800/x

Taking derivatives:

A´(x)  =  4πx  - 800/x²

A´(x) = 0                4πx  - 800/x² =0     ⇒ πx - 200/x² = 0

( πx³  -  200 )/ x²  = 0      

πx³  -  200 = 0      

x = 4 cm

and h = 400/πx²    ⇒  h = 78.6 cm

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Answer:

cos x = - 0.809, find the values of x. ( - 180≤ x ≥ 180 )

x = - 135° and x = 45°

Step-by-step explanation:

vertex form :

y = a(x - h)² + k

(h, k) are the coordinates of the vertex.

"a" is the orientation (up-down) and widening factor.

we use the given h, k and x, y (the given point) to calculate "a".

-5 = a(2 - 3)² + 1 = a(-1)² + 1 = a + 1

a = -6

and the full vertex form is then

y = -6(x - 3)² + 1

just a quick addition to the superb reply by "reinhard10158" above

[tex]~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{"a"~is~negative}{op ens~\cap}\qquad \stackrel{"a"~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\begin{cases} h=3\\ k=1 \end{cases}\implies y=a(x-3)^2 + 1\hspace{5em}\textit{we also know that} \begin{cases} x=2\\ y=-5 \end{cases} \\\\\\ -5=a(2-3)^2 + 1\implies -6=a(-1)^2\implies -6=a \\\\[-0.35em] ~\dotfill\\\\ ~\hfill {\Large \begin{array}{llll} y=-6(x-3)^2 + 1 \end{array}} ~\hfill[/tex]

Answer: 32 jackets

Step-by-step explanation:

add 1280 to 1176 then divide the sum by 75

Answer:

64

Step-by-step explanation:

do multiplication :)

The critical value used to test the hypothesis that the butterfly is the most challenging stroke to teach at the 1% significance level is 0.84

Given that,

Of the 62 swim teachers who participated in the survey, 37 claimed that teaching the butterfly stroke is the most challenging. Find the criticial value used to test the hypothesis that the butterfly is the most challenging stroke to teach at the 1% level of significance.

This is right tailed test

Z [tex]\alpha[/tex]  = Z 0.01 = 0.84   ( Using z table )  

Therefore, the critical value used to test the hypothesis that the butterfly is the most challenging stroke to teach at the 1% significance level is 0.84

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The sum of their ages 5 years ago is x + y -10.

The process of subtracting one number from another is known as subtraction.

Given that, Jean is x years old and her younger brother is y years old.

Age of Jean 5 years ago is:

x - 5

The age of Johnathan 5 years ago is:

y - 5

The sum of their ages 5 years ago is:

x - 5 + y - 5

= x + y -10

Hence, the sum of their ages 5 years ago is x + y -10.

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1. Radius is changing at the rate of 0.00042cm/sec when the radius is 75cm.

2.The implicit differentiation to determine dy/dx of relation sin(2x^2y^3)-3x^3=1 is 9x² - 4xy³ cos(2x² y³) / 6x²y² cos(2x² y³)

1. a) check the image attached

r= radius

h=height

b) Volume of conical tank ,V = (1/3)πr²h

= (1/3)πr²(2r)

V = 2/3πr³

c) Rate of change of volume with respect to time is denoted as dV/dt

Rate of change of radius is denoted as dr/dt

d) V = 2/3πr³

dV/dt =  2/3π (3r²) dr/dt

dV/dt = 2πr²  dr/dt

e) Rate of change in volume, dV/dt = 15cm³ / sec  ,  r =75cm

∴ 15 = 2π(75)² .  dr/dt

dr/dt = 15 / (2π(75)²)

dr/dt = 15/35342.92

dr/dt = 0.00042 cm/sec

∴Radius is changing at the rate of 0.00042cm/sec when the radius is 75cm.

2. sin(2x² y³) - 3x³ = 1

Differentiating :

cos(2x² y³) [2x² y³dy/dt + 2y³ 2x] - 9x² = 0

6x²y² dy/dx . cos(2x² y³) + 4xy³ cos(2x² y³) - 9x² = 0

dy/dx = 9x² - 4xy³ cos(2x² y³) / 6x²y² cos(2x² y³)

Hence, the implicit differentiation to determine dy/dx of relation sin(2x^2y^3)-3x^3=1 is 9x² - 4xy³ cos(2x² y³) / 6x²y² cos(2x² y³)

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Holding the effect of the distance from the quad constant, the estimated mean costs for parking on campus is $0.16 per hour more than parking off campus.

According to given data  determined parking meter rates throughout the campus area.

What is the hypothesis for regression?

For simple linear regression, the chief null hypothesis is H0 : β1 = 0, and the corresponding alternative hypothesis is H1 : β1 = 0. If this null hypothesis is true, then, from E(Y ) = β0 + β1x we can see that the population mean of Y is β0 for every x value, which tells us that x has no effect on Y .

Holding the effect of the distance from the quad constant, the estimated mean costs for parking on campus is $0.16 per hour more than parking off campus.

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The two methods are identical because by both using algebra tiles and factoring symbolically, one need to think of the factors.

Factoring simply has to do with the process of writing a number or another mathematical object as a product of different factors, and this is usually smaller or simpler objects that are of the same kind.

Factoring is a common mathematical process that's used for separating the factors, or numbers, which multiply to form another number. Some numbers are affected by multiple factors.

For example, in a situation when you multiply the factors 6 and 4, 8 and 3, 12 and 2, and 24 and 1, you get the number 24.

Algebra tiles are rectangular and square tiles that represent numbers and variables. Using algebra tiles allows us to solve our problems in a more visual manner. It allows us to see exactly what quantities we're dealing with. It's as if we're using building blocks to assist us.

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Answer:

Y=12

Step-by-step explanation:

ya bae

The value of f(2)-g(-7) is -34

A function is defined as a relation between a set of inputs having one output each.

Given are two functions, f(x) = -|3-8x| and g(x) = x²+4x

f(2) = -|3-8x2|

= -|3-16|

= -13

g(x) = (-7)²+4(-7)

= 49-28

= 21

f(2)-g(-7) = -13-21 = -34

Hence, The value of f(2)-g(-7) is -34

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The number of chocolate bars which Mr Banda bought is equal to 50 chocolate bars.

In order to determine the number of chocolate bars and the number of hot chips that Carter bought, we would have to write a system of linear equation and assign variables to the number of chocolate bars and number of candies respectively, and then translate the word problem into algebraic equation as follows:

Since chocolate bars cost $1.1 each and hot chips 0.40 each and Mr Banda spent $87 on buying 130 chocolate bars and hot chips, we would have the following system of linear equation:

1.1x + 0.40y = 87    ......equation 1

x + y = 130              ......equation 2

From equation 2, we have:

y = 130 - x        ......equation 3

Substituting equation 3 into equation 1, we have:

1.1x + 0.40(130 - x) = 87

1.1x + 52 - 0.4x = 87

0.7x + 52 = 87

0.7x = 87 - 52

0.7x = 35

x = 35/0.7

x = 50 chocolate bars.

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These rules of five letters (a, b, c, d, and e) and five numbers allow for 28,800 different passwords. Additionally, she makes sure that no two letters or two numbers are ever placed back-to-back.

The basic counting concept is a method for keeping track of all the potential outcomes in a scenario. According to this statement, there are n*m methods to carry out both of these acts if there are n ways to execute one activity and m ways to conduct another action after that.

Here,

Eloise has 10 alternatives since the password's initial character can be any letter or number. She has five alternatives, and her second selection must be from the set (letter or number) that has not previously been utilized. She has four choices remaining, and Choice 3 is from the same group as Choice 1. She gets four possibilities since Choice 4 is the second item in the same set as Choice 2. The choices 5 and 6 come from separate sets, each of which has three possibilities; the choices 7 and 8 come from separate sets, each of which has two options; and the choices 9 and 10 come from separate sets, each of which has only one option left.

10 5 4 4 3 3 2 2 1 1

10 ×5 ×4 ×4 ×3 ×3 ×2 ×2 ×1 ×1 = 28,800

28,800 different passwords conform to these rules of five letters (a, b, c, d, and e) and five numbers (2, 3, 4, 5, and 6). additionally, she always ensures that no pair of letters is consecutive and that no pair of numbers is consecutive.

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The different ways in which 6 candles be chosen 84 ways.

What is a combination?

The combination is a method of choosing things from a collection where the order of the choices is irrelevant.

Given, When we have 'n' types of elements and choose 'r' elements from them, then we have C(n+r-1, r) combinations.

n=4(4 types of candles), r=6(no of candles to pick)

Choose 6 different candles, then

=C(4+6-1,6)

=C(9,6)

= 9! / (9-6)! (6!)

= 9!/ 6! 3!

=84

The different ways in which 6 candles can be chosen are 84 ways.

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keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above

[tex]x-6y=24\implies -6y=-x+24\implies y=\cfrac{-x+24}{-6} \\\\\\ y=\cfrac{-x}{-6}+\cfrac{24}{-6}\implies y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{1}{6}} x-4\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{\cfrac{1}{6}} ~\hfill \stackrel{reciprocal}{\cfrac{6}{1}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{6}{1}\implies {\Large \begin{array}{llll} -6 \end{array}}}}[/tex]

Answer:

-6

Step-by-step explanation:

i just got this question and it wasn’t 1/6 it was -6

Correct option is " B " the is the test statistics is 1.341 when sample 1: n₁=16, x₁=44, and s₁=5 and sample 2:x₂ = 41, s₂ = 6, n₂ = 9.

Given that,

Assume you are conducting a two-sample test of averages and that the variances in the population are identical. The information below is ours:

Sample 1: n₁=16, x₁=44, and s₁=5.

Sample 2:x₂ = 41, s₂ = 6, n₂ = 9.

We have to find the test statistic.

We know that,

n₁ = 16

x₁ = 44

s₁ = 5

n₂ = 9

x₂ = 41

s₂ = 6

We have

t= (x₂ - x₁)/√(n₁-1)s₁²+(n₂-1)s₂²/n₁+n₂-2(1/n₁+1.n₂)

t = 1.341.

Therefore, Correct option is " B " the is the test statistics is 1.341 when sample 1: n₁=16, x₁=44, and s₁=5 and sample 2:x₂ = 41, s₂ = 6, n₂ = 9.

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The critical value for right tailed test is 2.998, the critical value for left tailed test is - 1.717.

First, we must know the following data,

df = degree of freedom = n - 1

α = significance level

Next, t test have three type of test which is

Then, to find critical value using the t distribution table search df value in left column of table and search the α in the top row of the table (depending on the type of test, use one tail row for this case). For example, see attached image.

Thus, the right tailed test critical value is 2.998, and the left tailed test critical value is - 1.717.

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