A contractor charges $35 for the installation of gutters, plus $1.50 per foot. If one installation cost $162.50, how many feet of gutter were required?

9514 1404 393

Answer:

  3/(√(3(x+h)) +√(3x))

Step-by-step explanation:

The conjugate of a binomial is the same sum, but with the sign changed. That is (a +b) and (a -b) are a conjugate pair: each is the conjugate of the other.

The purpose of multiplying expressions involving roots or complex numbers by their conjugate is to take advantage of the relation ...

  (a +b)(a -b) = a² -b²

Here. you want to eliminate the h from under the radical, so squaring the radical containing h is useful for the purpose. Hence the conjugate gets involved.

"Multiply by the conjugate" means multiply both the numerator and the denominator by the conjugate. (Effectively, multiply by 1.)

__

  \(\displaystyle\frac{f(x+h)-f(x)}{h}=\frac{\sqrt{3(x+h)}-\sqrt{3x}}{h}\\\\=\frac{(\sqrt{3(x+h)}-\sqrt{3x})(\sqrt{3(x+h)}+\sqrt{3x})}{h(\sqrt{3(x+h)}+\sqrt{3x})}=\frac{3(x+h)-3x}{h(\sqrt{3(x+h)}+\sqrt{3x})}\\\\=\frac{3h}{h(\sqrt{3(x+h)}+\sqrt{3x})}=\boxed{\frac{3}{\sqrt{3(x+h)}+\sqrt{3x}}}\)

_____

Additional comment

In the end, you will want the limit as h → 0, which will be 3/(2√(3x)). You will notice that h=0 no longer makes the denominator zero, so the limit is found by simple evaluation.

When the "binomial" is a complex number of the form a+bi, its conjugate is a-bi, regardless of the sign of b. That is, the conjugate of a complex number is found by negating its imaginary part. The sign of the real part is left alone.

The unknown sides of the right angle triangle using trigonometric ratios are as follows;

PR = 13.2 units

RQ = 17.6 units

A right angle triangle is a triangle that has one of its angles as 90 degrees.

The sides of the triangle can be found using trigonometric ratios.

Therefore,

sin 37° = opposite / hypotenuse

sin 37° = PR / 22

cross multiply

PR = 22 sin 37

PR = 22 × 0.60181502315

PR = 13.2399305093

PR = 13.2 units

cos 37 = adjacent / hypotenuse

cos 37 = RQ / 22

RQ = 22 cos 37

RQ = 17.569981221

RQ = 17.6 units

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Answer:

pretty hard

Step-by-step explanation:

Answer: 8 pieces

Step-by-step explanation:

- if there are 24 pieces of sushi in 6 packages then 24/6 = 4 so there’s 4 pieces of sushi in each package

- since there’s 4 pieces of sushi in each pack just do 4 x 2 = 8

- so there’s 8 pieces of sushi in 2 packages

hope this helps :)

Just put -2 in,where X is and get your answer that way

The property you are referring to is called the associative property of multiplication. According to this property, when multiplying three numbers, the grouping of the numbers does not affect the result. In other words, you can change the grouping of the factors without changing the product.

In the equation you provided: 3x(5x7) = (3x5)7

The associative property allows us to group the factors in different ways without changing the result. So, whether we multiply 5 and 7 first, or multiply 3 and 5 first, the final product will be the same.

Answer:

5x+4y=6-2x-3y=-1

x = 2
y = -1

Step-by-step explanation:

(taken from symbolab!)

Answer: 2 3/8

I got an A

Answer:

2 + 3/8

Step-by-step explanation:

Therefore we get 19 / 8 = 2 + 3/8 , which is written as 2 3/8 , called a mixed number. To convert 19/8 you do division; 8 into 19 goes 2 times with 3 left over; so 2 3/8.

Answer:422977164210 or it could be \(4.2297716421(10) ^{11}\)

Step-by-step explanation:

In a case whereby Anu was checking her emails and she casually told her friend siting next to her the ratio of unread emails in my inbox is ratio 4:1 where the total number of emails is 120  then the number of unread emails in her inbox is 96mails.

The concept that will be used here is ratio. An ordered pair of numbers a and b, represented as a / b, is a ratio if b is not equal to 0. A proportion is an equation that sets two ratios at the same value. For instance, you could express the ratio as follows: 1: 3 if there is 1 boy and 3 girls (for every one boy there are 3 girls)

The total mails in her box is 120

The we can represent the unread mails  as 4x and read mails be 1x

This can be expressed as :

(4x+1x)=120mails

5x=120mails

x=120/5

x=24

This implies that the uread mails(4×24)

=96mails

Then the number of the read mail =(1×24)=24mails

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Answer:

Step-by-step explanation:

use point-slope formula [y-y1 = m(x-x1) ] with the given info

P=(x1,y1) = (0,2)  and slope m = -1/4

y-2=-1/4(x-0)

y-2= -1/4x

now try to move things till you get the slope-intercept form [y=mx+b]

y= -1/4x+2

good all done :)

Answer:

5 is the correct answer ....

The simplified expression for ( x - y ) ( x - y ) ( x - y ) is x³ - x²y - xy² + y³.

To simplify ( x - y ) ( x - y ) ( x - y ), we can expand the expression using the distributive property of multiplication. We can also simplify the expression by combining like terms.

In using the distributive property of multiplication, we multiply each term in the first set of parentheses by each term in the second set of parentheses. This gives us:( x - y ) ( x - y ) ( x - y ) = ( x(x - y) - y(x - y) ) ( x - y )= ( x² - xy - xy + y² ) ( x - y )= ( x² - 2xy + y² ) ( x - y )

Now we can further simplify the expression by combining like terms. We can use the difference of squares formula to simplify the second set of parentheses.( x² - 2xy + y² ) ( x - y ) = ( x - y ) ( x - y )²= ( x - y ) ( x² - 2xy + y² )= x³ - x²y - 2xy² + xy² - xy² + y³= x³ - x²y - xy² + y³

Therefore, the simplified expression is x³ - x²y - xy² + y³.

Summary: To simplify ( x - y ) ( x - y ) ( x - y ), we expand the expression using the distributive property of multiplication. We simplify the expression by combining like terms. We use the difference of squares formula to simplify the second set of parentheses. Finally, we simplify the expression further by combining like terms. The simplified expression is x³ - x²y - xy² + y³.

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Answer: 450000, 160

Step-by-step explanation:

4.5 ⋅ 10^5 =

4.5*100000=

4*100000+0.5*100000=

400000+50000=450000

1.6 ⋅ 10^2=

1.6*100=

1*100+0.6*100=

100+60=160

The teacher Jill Exhibited the most consistent performance over the 5 year span with the help of the graph.

According to the statement

we have to find the performance span of the given teachers.

So, For this purpose, we know that the

From the given information and the figure:

The figure graph  attached with this question shows the performance of 5 teachers at Oakdale middle school.

It shows five year chart (2005 - 2009).  

Teachers are Jill, Ed,Ralph,Tina and Sam.First consider performance of Jill, performance vary only in 2006 and every one other  years the performance is consistent.In the case of all other teachers their performance vary inconsistently.  

And because of this the teacher Jill Exhibited the foremost consistent performance over the 5 year span.

So, The teacher Jill Exhibited the most consistent performance over the 5 year span with the help of the graph.

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Question:

Which teacher exhibited the most consistent performance over this 5 year span?

For more details please see the image below.

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Answer:

Given Figure has set of Stacked cubes.

To Draw the Base Plan, lets see its right side, Left Side and Front side view.

Right side:

It has 3 cubes in base.

2 cubes in 2nd layer clearly seen in figure and only 1 cube in 3rd layer.

Left Side:

It has 2 Cubes in base.

Both Cubes attached to 1st and 2nd cube of right side cubes.

Front Side:

It can be seen that, here base only has 2 rows of cubes.

Therefore,  1st Figure is correct Base Plan (or enclosed figure)

Note: Numbers in enclosed figure just shows the total no. of cubes in that stack.

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Answer:

1/8 : 1/2

Step-by-step explanation:

Is that the right answer? I’m not sure lol I don’t remember

56 would have a remainder of 2 using multiples of 3

56/3 = 18 remainder 2

56/6 = 9 remainder 2

56/9 = 6 remainder 2

56/18 = 3 remainder 2

56/27 = 2 remainder 2

Now divide 28 by the above numbers to see which ones have 1 as a remainder:

28/3 = 9 remainder 1

28/6 = 4 remainder 4

28/9 = 3 remainder 1

28/18 = 1 remainder 10

28/27 = 1 remainder 1

There are 3 numbers that will work.

Answer: c. 3

Answer:

c

Step-by-step explanation:

The y-values of the graph increases as the value of x increases, which

indicates that a characteristic of the base of the logarithm function.

Correct response:

The given points on the graph are;

The point where the graph starts = Quadrant 4

Direction of the graph = From quadrant 4 to quadrant 1

Points on the graph are;

(0.5, -0.4), (1, 0), and (6, 1)

The given options are;

\(y = \mathrm{log_{\frac{1}{6} } x}\)

\(y = \mathrm{log_{0.5} x}\)

\(y = \mathrm{log_1 x}\)

\(y = \mathrm{log_{6} x}\)

From the shape of the graph, in which, log x increases as x increases, therefore;

The base, b, of the logarithm is larger than 1, given that we have;

\(\mathbf{log_bx} = y\)

\(\mathbf{b^y} = x\)

From the given coordinate points, x increases as y increases, therefore;

b > 1

The possibly option is therefore, y = log₆x

Verifying, we have;

At x = -0.4, y = 0.5

\(b^y = x\)

\(6 ^{(-0.4)}\) ≈ 0.488

Therefore, the point (0.5, -0.4) is close to the graph of y = log₆x

At the point (1, 0), we have;

6⁰ = 1

Therefore, the point (1, 0), is on the graph of y = log₆x

At the point (6, 1), we have;

6¹ = 6

Therefore, the point (6, 1) is on the graph of y = log₆x

The function of the graph is therefore;

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Answer:

Step-by-step explanation:

P = 2(L + W)

P = 34

L = W + 5

34 = 2(W + 5 + W)

34 = 2(2W + 5)

34 = 4W + 10

34 - 10 = 4W

24 = 4W

24/4 = W

6 = W <=== 6 meters wide

L = W + 5

L = 6 + 5

L = 11 <=== 11 meters long

Answer:

$26,986.29

Step-by-step explanation:

We can use the formula for calculating the depreciation of an asset over time:

wor

\(\bold{D = P(1 - \frac{r}{100} )^t}\)

where:

D= the current value of the asset

P = the initial purchase price of the asset

r = the annual depreciation rate as a decimal

t = the number of years the asset has been in use

In this case, we have:

P = $39,600

r = 12% = 0.12

t = 3 years

Substituting these values into the formula, we get:

\(D= 39,600(1 - \frac{12}{100})^3\\D= 39,600(1 - 0.12)^3\\D= 39,600*0.88^3\\D= 39,600*0.681472\\D=26986.2912\)

Therefore, the car is worth approximately $26,986.29 after 3 years of depreciation at a rate of 12% per annum.

Answer:

$26,986.29

Step-by-step explanation:

As the car's value depreciates at a constant rate of 12% per annum, we can use the exponential decay formula to create a function for the value of the car f(t) after t years.

\(\boxed{f(t)=a(1-r)^t}\)

where:

In this case, the initial value is $39,600, and the rate of depreciation is 12% per year. Therefore, the function that models the value of the car after t years is:

\(f(t)=39600(1-0.12)^t\)

\(f(t)=39600(0.88)^t\)

To calculate the value of the car after 3 years, substitute t = 3 into the function:

\(\begin{aligned} f(3)&=39600(0.88)^3\\&=39600(0.681472)\\&=26986.2912\\&=26986.29\;(\sf 2\;d.p.)\end{aligned}\)

Therefore, the car is worth $26,986.29 after 3 years.

Check the picture below.

so the picture has a rectangle that is 8 units high and 12 units wide, and it has a couple of "empty" trapezoids, with a height of 5 and "bases" of 9 and 3.

now, if we just take the whole area of the rectangle and then subtract the area of those two trapezoids, what's leftover is the blue area.

\(\textit{area of a trapezoid}\\\\ A=\cfrac{h(a+b)}{2}~~ \begin{cases} h=height\\ a,b=\stackrel{parallel~sides}{bases}\\[-0.5em] \hrulefill\\ h=5\\ a=9\\ b=3 \end{cases}\implies \begin{array}{llll} A=\cfrac{5(9+3)}{2}\implies A=30 \end{array} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{\large Areas}}{\stackrel{rectangle}{(12\cdot 8)}~~ -~~\stackrel{\textit{two trapezoids}}{2(30)}}\implies 96-60\implies 36\)

Step-by-step explanation: It's important to understand that the sum of the measures of the interior angles of a triangle is always 180 degrees.

So let's first add up the measures of the two angles we know.

So we have 30 + 4 which is 75 degrees.

The missing angle will be equal to 180 minus 75.

So the missing angle measure is 105 degrees.

20. A rational function \(\frac{p(x)}{q(x)}\) (where \(p,q\) are polynomials in \(x\)) has a removable discontinuity at \(x=a\) if for some positive integer \(n\) we can factorize

\(p(x) = (x-a)^n p^*(x)\)

\(q(x) = (x-a)^n q^*(x)\)

\(\implies \dfrac{p(x)}{q(x)} = \dfrac{(x-a)^n p^*(x)}{(x-a)^n q^*(x)} = \dfrac{p^*(x)}{q^*(x)}\)

That is, we "remove" the discontinuity at \(x=a\) because while \(x\neq a\), we have \(\frac{x-a}{x-a}=1\). The discontinuity is still there, since \(\frac{p(a)}{q(a)}\) is undefined, but in the limit sense we can essentially ignore the factors of \(x-a\). The graph of \(\frac{p(x)}{q(x)}\) will have all the features of \(\frac{p^*(x)}{q^*(x)}\) aside from a hole at \(x=a\).

In this case, we have

\(f(x) = \dfrac{x+4}{x^2-x-20} = \dfrac{x+4}{(x+4)(x-5)}\)

and when \(x\neq-4\), we can cancel the factor of \(x+4\) so that

\(f(x) = \begin{cases}\dfrac1{x-5} & \text{if }x \neq -4 \\\\ \text{DNE} & \text{if }x = -4\end{cases}\)

("DNE" = "does not exist", i.e. is undefined. For some reason "undefined" wouldn't render...)

This means \(x=-4\) is a removable discontinuity.

We cannot do the same with the factor of \(x-5\), so in contrast \(x=5\) is a non-removable discontinuity.

21. The pieces of \(f(x)\) defined on \(x<2\) and \(x>2\) are themselves continuous since they are polynomials. Then the continuity of \(f(x)\) over the entire real line depends on the point where the pieces meet.

Here we have

\(f(x) = \begin{cases}x+3 & \text{if }x\le2 \\ cx+6 & \text{if }x>2\end{cases}\)

so the pieces meet at \(x=2\). Continuity at this point requires that the both limits from either side of \(x=2\) be the same. This means

\(\displaystyle \lim_{x\to2^-} f(x) = \lim_{x\to2} (x+3) = 2+3 = 5\)

\(\displaystyle \lim_{x\to2^+} f(x) = \lim_{x\to2} (cx+6) = 2c + 6\)

Solve for \(c\).

\(2c + 6 = 5 \implies 2c = -1 \implies \boxed{c = -\dfrac12}\)

Answer:

yeah is that all?

Step-by-step explanation:

i feel like I'd understand if there was more to this

Answer:

B. 1 + ln 2 - ln x

General Formulas and Concepts:

Algebra II

Step-by-step explanation:

Step 1: Define

Identify

\(\displaystyle ln(\frac{2e}{x})\)

Step 2: Simplify

Answer:

125.9

Step-by-step explanation:

To round 125.892 to the nearest tenth consider the hundredths’ value of 125.892, which is 9 and equal or more than 5. Therefore, the tenths value of 125.892 increases by 1 to 9.

125.892 rounded to the nearest tenth = 125.9