Abbot Tools is designing a new toolbox. The length of the toolbox will be three times its width, w, and the length of the diagonal of the top cover will be 30 inches.

Use this information to determine the length, l, of the box. Select the correct answer from the drop-down menu.

Answer:

[tex](x-2)^5 = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32[/tex]

Step-by-step explanation:

Binomial Theorem

[tex](a+b)^n=a^n+\dfrac{n!}{1!(n-1)!}a^{n-1}b+\dfrac{n!}{2!(n-2)!}a^{n-2}b^2+...+\dfrac{n!}{r!(n-r)!}a^{n-r}b^r+...+b^n[/tex]

Given expression:

[tex](x-2)^5[/tex]

Compare with the Binomial Theorem:

[tex]a=x, \quad b=-2, \quad n=5[/tex]

Substitute the values into the formula:

[tex]\begin{aligned}(x-2)^5 & = x^5+\dfrac{5!}{1!4!}x^{4}(-2)+\dfrac{5!}{2!3!}x^{3}(-2)^2+\dfrac{5!}{3!2!}x^{2}(-2)^3+\dfrac{5!}{4!1!}x^{1}(-2)^4+(-2)^5\\\\& =x^5+\dfrac{120}{24}x^{4}(-2)+\dfrac{120}{12}x^{3}(4)+\dfrac{120}{12}x^{2}(-8)+\dfrac{120}{24}x(16)-32\\\\& =x^5-\dfrac{240}{24}x^{4}+\dfrac{480}{12}x^{3}-\dfrac{960}{12}x^{2}+\dfrac{1920}{24}x-32\\\\& =x^5-10x^{4}+40x^{3}-80x^{2}+80x-32\end{aligned}[/tex]

Pascal's Triangle

Pascal's triangle is an arrangement of binomial coefficients in triangular form.  Each row has 1 at both ends, and each remaining number in the row is found by adding together the two numbers directly above it.

[tex]\phantom{-b)))))))}1\\\phantom{-b)))))}1 \:\quad 1\\\phantom{-b)))}1 \:\quad 2\: \quad 1\\\phantom{-b)}1 \:\quad 3\: \quad 3\: \quad 1\\\phantom{))}1 \quad 4 \:\: \quad 6 \quad \;4 \: \quad 1\\1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1[/tex]

(Note: The initial row with the single number 1 is row zero)

Each term in the binomial expansion using Pascal's triangle is the product of a coefficient, a with an exponent, and b with an exponent.

As we move through each term from left to right, the exponent of a decreases from the exponent of the expression down to zero, and the exponent of b increases from zero up to the exponent of the expression.

The coefficients of each term are the numbers which appear in the number of the row that corresponds with the exponent of the expression.

From the given expression, the power that we are expanding the bracket to is 5, so we need to look at line 5 of Pascal's triangle, which is:

1 5 10 10 5 1

Therefore, the expansion using Pascal's triangle to the power 5 of is:

[tex](a+b)^5=1a^5b^0+5a^4b^1+10a^3b^2+10a^2b^3+5a^1b^4+1a^0b^5[/tex]

Substituting the values of a and b:

[tex]\begin{aligned}(x-2)^5 & = 1x^5(-2)^0 + 5x^4(-2)^1 + 10x^3(-2)^2 + 10x^2(-2)^3 + 5x^1(-2)^4 + 1x^0(-2)^5\\\\ & = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32\end{aligned}[/tex]

Answer:

This is always true so there are infinite solutions

Step-by-step explanation:

2(x+4)=4x+3-2x+5

To find the solution, the first step is to distribute

2x+8 =4x+3-2x+5

Combine like terms

2x+8= 2x+8

Subtract 2x from each side

2x+8 -2x = 2x+8-2x

8 = 8

This is always true so there are infinite solutions

A line segment can be made up of the addition of different fractions of the line on a straight path i.e a given line segment can be broken into smaller parts.

Therefore, the required proof that EG = HK is shown below:

Given that: EF = JK

                  FG = HJ

But,

EG = EF + FG .............. 1

and

HK = HJ + JK ............. 2

From equation 1, we have:

EG = EF + FG

     = JK + HJ (since EF = JK, and FG = HJ)

So that,

EG = JK + HJ ........... 3

Also from equation 2, we have:

HK = HJ + JK

     = FG + EF (since EF = JK, and FG = HJ)

So that,

HK = FG + JK ........... 4

Therefore, comparing equations 1, 2, 3, and 4, it can be concluded that;

EG = HK

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Answer:

12.5

30

68.5

Step-by-step explanation:

Hello !

Answer:

When x = -3 or x = 3

Step-by-step explanation:

[tex] \frac{ {x}^{2} - 9}{ {x}^{2} + 6x + 8} = 0[/tex]

[tex]\Leftrightarrow \frac{ {x}^{2} - 9 }{ {x}^{2} + 6x + 8} \times ( {x}^{2} + 6x + 8) = 0 \times ( {x}^{2} + 6x + 8)[/tex]

[tex]\Leftrightarrow {x}^{2} - 9 = 0[/tex]

[tex]\Leftrightarrow(x - 3)(x + 3) = 0[/tex]

[tex]\Leftrightarrow \: x - 3= 0 \: or \: x + 3 = 0[/tex]

[tex]\Leftrightarrow \: x = 3 \: or \: x = - 3[/tex]

[tex]\boxed{S = {-3 \: \: or \: \: 3}}[/tex]

Answer:

[tex](x+5)^{3}[/tex]

Step-by-step explanation:

Hope this helps

The expression for the word problem is ( x + 5 )².

Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.

The given word problem is the cube of the sum of a number and 5. The expression will be formed as below:-

Let the unknown number be x.

E = ( x + 5 )²

Therefore, the expression for the word problem is ( x + 5 )².

The complete question is given below:-

The cube of the sum of a number and 5. Write the mathematical expression.

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The Intervals of increasing and decreasing are written in terms of the 'x' value, and you will use the symbol for  'x' with the 'y' coordinate from the vertex.

The increasing and decreasing  intervals are written in the terms of the 'x' value of the function

They are represented thus;

Increasing interval =  f (x) ≤ f (y)

Decreasing interval = f (x) ≥ f (y)

Thus, the Intervals of increasing and decreasing are written in terms of the 'x' value, and you will use the symbol for  'x' with the 'y' coordinate from the vertex.

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Answer:

1/2 log2

Step-by-step explanation:

I hope you understand

Answer:

[tex]x & = \dfrac{1}{2} \ln 2[/tex]

Step-by-step explanation:

Given functions:

[tex]\begin{cases}f(x)=x^2+4x\\g(x)=1+e^{2x}\end{cases}[/tex]

Function composition is an operation that takes two functions and produces a third function.

Therefore, the given composite function fg(x) means to substitute the function g(x) in place of the x in function f(x):

[tex]\begin{aligned}f[g(x)] & = [g(x)]^2 + 4[g(x)]\\& = \left(1+e^{2x}\right)^2+4\left(1+e^{2x}\right)\\& = 1+2e^{2x}+(e^{2x})^2+4+4e^{2x}\\& = e^{4x}+6e^{2x}+5\end{aligned}[/tex]

[tex]\textsf{Let }u=e^{2x}[/tex]

[tex]\implies e^{4x}+6x^{2x}+5=u^2+6u+5[/tex]

Set the composite function to 21 and factor:

[tex]\begin{aligned}f[g(x)] & = 21\\\implies u^2+6u+5 & = 21\\u^2+6u-16 & = 0\\u^2+8u-2u-16 & = 0\\u(u+8)-2(u+8) & = 0\\(u-2)(u+8) & = 0\\\end{aligned}[/tex]

Apply the zero-product property:

[tex]\implies (u-2)=0 \implies u=2[/tex]

[tex]\implies (u+8)=0 \implies u=-8[/tex]

[tex]\textsf{Substitute back in }u=e^{2x}:[/tex]

[tex]\implies e^{2x}=2[/tex]

[tex]\implies e^{2x}=-8[/tex]

As we cannot take logs of negative numbers, the only solution is:

[tex]\begin{aligned}e^{2x} & = 2\\\ln e^{2x} & = \ln 2\\2x \ln e & = \ln 2\\2x(1) & = \ln 2\\2x & = \ln 2\\\implies x & = \dfrac{1}{2} \ln 2\end{aligned}[/tex]

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Options A,B and D. The correction about the area of the triangle are

Firstly this is an equilateral triangle

3 of the sides have the length of 8.7cm

b = half of 8.7

= 4.35cm

To get A we have

a² + b² = c²

a² + 4.35² = 5²

When we solve this out we would have

a² = 25 - 18.9225

a ≈ 2.5 cm.

Hence we have proved that the first option is correct.

For B,

In an equilateral triangles, each of the angles = 60 degrees

60/2 = 30 is the angle that is made from the base

a / 4.35 = tan 30

then a = 4.35 tan 30

This would also give us an approximate of 2.5

For D

The calculations we have done based on the statements in A and B shows that D is correct as 2.5.

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The midpoint of the segment is (-15/2, -15/2)

The complete question is in the attached image

The points are given as:

(-8, -7) and (-7, -8)

The midpoint is calculated as:

(x,y) = 1/2 * (x1 + x2, y1 + y2)

So, we have:

(x,y) = 1/2 * (-8 - 7, -7 - 8)

Evaluate the difference

(x,y) = 1/2 * (-15, -15)

Evaluate the product

(x,y) = (-15/2, -15/2)

Hence, the midpoint of the segment is (-15/2, -15/2)

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Answer:

y=7x

Step-by-step explanation:

The equation of the with a slope of m and a y-intercept of b is y=mx+b.

Answer:

45 dandelion × 15 rows= 675

Step-by-step explanation:

IR = 9√3

E = 60°

I = 30°

R = 90°

• Find ER.

ER/sin(I) = IR/sin(E)

ER/sin(30°) = 9√3/sin(60°)

ER/(1/2) = 9√3/(1/2 √3)

ER/(1/2) = (9√3 . 2)/√3

ER/(1/2) = 18√3/√3

ER/(1/2) = 18 → IR/sin(E)

ER = 18 . 1/2

ER = 9 is the answer

• Find IE (There are 2 ways).

First, using a Sinus Rule.

IE/sin(R) = IR/sin(E)

IE/sin(90°) = 9√3/sin(60°)

IE/1 = 9√3/(1/2 √3)

IE = (9√3 . 2)/√3

IE = (18√3)/√3

IE = 18 is the answer

Second, using a Pythagoras Theorem.

IE = √(IR² + ER²)

IE = √((9√3)² + 9²)

IE = √(81.3 + 81)

IE = √(243 + 81)

IE = √324

IE = 18 is the answer

Answer:  < -4/5,  3/5>

This is equivalent to writing < -0.8, 0.6 >

======================================================

Explanation:

Draw an xy grid and plot the point (-4,3) on it. Draw a segment from the origin to this point. Then draw a vertical line until reaching the x axis. See the diagram below.

We have a right triangle with legs of 4 and 3. The hypotenuse is [tex]\sqrt{4^2+3^2} = \sqrt{16+9} = \sqrt{25} = 5[/tex] through use of the pythagorean theorem.

We have a 3-4-5 right triangle.

Therefore, the vector is 5 units long. This is the magnitude of the vector.

Divide each component by the magnitude so that the resulting vector is a unit vector pointing in this same direction.

Therefore, we go from < -4, 3 > to < -4/5,  3/5 >

This is equivalent to < -0.8, 0.6 > since -4/5 = -0.8 and 3/5 = 0.6

Side note: Unit vectors are useful in computer graphics.

Answer:

A.) x = 3

Step-by-step explanation:

This equation is an example of a perfect square quadratic. This means that the quadratic can be simplified into two of the same factors.  

How can you factor the equation? The simplest way is to ask yourself, which two numbers multiply to 9 while also adding to -6? The two numbers which satisfy this question are -3 and -3. Knowing this, you can construct your factors, simplify, and then solve for "x".

x² - 6x + 9 = 0                    <----- Expanded equation

(x - 3)(x - 3) = 0                  <----- Factored equation

(x - 3)² = 0                          <----- Combine both factors

x - 3 = 0                             <----- Take square root of both sides

x = 3                                 <----- Add 3 to both sides

The system of equations are x+ y =16 and 2y=x.

Algebra is the part of mathematics that helps represent problems or situations in the form of mathematical expressions

let the two points shots be x

let the three points shots be y.

According to question,

x+ y =16

and,

2y=x

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Answer:

The answer is B.

Step-by-step explanation:

I learned this last year.

Answer:

1

Step-by-step explanation:

any number (except zero) raised to the 0th power is equal to 1.

Answer:

1

Step-by-step explanation:

(10 - 11)⁰ = ( -1)⁰ = 1

We have a⁰ = 1

The function that has the given properties below is y = -4sin(x).

It should be noted that a function can be used to illustrate the relationship between variables.

In this case, the domain is the set of all real numbers, the amplitude is 4, and one x-intercept is (2 pi, 0).

The appropriate function is y = -4sin(x).

The graph is attached.

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Ten gave the check of $5768 to the bank for payment.

Simple interest is calculated based on a loan's principal or the initial deposit into a savings account. Simple interest doesn't compound, so a borrower will never have to pay interest on the interest already accumulated because a creditor will only pay interest on the principal amount.

Formula for simple interest is;-

Simple interest = (principal amount×rate of interest×time duration in years)/100

S.I = (P×R×T)/100

Calculation of the total amount paid by Ted to the bank-

The principal amount is  $5,600.

The rate of interest is 9%.

Time duration is 4 months equals 4/12 = 1/3 years.

S.I = (5,600×9×1)/(3×100)

S.I  = 168

The simple interest on amount is $168.

Thus, the total amount is S.I + principal amount

Total amount = 168 + 5600

                      = 5768

Therefore, the total amount paid by the Ted to the bank is $5768.

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Answer:

-∞ < x < 4

Step-by-step explanation:

→ The graph touches the y axis at 4, therefore 4 part of the range. As h ( x  ) has a vertical asymptote it will never touch the line x = 1, therefore -∞.

Answer:

C

Step-by-step explanation:

In option c, the first block is divided into 3 pieces where 2 parts are shaded whereas the second block is divided into 6 pieces where 3 parts are shaded

For both to be equivalent, both of them should be the same

2/3 = x/6

x = 4

4 pieces should be shaded in in the second block to be equal to the first one, in the diagram only 3 parts are shaded thus the answer is c

Plug the second equation (the one already solved for y) into the first equation

2x - (2x + 3) = 7

2x - 2x - 3 = 7

-3 = 7

Since the 2x's canceled out and left a false statement, this system of equations has no solution.

Hope this helps!

The parameter to be tested is z.

We would set up the hypothesis test.

For the null hypothesis,

µ ≥ 4.2

For the alternative hypothesis,

µ < 4.2

It is a left tailed test.

Since the population standard deviation is given, z score would be determined from the normal distribution table. The formula is

z = (x - µ)/(σ/√n)

where

x = sample mean

µ = population mean

σ = population standard deviation

n = number of samples

From the information given,

µ = 4.2

x = 4.3

σ = 0.4

n = 37

z = (4.3 - 4.2)/(0.4/√37) = 1.53

Looking at the normal distribution table, the probability corresponding to the z score is 0.933

The p value gotten is to the right of the normal curve. Since it is a left tailed test, we need the p value to the left of the curve. Therefore,

p = 1 - 0.933 = 0.067

Since alpha, 0.05 < than the p value, 0.067, then we would fail to reject the null hypothesis. Therefore, At a 5% level of significance, there is no significant evidence that mean GPA of these graduates does not exceed 4.30

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The statements that are true are

From the question, we are to determine the statements that are true

In the given number line,

A =  -2 1/6

B = - 1/2

C = 1/3

D = 1 2/3

B - C = -1/2 - 1/3

B - C = -5/6

∴ B - C ≠ 1/6

A + C = -2 1/6 + 1/3

A + C = -1 5/6

∴ The sum of A and C is negative

A + D = -2 1/6 + 1 2/3

A + D = -1/2

B = -1/2

∴ A + D = B

1 2/3 ÷ - 1/2 = -10/3

∴ The quotient of D and B is -10/3

A×B = -2 1/6 × -1/2

A×B = 13/12

Product of A and C

A×C = -2 1/6 × 1/3

A×C = -13/18

∴ The product of A and B > the product of A and C

|B| = 1/2

|C| = 1/3

|B| > |C|

|B| is NOT less than |C|

Hence, the statements that are true are

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A) The outcome for Event X or Y is expressed as; X ∪ Y = {D, A, B, E}

B) The outcome of Event X and Y is; X ∩ Y = {A, B, E}

We are given the events;

Event X; The letter selected is found in the word "BEAD"

Event Y; The letter selected comes after D

Thus;

A) The outcome for Event X or Y is expressed as;

X ∪ Y = {D, A, B, E}

B) The outcome of Event X and Y is;

X ∩ Y = {A, B, E}

C) Complement of Event Y is;

Y^(c) = {D}

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The number which would correspond to the position of the word; $BAB$ is; $2×1×2$ = 4.

As evident in the task content, by observation, if follows that since; $AAA$ is Word $1 and $CCC$ is Word $27$.

Then, it can be inferred that A = 1 , B = 2 and C = 3.

On this note, the number which would correspond to the position of the word; $BAB$ is; $2×1×2$ = 4.

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Answer:

B) 18%

Step-by-step explanation:

100% - 36% - 29% - 17% = 18%

Answer:

the answer would be B) 18%

Answer: 10

Step-by-step explanation:

By the geometric mean theorem,

[tex]\frac{BD}{5}=\frac{20}{BD}\\\\(BD)^2 = 100\\\\BD=10[/tex]

The length of side BD, given that AD = 5 and CD = 20 is 10 (Option A)

Since the triangles are similar, we can obtain the length of side BD as illustrated below:

CD / BD = BD / AD

20 / BD = BD / 5

Cross multiply

BD × BD = 20 × 5

BD² = 100

Take the square root of both sides

BD = √100

BD = 10

Thus, the length of side BD is 10

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Answer:

The linear equation for tip barista gets in a day is f(n)=0.5n+20.

Step-by-step explanation:

It is given, at a noon a barista notice that she has $20 in tip jar, if she makes an average of $0.50 from each customer,

It is required to find how much will she have in her tip jar if she serves n more customers during her shift. m=0.5. Bases on this write the equation for n more customers.

Step 1 of 1

It is given, at a noon a barista notice that she has $20 in tip jar, if she makes an average of $0.50 from each customer,

To solve this let f(n) is the total tip barista has in her tip jar after a day.

When n is equal to 0 , that barista has $20,

Then if barista make 0.5 from each customer, then for the linear function m=0.5,

Bases on this write the equation for n more customers,

[tex]$$f(n)=0.5 n+20 .$$[/tex]