Mr. Case is placing boxes along one of the walls in his attic. If the wall is 11 1⁄3 feet long and each box is 2 1⁄2 feet long, how many boxes can he fit along the attic wall?

For this case we have the following expression:

[tex]3 + 2 \sqrt {2} + \frac {1} {3 + 2 \sqrt {2}} =[/tex]

We rationalize the second term multiplying by:

[tex]\frac {3-2 \sqrt {2}} {3-2 \sqrt {2}}:[/tex][tex]\frac {1} {3 + 2 \sqrt {2}} * \frac {3-2 \sqrt {2}} {3-2 \sqrt {2}} = \frac {3-2 \sqrt {2} } {9-6 \sqrt {2} +6 \sqrt {2} -4 (\sqrt {2}) ^ 2} = \frac {3-2 \sqrt {2}} {9-4 * 2} = \frac {3-2 \sqrt {2}} {1} = 3-2 \sqrt {2}[/tex]

So, we have:

[tex]3 + 2 \sqrt {2} + 3-2 \sqrt {2} = 3 + 3 = 6[/tex]

Answer:

6

Hey there! :)

y = 3/4x - 2

To find the slope and y-intercept, simply use slope-intercept form to understand this.

Slope-intercept form is : y = mx + b ; where m = slope, b = y-intercept.

Therefore, you must ask yourself : which value is in the "m" spot and which value is in the "b" spot in our given equation?

Using this logic, we can come to the conclusion that :

Slope = 3/4 & Y-intercept = -2

~Hope I helped! :)

Answer:

D

Step-by-step explanation:

-6(2)=-12

5^-12

can't have negative power

1/5^12

Answer: OPTION D

Step-by-step explanation:

We need to remember that, according to the Negative exponents rule:

[tex]a^{-n}=\frac{1}{a^n}[/tex]

Then, we can rewrite the expression in the following form:

[tex](5^{-6})^2=(\frac{1}{5^6})^2[/tex]

We also need to remember the Power of a power property, which states that:

[tex](a^n)^m=a^{(nm)}[/tex]

Then, applying this property, we get:

[tex]\frac{1}{5^{(6*2)}}=\frac{1}{5^{12}}[/tex]

This matches with the Option D

[tex]\bf ~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{1}{~~y^{-\frac{1}{4}}~~}\implies y^{\frac{1}{4}}[/tex]

Answer:

It would be the same as the similar polygons length. so 6:11.

Step-by-step explanation:

Hope i have helped you!

the answer to your equation is x=2

To show your work I’ll do

37x-14=60

37x=60+14

37x=74

X=2

[tex]\sigma_A=\sum_{1}^{3}=180\Longrightarrow\sum{\frac{180}{3}}=\boxed{60}[/tex]

Answer: all together a triangle = 180 degrease that means if there are three angles then each one would = 60 degrease each

these inequalities can be solved just like equations

a. x - 7 > 25

Answer:

Option b. 12

Step-by-step explanation:

This exercise asks us to find the derivative of a function using the definition of a derivative.

Our function is [tex]f(x) = x^{3}[/tex]. Therefore:

[tex]f(2+h) = (2+h)^{3}[/tex]

[tex]f(2) = (2)^{3} = 8[/tex]

Then:

[tex]\lim_{h \to \0} \frac{f(2+h)-f(2)}{h}=\lim_{h \to \0} \frac{(2+h)^{3}-8}{h}[/tex]

Expanding:

[tex]\lim_{h \to \0} \frac{(2+h)^{3}-8}{h} =\lim_{h \to \0} \frac{8+ h^{3} +6h(2+h) -8}{h} =\lim_{h \to \0} \frac{h^{3} +6h(2+h)}{h}[/tex]

[tex]\lim_{h \to \0} \frac{h^{3}+ 6h(2+h)}{h} =\lim_{h \to \0} h^{2} + 6(2+h) [/tex]

Now, if x=0:

[tex]\lim_{h \to \0} \frac{f(2+h)-f(2)}{h} = (0)^{2} +6(2+0) = 12[/tex]

Answer:

The vertex form is y = -3(x + 2)² + 10

Step-by-step explanation:

* Lets revise how to put the quadratic in the vertex form

- The general form of the quadratic is y = ax² + bx + c, where

 a , b , c are constants

# a is the coefficient of x²

# b is the coefficient of x

# c is the numerical term or the y-intercept

- The vertex form of the quadratic is a(x - h)² + k, where a, h , k

 are constants

# a is the coefficient of x²

# h is the x-coordinate of the vertex point of the quadratic

# k is the y-coordinate of the vertex point of the quadratic

- We can find h from a and b ⇒ h = -b/a

- We find k by substitute the value of h instead of x in the general form

 of the quadratic

 k = ah² + bh + c

* Now lets solve the problem

∵ y = -3x² - 12x - 2

∵ y = ax² + bx + c

∴ a = -3 , b = -12

∵ h = -b/2a

∴ h = -(-12)/2(-3) = 12/-6 = -2

- Lets find k

∴ k = -3(-2)² - 12(-2) - 2 = -3(4) + 24 - 2 = -12 + 24 - 2 = 10

* Lets writ the vertex form

∵ y = a(x - h)² + k

∵ a = -3 , h = -2 , k = 10

∴ y = -3(x - -2)² + 10

∴ y = -3(x + 2)² + 10

* The vertex form is y = -3(x + 2)² + 10

ANSWER

The vertex form is:

[tex]y = - 3{(x + 2)}^{2} +10[/tex]

EXPLANATION

The given equation is:

[tex]y = - 3 {x}^{2} - 12x - 2[/tex]

[tex]y = - 3( {x}^{2} + 4x) - 2[/tex]

We add and subtract the square of half the coefficient of x.

[tex]y = - 3( {x}^{2} + 4x + {2}^{2} ) - - 3( {2})^{2} - 2[/tex]

We simplify to get,

[tex]y = - 3{(x + 2)}^{2} + 3(4)- 2[/tex]

[tex]y = - 3{(x + 2)}^{2} +10[/tex]

This is in the vertex form.

Answer:

A right triangle

Step-by-step explanation:

Suppose a, b, c are the sides of a triangle,

If a² = b² + c² or b² = a² + c² or c² = a² + b²

Then the triangle is called a right angled triangle,

If a² + b² > c², a² + c² > b², b² + c² > a²

Then the triangle is called an acute triangle,

If a = b = c

Then the triangle is called an equilateral triangle,

If a² + b² < c², where c is the largest side of the triangle,

Then the triangle is called an obtuse triangle,

Now, In triangle RST,

By the distance formula,

[tex]RS=\sqrt{(3-1)^2+(-1-(-3))^2}=\sqrt{2^2+2^2}=\sqrt{4+4}=\sqrt{8}\text{ unit }[/tex]

Similarly,

ST = √40 unit,

TR = √32 units,

Since, ST² = RS² + TR²

Hence, by the above explanation it is clear that,

Triangle RST is a right angled triangle,

First option is correct.

Answer:

this isa right triangle!!!

Step-by-step explanation:

this isa right triangle!!!

bearing in mind that perpendicular lines have negative reciprocal slopes hmmmmm wait a second, what's the slope of that line above anyway?

[tex]\bf y=\stackrel{\stackrel{m}{\downarrow }}{-\cfrac{1}{3}} x+5\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}[/tex]

therefore any perpendicular line to that

[tex]\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope}{-\cfrac{1}{3}}\qquad \qquad \qquad \stackrel{reciprocal}{-\cfrac{3}{1}}\qquad \stackrel{negative~reciprocal}{+\cfrac{3}{1}\implies 3}}[/tex]

so, we're really looking for the equation of a line whose slope is 3 and runs through (1, -10)

[tex]\bf (\stackrel{x_1}{1}~,~\stackrel{y_1}{-10})~\hspace{10em} slope = m\implies 3 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-(-10)=3(x-1) \\\\\\ y+10=3x-3\implies y=3x-13[/tex]

Answer:

use M=(Y-Y)/(X-X)format and see what you get

Step-by-step explanation:

so it should be

-10-3/1-1

Im not sure how to explain this without having the proper set up

(2,2) = (x,y)

(1,1) = (x,y)

so if we set this up right

-13/0....

Answer:

B. {-1, 3, 7, 11, 15}

Step-by-step explanation:

The given function is [tex]f(x)=2x-1[/tex]

To find the corresponding odd numbers, we substitute the even numbers as x-values into the function.

When x=0,  [tex]f(0)=2(0)-1=-1[/tex]

When x=2,  [tex]f(2)=2(2)-1=3[/tex]

When x=4,  [tex]f(4)=2(4)-1=7[/tex]

When x=6,  [tex]f(6)=2(6)-1=11[/tex]

When x=8,  [tex]f(8)=2(8)-1=15[/tex]

Therefore the set of odd numbers corresponding to this set of even numbers are  {-1, 3, 7, 11, 15}

Answer:

B. {-1, 3, 7, 11, 15}

Step-by-step explanation:

In order to find the set of corresponding odd numbers then we have to put the even numbers one by one as we already know that x can only have even values..

So,

Putting x = 0

2(0) - 1

=0-1

= -1

Putting x=2

2(2) - 1

=4-1

=3

Putting x=4

2(4)-1

=8-1

=7

Putting x= 6

2(6) -1

=12-1

=11

Putting x=8

2(8)-1

=16-1

=15

So the corresponding set for  {0, 2, 4, 6, 8} is {-1, 3, 7, 11, 15}

Hence, option B is the correct answer ..

Karina needs to pay $35 for 8 weeks to pay the amount of her earrings.

A unit rate is defined as a ratio that compares the first quantity to one unit of the second quantity.

Given that, Karina put a $300 pair of earrings on lay away by making a 10% down payment and agreeing to pay $35 a week.

10% of 300 = 300x10/100 = 30

Therefore, she has already paid $30 for her earrings

Amount she needs to pay = 300-30 = $270

Let x be the total weeks in which she pays her remaining amount,

Establishing the equation,

35x = 270

x = 270/35

x = 7.71 ≈ 8

Hence, Karina needs to pay $35 for 8 weeks to pay the amount of her earrings.

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[tex]\bf \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \sqrt{c^2-a^2}=b \qquad \begin{cases} c=\stackrel{hypotenuse}{13}\\ a=\stackrel{adjacent}{12}\\ b=opposite\\ \end{cases} \\\\\\ \sqrt{13^2-12^2}=b\implies \sqrt{25}=b\implies 5=b[/tex]

Answer:

5

Step-by-step explanation:

a^2 + b^2 = c^2

13 is the hypotenuse because it is the longest side.

12^2 + b^2 = 13^2

144 + b^2 = 169

subtract the 144 from both sides to isolate the variable you are solving for

b^2 = 25

Square root both sides to solve

b = 5

Yhats how to get the answer, but it said I have to put words so ignore this.

considering the total figure,

lenght = 7+x

height =10

Area= L×B

106=10(7+x)

106=70+10x

106-70=10x

36=10x

x=3.6cn

To calculate the size of x when the area of a compound shape is 106 cm², specific information about the shape's dimensions or the relationship of x to the area is needed. Once we have a formula, we can solve for x and express it to the correct number of significant figures.

The student is tasked with calculating the size of x given that the area of a compound shape is 106 cm². To solve for x, we must have a description of the shape with its dimensions, or a formula relating x to the area. Without additional information about the shape or how x is defined within the context of this shape, we cannot provide a numerical answer.

If x represents a length or width of a rectangular part of the compound shape, we might use the formula for the area of a rectangle, A = l×w, to solve for x by dividing the area by the known dimension. However, if the compound shape includes circles, triangles, or other geometric figures, different area formulas would be required. Once we have the correct formula, we would isolate x and calculate its value ensuring the answer is given to the correct number of significant figures as specified by the area given by the question.

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Answer

2/5

Hoped this helped

Answer:

2/5

Step-by-step explanation:

3

____

7 1/2

3

____

15/2

= 2/5

X equals negative B and plus and minus the square root of b squared minus 4ac all over 2a

Answer:

Step-by-step explanation:

The quadratic formula is as follows:

[tex]\frac{-b+/-\sqrt{b^2-4ac} }{2a}[/tex]

It is used for factoring second degree polynomials ONLY!!!  That's why it's called the quadratic formula...quadratics are second degree polynomials.  The a, b and c are found in the standard form of a quadratic equation:

[tex]y=ax^2+bx+c[/tex]

Just plug them into the formula, but be careful with the "-4ac".  Sometimes that mistakenly leads to an unwanted negative sign under you radical.  Unless you are dealing with imaginary numbers, your solutions should be real.

[tex]\bf ~\hspace{5em} \textit{ratio relations of two similar shapes} \\\\ \begin{array}{ccccllll} &\stackrel{\stackrel{ratio}{of~the}}{Sides}&\stackrel{\stackrel{ratio}{of~the}}{Areas}&\stackrel{\stackrel{ratio}{of~the}}{Volumes}\\ \cline{2-4}&\\ \cfrac{\stackrel{similar}{shape}}{\stackrel{similar}{shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array}~\hspace{6em} \cfrac{s}{s}=\cfrac{\sqrt{Area}}{\sqrt{Area}}=\cfrac{\sqrt[3]{Volume}}{\sqrt[3]{Volume}} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]

[tex]\bf \cfrac{\textit{sphere A}}{\textit{sphere B}}\qquad \stackrel{\stackrel{sides'}{ratio}}{\cfrac{1}{4}}\qquad \qquad \stackrel{\stackrel{sides'}{ratio}}{\cfrac{1}{4}}=\stackrel{\stackrel{volumes'}{ratio}}{\cfrac{\sqrt[3]{288}}{\sqrt[3]{v}}}\implies \cfrac{1}{4}=\sqrt[3]{\cfrac{288}{v}}\implies \left( \cfrac{1}{4} \right)^3=\cfrac{288}{v} \\\\\\ \cfrac{1^3}{4^3}=\cfrac{288}{v}\implies \cfrac{1}{64}=\cfrac{288}{v}\implies v=18432[/tex]

Answer:

the answer is C

Step-by-step explanation:

Answer:

The rate of decline is [tex]r=0.1455[/tex]  or [tex]r=14.55\%[/tex]

Step-by-step explanation:

we know that

The  formula to calculate the depreciated value  is equal to  

[tex]D=P(1-r)^{t}[/tex]  

where  

D is the depreciated value  

P is the original value  

r is the rate of depreciation  in decimal  

t  is Number of Time Periods  

in this problem we have  

[tex]P=\$19,400\\D=\$12,105\\t=3\ years[/tex]

substitute in the formula above and solve for r

[tex]\$12,105=\$19,400(1-r)^{3}[/tex]  

Simplify

[tex](12,105/19,400)=(1-r)^{3}[/tex]  

[tex](1-r)=\sqrt[3]{(12,105/19,400)}[/tex]

[tex]r=1-\sqrt[3]{(12,105/19,400)}[/tex]

[tex]r=0.1455[/tex]

Convert to percentage

[tex]r=14.55\%[/tex]

Answer:

108 cm²

Step-by-step explanation:

The lateral area of this pyramid is the addition of the six triangle areas formed on its lateral side.

The area of each triangle is:

Area = (1/2)*side*slant

Area = (1/2)*3*12 = 18 cm²

Then the lateral area is 6*18 = 108 cm²

Answer:

108cm2

Step-by-step explanation:

Answer:

x = 20; the labeled angles are 80° and 100° ⇒ the last answer

Step-by-step explanation:

* Lets revise some facts about a parallelism

- If two lines are parallel, and intersected by a line, there are 3

 pair of angles are formed

1) Alternate angles equal in measures (corners of Z letter)

2) Corresponding angles equal in measure (corners in F letter)

3) Interior supplementary angles their sum is 180° (corners of U letter)

* Now lets solve the problem

∵ There are two parallel lines intersect by another line

- The formed U letter

∴ The labeled angles are interior supplementary angles

∴ Their sum = 180°

∴ 5x + 4x = 180° ⇒ add the like terms

∴ 9x = 180° ⇒ divide both sides by 9

∴ x = 20°

* Lets find the measure of the labeled angles

∵ The measure of one of them is 4x°

∴ Its measure = 4(20) = 80°

∵ The measure of the other is 5x°

∴ Its measure = 5(20) = 100°

Final answer:

The recursive rule for the geometric sequence 1/2, -2, 8, -32 is defined by the first term a1 = 1/2, and for n > 1, the nth term is given by aₙ = aₙ₋₁ × (-4), illustrating the sequence's common ratio of -4.

Explanation:

The question asks for the recursive rule for a geometric sequence: 1/2, -2, 8, -32. To find the recursive rule, we need to determine the common ratio by dividing each term by its preceding term. For instance, dividing -2 by 1/2 gives -4, and similarly dividing 8 by -2 also gives -4. This shows that the common ratio (r) is -4. A geometric sequence can be defined recursively by specifying the first term and the rule for obtaining each term from the previous one.

The first term, represented as a1, is 1/2. Given that the common ratio is -4, the recursive rule for finding the nth term (an) can be expressed as aₙ = aₙ₋₁ ×  r, where n > 1. Substituting the common ratio, we get aₙ = aₙ₋₁ ×  (-4) for n > 1.

Answer: -2 is the correct answer

Step-by-step explanation: All the other values in the options represent an input (x) instead of an output f(x). Al the values in the chart that are on your left are x inputs and all the values on the right are f(x) outputs Leaving you with -2 as the only correct answer.

Hope this helps have a good one!

The value is an output of the function is -2.

What is function?

An expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

Here -6,4 and 7 are in column of x which they are the inputting value for function f(x).

The number -2 is on column of f(x) which means it is output corresponding to the input value 3.

Hence, -2 is the output value for f(x).

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Answer:

x = 6

Step-by-step explanation:

(4x - 3)/3 = 7

4x - 3 = 21

4x = 24

x = 6

Answer:

6

Step-by-step explanation:

I'm going to assume you mean (4x - 3) / 3 = 7

If that is not correct, could you please correct it.

Multiply both sides by 3

3*(4x - 3) / 3 = 7*3

The 3s cancel on the left

4x - 3 = 21  

Add 3 to both sides.

4x - 3 + 3 = 21 + 3

combine

4x = 24

Divide by 4

4x/4 = 24/4

x = 6

It was reflected. this is bc it is the same shape, just flipped.

Answer:Reflection about the y-axis followed by a translation to the left by 2 units

Step-by-step explanation:I took the test

Answer:

(3, -7) are the coordinates

Answer:

no solutions

Step-by-step explanation:

Answer:

Exactly one solution.

Step-by-step explanation:

We have been given a system of equations. We are asked to find the number of solutions for our given system.

[tex]y=x+2...(1)[/tex]

[tex]6x-4y=-10...(2)[/tex]

We can see that equation (1) in slope-intercept form of equation.

We will convert equation (2) in slope-intercept form of equation as shown below:

[tex]6x-6x-4y=-10-6x[/tex]

[tex]-4y=-6x-10[/tex]

Divide both sides by [tex]-4[/tex].

[tex]\frac{-4y}{-4}=\frac{-6x}{-4}+\frac{-10}{-4}[/tex]

[tex]y=\frac{3}{2}x+\frac{5}{2}[/tex]

We can see that slopes of both lines are different, therefore, our given lines will intersect exactly at one point and our given system of equations has exactly one solution.