A collection of quarters and nickels contains at least 42 coins and is worth at most $8.00. If the collection contains 25 quarters, how many nickels can be in the collection? Let x = the number of quarters. Let y = the number of nickels. There are at least nickels, but no more than nickels.

Answer:

Area of the biggest square: 25 m²

Area of the second biggest square: 16 m²

Area of smallest square: 9 m²

Area of triangle: 6 m²

[tex]d. \: y = 6 + 3x \\ \\ 1. \: 3x + 6 = y \\ 2. \: 3x = y - 6 \\ 3. \: \frac{3x}{3} = \frac{y - 6}{3} \\ x = \frac{1}{3} y - 2[/tex]

option A

f(x) = (x – 1)2 + 3

Given in the question a function,

f(x) = 4 + x² – 2x

Step 1

f(x) = 4 + x² – 2x

here a = 1

        b = -2

        c = 4

Step 2

x = -b/2a

h = -(-2)/2(1)

h = 2/2

h = 1

Step 3

Find k

k = 4 + 1² – 2(1)

k = 3

Step 4

To convert a quadratic from y = ax² + bx + c form to vertex form,

y = a(x - h)²+ k

y = 1(x - 1)² + 3

y = (x - 1)² + 3

Exponential form, would be the number of times X gets multiplied by itself.

(x)(x)(x)(x)  there are 4 x's, so the exponential form would be x^4

(x)(x)(x)(x) in exponential form can be written as

[tex]\rm \bold{x^4}}[/tex]

According to the properties of exponent with same base number the powers/exponents of the number are added

This can be simply expressed in the form as formulated in equation (1)

[tex]\rm a ^x \times a^y = a ^{x +y} ............(1)[/tex]

here a = base  number

x and y are the exponents of number  a  

According to the  given question same number "x" is multiplied 4 times

let  the given expression be represented by a variable "y"  

[tex]\rm y = x\times x \times x \times x ...........(2)[/tex]

Equation (2)  can be simply written as follows

[tex]\rm y = x^1 \times x^1 \times x ^1 \times x^1 \\y = x ^{(1+1+1+1)} \\\bold{y = x ^4}[/tex]

So we can conclude that (x)(x)(x)(x) in exponential form can be written as

[tex]\rm \bold{x^4}}[/tex]

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Answer:

The measure of angle ERK is 55°

Step-by-step explanation:

step 1

Find the measure of arc EK

we know that

The diameter divide the circle into two equal parts

In this problem

EOR is a diameter

see the attached figure to better understand the problem

so

arc EK + arc RK=180°

substitute the given values

arc EK + 70°=180°

arc EK=180°-70°=110°

step 2

Find the measure of angle ERK

we know that

The inscribed angle is half that of the arc it comprises.

m∠ERK=(1/2)[arc EK]

substitute  

m∠ERK=(1/2)[110°]=55°

The measure of <ERK is 55 degrees

Given the following parameters

arcRK = 70 degrees

Determine the measure of arcEK
arcEK + arcRK + 180 = 360

arcEK + 70 + 180 = 360

arcEK + 250 = 360

arcEK = 110 degrees

<ERK = 1/2 arcEK

<ERK = 1/2(110)
<ERK = 55 degrees

Hence the measure of <ERK is 55 degrees

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Check the picture below.

Answer:

[tex]2.05\ miles[/tex]  or  [tex]2\frac{1}{20}\ miles[/tex]

Step-by-step explanation:

we know that

To calculate the total distance Barb walked, add the distance to her friend's house plus the distance to the library.

so

[tex]1.3+\frac{3}{4}[/tex]

Remember that

[tex]1.3=\frac{13}{10}[/tex]

substitute

[tex]\frac{13}{10}+\frac{3}{4}=\frac{13*2+5*3}{20}[/tex]

[tex]=\frac{41}{20}\ miles[/tex]

[tex]=2.05\ miles[/tex]

Convert to mixed number

[tex]\frac{41}{20}=\frac{40}{20}+\frac{1}{20}=2\frac{1}{20}\ miles[/tex]

Answer:

The function f(x) is a geometric sequence

Step-by-step explanation:

If we let the first value of this function be denoted by y, then the second value will grow by;

12% of y

= (12/100)*y = 0.12y

The second value will thus be;

y + 0.12y = 1.12y

The third value will grow by;

12% of 1.12y

= (12/100)*1.12y = 0.12(1.12y)

The third value will thus be;

1.12y + 0.12(1.12y)

= 1.12y(1 + 0.12)

= 1.12y * 1.12 = [tex]1.12^{2}y[/tex]

The function f(x) will thus have the sequence;

y, 1.12y, [tex]1.12^{2}y[/tex], ans so on. This is clearly a geometric sequence since we have a common ratio of 1.12.

Answer: B

Ive done the test before, this was correct. glad i could help

Answer:

A and C

Step-by-step explanation:

Just plug in the point into the equations:

a) 14 = 2(4.5) + 5

14 = 9 + 5   14 = 14 A is correct

b) 14 = 3(4.5) + 1.5

14 = 13.5 + 1.5   14 ≠ 15 B is not correct

c) 14 = 4(4.5) - 4

14 = 18 - 4   14 = 14 C is correct

d) is not correct 12 is already to large and 10 is not a negative so it is far to large

Answer:

C

Step-by-step explanation:

Answer:

a. 6km

b. 6.185 km

c. 18.37 km

Step-by-step explanation:

The question is on finding the distance of between two points

The general formulae is given by;

[tex]d= \sqrt{(X2-X1)^2+(Y2-Y1)^2}[/tex]

Where d is the distance

Given that;

1 unit on grid = 1.5 km

a. Finding distance between the library and Town A

A (-2,3)  and library (2,3)

[tex]d= \sqrt{(2- -2)^2  + (3-3)^2} \\\\= \sqrt{4^2} \\= 4[/tex]

Actual distance = 4 × 1.5 = 6 km

b. Distance between the library and Town B is

library (2,3)  and town B (1, -1)

[tex]d=\sqrt{(1-2)^2+ (-1-3)^2} \\\\\\d=\sqrt{-1^2 +-4^2}\\ \\\\d=\sqrt{1+16} \\\\\\d=\sqrt{17}  = 4.123[/tex]

Actual distance = 4.123×1.5 =6.185 km

c. First find the distance between Town B and Town C

Town B (1, -1) and Town C (-3,-2)

[tex]d=\sqrt{(-3-1)^2 + (-2--1)^2} \\\\d=\sqrt{-4^2 + -1^2} \\\\d=\sqrt{16+1} \\\\d=\sqrt{17} \\\\d=4.123[/tex]

Actual distance= 4.123×1.5 =6.185 km

Total distance traveled by Amelia = 6 +6.185 +6.185 =18.37 km

Let x represent those who both football and basketball

The given information can be illustrated in a Venn diagram as shown in the attachment.

We solve the equation below to find the value of x.

[tex](53-x)+x+(24-x)+31=101[/tex]

[tex]\implies -x+x-x=101-53-31-24[/tex]

[tex]\implies -x=-7[/tex]

[tex]\implies x=7[/tex]

From the second diagram;

25. [tex]P(Basketball)=\frac{17}{101}[/tex]

26. [tex]P(Football)=\frac{46}{101}[/tex]

27. [tex]P(Football\: \cap\:Basketball)=\frac{7}{101}[/tex]. This is because 7 play both Football and Basketball.

28. [tex]P(Football\: \cup\:Basketball)=\frac{46}{101}+\frac{17}{101}-\frac{7}{101}=\frac{56}{101}[/tex]. This is because there is intersection.

29. [tex]P(Neither\: \cup\:Both)=\frac{31}{101}+\frac{7}{101}=\frac{38}{101}[/tex]. The two events are mutually exclusive.

If Tatiana ran the marathon with an average speed of 0.09 miles per minute. 5.4 was her speed to the nearest mile per hour

The rate of change of position of an object in any direction. Speed is measured as the ratio of distance to the time in which the distance was covered.

Given,

Tatiana ran the marathon with an average speed of 0.09 miles per minute

We know that a hour has 60 minutes.

Speed to the nearest mile per hour we will calculate by multiplying 0.09 with 60

Zero point zero nine times of sixty.

0.09×60

Five point four miles per hour.

5.4 miles/hour

Hence 5.4 miles/hour is Tatiana speed to the nearest mile per hour

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In Bar Graphs;

- Bars have equal space

- One the y-axis, we have numbers & on the x-axis, we have data which can be anything.

In Histograms;

- Bars are fixed

- On the y-axis, we have numbers & and on the x-axis, we have data which in continuous & will always be number.

An easy way you can remember the difference is looking at the spaces of the bars.

A bar graph has gaps

A histogram has no gaps.

Answer:

(x+3)² + (y+4)²=25

Step-by-step explanation:

The question is on equation of a circle

The distance formula is given by;

√(x-h)²+ (y-k)²=r

The standard equation of  circle is given as ;

(x-h)²+ (y-k)²=r²

The equation of this circle with center (-3, -4) and radius 5 will be;

(x--3)² + (y--4)²=5²

(x+3)² + (y+4)²=25

ANSWER

[tex]{(x + 3)}^{2} + {(y + 4)}^{2} = 25[/tex]

EXPLANATION

The equation of a circle with center (h,k) and radius r units is given by:

[tex]{(x - h)}^{2} + {(y - k)}^{2} = {r}^{2} [/tex]

From the given information the center of the circle is (-3,-4) and the radius is r=5 units.

We substitute the known values to obtain:

[tex]{(x - - 3)}^{2} + {(y - - 4)}^{2} = {5}^{2} [/tex]

We simplify to get:

[tex]{(x + 3)}^{2} + {(y + 4)}^{2} = 25[/tex]

Therefore the equation of the circle in standard form is:

[tex]{(x + 3)}^{2} + {(y + 4)}^{2} = 25[/tex]

Answer:

[tex]\large\boxed{y=\dfrac{1}{3}x+2\dfrac{1}{3}}[/tex]

Step-by-step explanation:

[tex]\text{Let}\ k:y=_1x+b_1\ \text{and}\ l:y=m_2x+b_2.\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-\dfrac{1}{m_1}\\============================\\\\\text{We have}\ y=-3x+1\to m_1=-3.\\\\\text{Therefore}\ m_2=-\dfrac{1}{-3}=\dfrac{1}{3}.\\\\\text{The equation of the searched line:}\ y=\dfrac{1}{3}x+b.\\\\\text{The line passes through }(2,\ 3).[/tex]

[tex]\text{Put the coordinates of the point to the equation.}\ x=2,\ y=3:\\\\3=\dfrac{1}{3}(2)+b\\\\3=\dfrac{2}{3}+b\qquad\text{subtract}\ \dfrac{2}{3}\ \text{from both sides}\\\\b=2\dfrac{1}{3}[/tex]

Answer:

  y = 1/3(x -2) +3

Step-by-step explanation:

The slope of the given line is the coefficient of x, -3. The slope of the perpendicular line will be the negative reciprocal of that: -1/-3 = 1/3. The line through a point (h, k) with slope m can be written in point-slope form as ...

  y = m(x -h) +k

For m=1/3, (h, k) = (2,3), the equation of the line is ...

  y = (1/3)(x -2) +3

Answer:

2/10

Step-by-step explanation:

First you - 20 from 40 (aka 2/10 - 4/10) and you will get 20 (aka 2/10).

To find out what fraction more of the pack Dan used on Saturday compared to Sunday, subtract 2/10 from 4/10.

To find out what fraction more of the pack Dan used on Saturday compared to Sunday, we need to subtract the amount used on Sunday from the amount used on Saturday and express it as a fraction of the original pack.

On Saturday, Dan used 4/10 of the pack. On Sunday, he used 2/10 of the pack. To find the fraction more, we subtract 2/10 from 4/10:

4/10 - 2/10 = 2/10

Therefore, Dan used 2/10 more of the pack on Saturday compared to Sunday.

If the parabola looks like an “n,” your vertex will be a maximum. If the parabola looks like a “u,” the vertex will be a minimum.

To determine if a vertex is a minimum or maximum, evaluate the behavior of the function at that point. If the function is increasing before the vertex and decreasing after, it is a minimum point. If the function is decreasing before the vertex and increasing after, it is a maximum point.

In mathematics, a vertex is a point where two or more lines, curves, or edges meet. When determining if a vertex is a minimum or maximum, we need to look at the behavior of the function or equation at that point.

If the function is increasing before the vertex and decreasing after the vertex, then the vertex is a minimum point. Conversely, if the function is decreasing before the vertex and increasing after the vertex, then the vertex is a maximum point.

For example, consider the parabola y = x^2. The vertex of this parabola is at (0, 0).

Since the parabola opens upwards and the function values increase on either side, the vertex is a minimum point.

Check the picture below, notice is simply a 12x36 rectangle = 432 cm².

Answer:

432 is correct as in this problem the cross section is identical in size and thus area to the face CDHG.

Step-by-step explanation:

The 95% confidence interval for the difference of the two sample means, given a mean difference of 22 and a standard deviation of 10, ranges from 2.4 to 41.6.

The problem provided involves the concept of confidence intervals in statistics. When working with two sample means and you want to find the 95% confidence interval of the difference, the standard deviation of the difference is essential. The difference of two sample means is 22 and the standard deviation of this difference is estimated to be 10.

The 95% confidence interval for a mean can be calculated using the formula:
Confidence Interval = mean difference ± (Z-score * standard deviation).

With a 95% confidence interval, our Z-score (also known as the critical value) is approximately 1.96 (from Z tables or any statistical calculator). Thus, substituting the provided figures into the formula, we have:
Confidence Interval = 22 ± (1.96 * 10).

This gives us a confidence interval range of: 22 - 19.6 to 22 + 19.6, thus the 95% confidence

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The difference of the means of two populations at a 95% confidence interval is between 2.4 and 41.6.

To find the difference of the means of the two populations at a 95% confidence interval, we can use the formula:

CI = (difference of sample means) ± (critical value) × (standard deviation of the difference of sample means)

In this case, the difference of the sample means is 22 and the standard deviation of the difference of the sample means is 10. The critical value for a 95% confidence interval is approximately 1.96.

Using these values, we can calculate the confidence interval as follows:

CI = 22 ± (1.96) × 10

Simplifying the expression, the confidence interval is (2.4, 41.6).

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Answer: 8, 40, 80, 120

1 gallon is equal to 8 pints

The number of pints Jordan needs to fill the fish tank of [tex]15[/tex] gallons is as follow:

gallons :[tex]1 \ \ \ \ \ \ \ \ \ 5\ \ \ \ \ \ \ \ \ \ 10\ \ \ \ \ \ \ \ \ \ \ 15[/tex]

pint       :[tex]8 \ \ \ \ \ \ \ \ \ 40\ \ \ \ \ \ \ \ \ \ 80\ \ \ \ \ \ \ \ \ \ \ 120[/tex]

" Number is defined as the count of a given quantity."

According to the question,

Capacity of fish tank [tex]= 15\ gallons[/tex]

Capacity of a container used to fill the fish tank [tex]= 1 \ pint[/tex]

Standard relation gallons to number of pints

[tex]1 \ gallon = 8\ pints[/tex]

Number of pints required :

[tex]1 \ gallon = 8\ pints[/tex]

[tex]5\ gallon = (8\times 5)\ pints[/tex]

             [tex]= 40 \ pints[/tex]

[tex]10\ gallons = (10 \times 8) \ pints[/tex]

                [tex]= 80\ pints[/tex]

[tex]15\ gallon = (15 \times 8) \ pints[/tex]

               [tex]= 120 \ pints[/tex]

Hence, the number of pints Jordan needs to fill the fish tank of [tex]15[/tex] gallons is as follow:

gallons :[tex]1 \ \ \ \ \ \ \ \ \ 5\ \ \ \ \ \ \ \ \ \ 10\ \ \ \ \ \ \ \ \ \ \ 15[/tex]

pint       :[tex]8 \ \ \ \ \ \ \ \ \ 40\ \ \ \ \ \ \ \ \ \ 80\ \ \ \ \ \ \ \ \ \ \ 120[/tex]

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Answer:

2.375 has the same value as 2 and 3/8.

19/8 also has the same value as 2 3/8.

For this case we have the following expression:

[tex]\sqrt {64}[/tex]

We have to:

[tex]64 = 8 * 8 = 8 ^ 2[/tex]

By definition of properties of powers and roots we have to meet:

[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}[/tex]

Then, rewriting the expression we have:

[tex]\sqrt {8 ^ 2} = 8 ^ {\frac {2} {2}} = 8 ^ 1 = 8[/tex]

Thus, we have that the result is a whole number "8".

Answer:

whole number

Answer:

2 and 45/60 which simplifies to 2 and 3/4

Answer:

a) 1,440 ways

b) 59,280 or 64,000

Step-by-step explanation:

a) Aircraft boarding.

8 people, 2 in first class, boarding first, then 8 economy class.

The 2 people in first class board first, but they can board as AB or BA... so 2 ways here.

For the 6 economy class passengers, we have a permutation of 6 out of 6, so 720, as follows:

[tex]P(6,6) = \frac{6!}{(6 - 6)!} = 6! = 720[/tex]

Since the two are independent, we multiply them to have a global number of ways: 2 * 720 = 1,440 different ways for the 8 passengers to board that plane.

b) combination lock.

Here we do have a little problem... the question doesn't specify if the 3 numbers are different numbers of not.  So, we'll calculate both:

Numbers go from 1 to 40 inclusively... so 40 possibilities.

Normally, in a combination lock, the numbers are different, so let's start with that one:

First number: 40 options available

Second number: 39 options available (cannot take the first one again)

Third number: 38 different options (can't take First or Second number again)

Overall, we then have 40 * 39 * 38 = 59,280 different lock combinations.

If we can pick pick the same number twice:

First number: 40 options available

Second number: 40 options available

Third number: 40 options available

Overall 40 * 40 * 40 = 64,000 different lock combinations

supplementary angles add to equal 180. so x + 57 = 180. solve for x and you get 123.

Answer:

(5, -10)

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Equality Properties

Algebra I

Step-by-step explanation:

Step 1: Define Systems

-9x - 6y = 15

9x - 10y = 145

Step 2: Solve for y

Elimination

Step 3: Solve for x

Step 4: Check

Graph the systems of equations to verify the algebraically solved solution set is the solution.

Where the 2 lines intersect is the solution set.

We see graphically that we get (5, -10).

∴ (5, -10) or x = 5 and y = -10 is the solution to our systems

Answer:

- 56 - 7k

Step-by-step explanation:

Given

- 7(8 + k)

Each term in the parenthesis is multiplied by - 7

= (- 7 × 8) + (- 7 × k)

= - 56 + (- 7k)

= - 56 - 7k

To solve the expression -7(8 + k), the distributive property is used. Multiplying -7 to each of the terms within the parentheses yields -56 - 7k. This is an example of product multiplying monomials.

The given expression is -7(8 + k). To find the product, you should use the distributive property. This property states that the multipliers of a sum or difference, multiplied separately by each addend or minuend, sum to the product. So -7 * 8 gives -56 and -7 * k gives -7k. Hence, the expression becomes -56 - 7k. This process is a demonstration of product multiplying monomials.

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Answer:

A. 10cm

B. 8 times

Step-by-step explanation:

The question is on volume of a conical container

Volume of a cone= [tex]\pi r^{2} h/3[/tex]

where r is the radius of base and h is the height of the cone

Given diameter= 12 cm, thus radius r=12/2 =6 cm

[tex]v=\pi r^2h/3 \\120\pi =\pi *6*6*h/3\\120\pi =12\pi h\\10=h[/tex]

h=10 cm

B.

If height and diameter were doubled

New height = 2×10 =20 cm

New diameter = 2×12 = 24, r=12 cm

volume = [tex]v=\pi r^2h/3\\v=\pi *12*12*20/3\\v=960\pi[/tex]

To find the number of times we divide new volume with the old volume

[tex]N= 960\pi /120\pi \\\\N= 8[/tex]

Answer: The height of the container is 10 centimeters. If its diameter and height were both doubled, the container's capacity would be 8 times its original capacity.

Step-by-step explanation:

The volume of a cone can be calculated with this formula:

[tex]V=\frac{\pi r^2h}{3}[/tex]

Where "r" is the radius and "h" is the height.

We know that the radius is half the diameter. Then:

[tex]r=\frac{12cm}{2}=6cm[/tex]

We know the volume and the radius of the conical container, then we can find "h":

[tex]120\pi cm^3=\frac{\pi (6cm)^2h}{3}\\\\(3)(120\pi cm^3)=\pi (6cm)^2h\\\\h=\frac{3(120\pi cm^3)}{\pi (6cm)^2}\\\\h=10cm[/tex]

The diameter and height doubled are:

[tex]d=12cm*2=24cm\\h=10cm*2=20cm[/tex]

Now the radius is:

[tex]r=\frac{24cm}{2}=12cm[/tex]

And the container capacity is

[tex]V=\frac{\pi (12cm)^2(20cm)}{3}=960\pi cm^3[/tex]

Then, to compare the capacities, we can divide this new capacity by the original:

 [tex]\frac{960\pi cm^3}{120\pi cm^3}=8[/tex]

Therefore,  the container's capacity would be 8 times its original capacity.

Answer:

Inverse Variation

Step-by-step explanation:

ANSWER

A. x-8

EXPLANATION

The given functions are:

[tex]f(x) = {x}^{2} - 11x + 24[/tex]

We factor this to get,

[tex]f(x) = (x - 8)(x - 3)[/tex]

and

[tex]g(x) = x - 3[/tex]

[tex]( \frac{f}{g} )(x) = \frac{f(x)}{g(x)} [/tex]

[tex]( \frac{f}{g} )(x) = \frac{ {x}^{2} - 11x + 24}{x - 3} \: for\: x \ne3[/tex]

[tex]( \frac{f}{g} )(x) = \frac{(x - 8)(x - 3)}{x - 3} [/tex]

Cancel the common factors to get,

[tex]( \frac{f}{g} )(x) = x - 8[/tex]

Answer: OPTION A

Step-by-step explanation:

You need to divide the function f(x) by the function g(x):

Then:

[tex](\frac{f}{g})(x)=\frac{x^2-11x+24}{x-3}[/tex]

Now, you need to simplify:

Factor the numerator. Find two numbers whose sum be -11 and whose product be 24. Theses numbers are -8 and -3. Then you get:

[tex](\frac{f}{g})(x)=\frac{(x-8)(x-3)}{x-3}[/tex]

Remember that:

[tex]\frac{a}{a}=1[/tex]

Then, you get that the expresson that is equal to  [tex](\frac{f}{g})(x)[/tex] is:

 [tex](\frac{f}{g})(x)=(x-8)[/tex]