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For this case we have that by definition, the distance between two points is given by:

[tex]d = \sqrt {(x_ {2} -x_ {1}) ^ 2+ (y_ {2} -y_ {1}) ^ 2}[/tex]

We have the following points:

[tex](x_ {1}, y_ {1}) = (6, -2)\\(x_ {2}, y_ {2}) = (- 3,4)[/tex]

Substituting we have:

[tex]d = \sqrt {(- 3-6) ^ 2 + (4 - (- 2)) ^ 2}\\d = \sqrt {(- 3-6) ^ 2 + (4 + 2) ^ 2}[/tex]

Answer:

Option B

The distance between the points (-3, 4) and (6, -2) is calculated using the distance formula from the Pythagorean Theorem, resulting in approximately 10.82 units.

To calculate the distance between two points in a coordinate system, you can use the distance formula derived from the Pythagorean Theorem. This is expressed as:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Given the points (-3, 4) and (6, -2), you can plug these into the formula as follows:

d = √((6 - (-3))² + (-2 - 4)²)

d = √((6 + 3)² + (-6)²)

d = √(9² + (-6)²)

d = √(81 + 36)

d = √(117)

d ≈ 10.82

This result means the distance between the points (-3, 4) and (6, -2) is approximately 10.82 units.

Answer:

Step-by-step explanation:

"Find the values of x that satisfy 3x - 2x^2 = 7."  Please do not use " × " to represent a variable; " × " is an operator, the "multiply" operator.

Rearrange these three terms in descending order by powers of x:

-2x^2 + 3x - 7 = 0.   Here the coefficients are a = -2, b = 3 and c = -7, and so the discriminant of this quadratic is b^2-4ac, or 9 - 4(-2)(-7), or 9 - 56, or -47.

Because the discriminant is negative, we'll have two different complex roots here.  The quadratic formula becomes

       -3 ± i√47         -3 ± i√47

x = ----------------- = -------------------

        2(-2)                     -4

The question involves solving a quadratic equation using the quadratic formula. Substituting a, b, and c values into the formula will give us the solutions for x, with the discriminant indicating the nature of the roots.

The question asks for the solution to the quadratic equation 3x - 2x² = 7.

To solve this equation, we first move all terms to one side of the equal sign to set the equation to zero:

2x² - 3x + 7 = 0

Since this is a quadratic equation, we can use the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

In this case, a = -2, b = 3, and c = -7. We substitute these values into the quadratic formula to find the values of x:

x = [(-3) ± √((3)² - 4(-2)(-7))] / (2(-2))

The discriminant, √(b² - 4ac), will determine the nature of the roots.

Depending on the value of the discriminant, the solutions may be real and distinct, real and equal, or complex.

Answer:

B. 3b - 2 = 4.

Step-by-step explanation:

Let us check each equation by plugging in the value of b = 2 to see if it is true.

A. 2 × 2 + 24 = 4 + 24 = 28.

28 ≠ 30 . So A is not true.

B. 3 × 2 - 2 = 6 - 2 = 4.

4 = 4. B is true!

C. 2 + 4 = 6.

6 ≠ 8. C is not true.

D. 2 × 2 - 3 = 4 - 3 = 1.

1 ≠ 0. D is not true.

I hope this helps!

Answer:

150 yards.

Step-by-step explanation:

1 yard = 3 feet

To find how many yards 450 feet is, divide 450 by 3.

450/3 = 150

So, 150 yards. :)

The air that can circulate per​ minute will be 150 cubic yards.

The term “volume” refers to the amount of three-dimensional space taken up by an item or a closed surface. It is denoted by V and its SI unit is in cubic cm.

A typical air conditioner can move 450 cubic feet of air per minute. It can circulate 150 cubic yards of air per minute.

Unit conversion;

1 yard = 3 feet

1 feet = 1/3  yard

Volume in the cubic yard is calculated as;

450 feet = 450/3

450 feet = 150 cubic yard

Hence, the air that can circulate per​ minute will be 150 yards.

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Answer:

27

Step-by-step explanation:

Evaluate the [tex]\sqrt[4]{81}[/tex] = 3

Since [tex]3^{4}[/tex] = 81

We are noe left to evaluate (3)³ = 27

Answer:

31%

Step-by-step explanation:

The total this student must pay is $900 + $400 = $1300.

Dividing the cost of books ($400) by the total ($1300) yields:

$400

---------- = 0.3077, or approx. 31%

$1300

You have to make a part over whole proportion ([tex]\frac{part}{whole}=\frac{part}{whole}[/tex]

One fraction will be the unknown percent over 100 (the "whole" of all percents are always 100)

The other fraction will be cost of books over total education cost ([tex]\frac{400}{1300}[/tex]

Proportion:

[tex]\frac{400}{1300}=\frac{x}{100}[/tex]

Now cross multiply

400 * 100 = 1300 * x

40000 = 1300x

Solve for x by dividing 1300 to both sides

40000/13000 = 1300x/1300

x = 30.76923

I will round to the nearest tenth...

30.8% is the percent spent on books

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

8 inches

Step-by-step explanation:

We are given:  2L+2W=38  or simplified version: L+W=19

                          L=3+W

So plug 2nd equation into first, like so, (3+W)+W=19

                                                                 3+W+W=19

                                                                 3+2W=19

                                                                     2W=16

                                                                       W=8

The width of rectangle is,

⇒ W = 8 inches

We have to given that,

The perimeter of a rectangle is 38 inches.

And, the length is 3 inches more than the width.

Let us assume that,

Width = W

Length = W + 3

Hence, We get;

2 (W + W + 3) = 38

2 (2W + 3) = 38

4W + 6 = 38

4W = 38 - 6

4W = 32

W = 32/4

W = 8

Therefore, The width of rectangle is,

⇒ W = 8 inches

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Answer:

B. 0.4

Step-by-step explanation:

Use the definition of the probability

[tex]Pr=\dfrac{\text{Number of all favorable outcomes}}{\text{Number of all possible outcomes}}[/tex]

You have to find the probability that a randomly selected reference book is hard cover. Hence, from the table

So, the probability is

[tex]Pr=\dfrac{10}{25}=\dfrac{40}{100}=0.4[/tex]

Hence, the probability that a randomly selected reference book is a hardcover is:

                             0.4

Let A denote the event that the book selected is a reference book.

and B denote the event that the book  is hardcover.

Let P denote the probability of an event.

We are asked to find:

                  P(B|A)

We know that:

[tex]P(B|A)=\dfrac{P(A\bigcap B)}{P(A)}[/tex]

From the table we have:

[tex]P(A)=\dfrac{25}{60}=\dfrac{5}{12}[/tex]

and

[tex]P(A\bigcap B)=\dfrac{10}{60}=\dfrac{1}{6}[/tex]

Hence, we have:

[tex]P(B|A)=\dfrac{\dfrac{1}{6}}{\dfrac{5}{12}}\\\\\\P(B|A)=\dfrac{2}{5}\\\\\\P(B|A)=0.4[/tex]

           Hence, the answer is:

                 0.4

Answer:

ITS NEGATIVE 10

Step-by-step explanation:

Answer:

x = 75 and y = -72

Step-by-step explanation:

It is given that,

1/5 x + 1/8 y = 1   ------(1)

1/2 x − 1/3 y = 1  -------(2)

To find the solutions of the system of equations

Step 1: eq(1) * 5 ⇒

x + 5/8y = 5  ----(3)

Step 2:  eq(2) * 2 ⇒

x - 2/3y = 2  -----(4)

Step 3: eq(3) - eq(4) ⇒

x + 5/8y = 5  ----(3)

x - 2/3y = 2  -----(4)

0 +(5/8 - 2/3)y = 3

 -1/24 y = 3

y = -24*3 = -72

Step 4: Substitute the value of y in eq(1)

1/5 x + 1/8 y = 1   ------(1)

1/5 x + 1/8 (-72) = 1   ------(1)

1/5 x  - 24 = 1

1/5 x = 25

x = 5*25 = 75

Therefor x = 75 and y = -72

Answer:

[tex]x = \dfrac{110}{31}; \qquad y = \dfrac{72 }{31}[/tex]

Step-by-step explanation:

I am guessing that your two equations are

(1) ⅕x + ⅛y = 1

(2) ½x - ⅓ y = 1

To get rid of fractions, I would multiply each equation by the least common multiple of its denominators.

[tex]\begin{array}{rcrl}(3) \qquad 8x + 5y & = & 40 & \text{Multiplied (1) by 40}\\(4) \qquad 3x - 2y & = & 6 & \text{Multiplied (2) by 6}\\\end{array}[/tex]

We can solve this system of equations by the method of elimination.

[tex]\begin{array}{rcrl}(5) \qquad \, \, 16x + 10y & = & 80 & \text{Multiplied (3) by 2}\\(6) \qquad \, \: 15x - 10 y & = & 30 & \text{Multiplied (4) by 5}\\31x & = & 110 & \text{Added (5) and (6)}\\\\(7)\qquad\qquad \qquad x & = & \dfrac{110 }{31} & \text{Divided each side by 31}\\\end{array}[/tex]

[tex]\begin{array}{rcrl}3 \left (\dfrac{110}{31} \right) - 2y & = & 6 & \text{Substituted (7) into (4)}\\\\ (5) \qquad16x + 10y & = & 80 & \text{Multiplied (3) by 2}\\\\(6)\qquad 15x - 10 y & = & 30 & \text{Multiplied (4) by 5}\\\\31x & = & 110 & \text{Added (5) and (6)}\\\\(7)\qquad \qquad \qquad x & = & \dfrac{110 }{31} & \text{Divided each side by 31}\\\\3 \left(\dfrac{110}{31} \right ) - 2y & = & 6 & \text{Substituted (7) into (4)}\\\\\end{array}\\\\[/tex]

[tex]\begin{array}{rcll}\dfrac{330}{31} - 2y & = & 6 &\\\\-2y & = & 6 - \dfrac{330}{31} &\\\\y & = & \dfrac{165}{31} -3 & \text{Divided each side by -2}\\\\ & = & \dfrac{165 - 93}{31} &\\\\ & = & \dfrac{72}{31} &\\\\\end{array}\\\\\therefore x = \dfrac{110}{31}; \qquad y = \dfrac{72 }{31}[/tex]

The diagram below shows the graphs of your two functions intersecting at (3.548, 2.323). These are the decimal equivalents of your fractional coordinates.

Answer:

The ten millions place

Step-by-step explanation:

because the three after it is the unit million then the 8 being before the 3 would definitely be it's tens

Answer:

a = 6, b = 4, c = 2

Step-by-step explanation

see attached

Answer:

The required values are a=6, b=4 and c=2.

Step-by-step explanation:

The given expression is

[tex]\sqrt{x^{12}y^{9}z^{5}}=(x^{a}y^bz^c)\sqrt{yz}[/tex]          .... (1)

It can be written as

[tex]\sqrt{x^{12}\cdot y^{8}\cdot y\cdot z^{4}\cdot z}[/tex]

[tex]\sqrt{x^{12}\cdot y^{8}\cdot z^{4}\cdot y\cdot z}[/tex]

[tex]\sqrt{(x^{6})^2\cdot (y^{4})^2\cdot (z^{2})^2\cdot y\cdot z}[/tex]     [tex][\because (a^m)^n=a^{mn}][/tex]

[tex]\sqrt{(x^{6}y^4z^2)^2\cdot y\cdot z}[/tex]          [tex][\because a^xb^x=(ab)^x][/tex]

[tex](x^{6}y^4z^2)\sqrt{yz}[/tex]               .... (2)       [tex][\because \sqrt{x^2}=x][/tex]

From (1) and (2), we get

[tex]a=6,b=4,c=2[/tex]

Therefore the required values are a=6, b=4 and c=2.

Answer:

x=1.25

Step-by-step explanation:

We want to solve [tex]\log_3(x+1)=\log_6(5-x)[/tex] by graphing.

We let  [tex]y=\log_3(x+1)[/tex] and also

[tex]y=\log_6(5-x)[/tex].

We graph both equations to obtain the graph shown in the attachment.

The two graphs intersect at (1.25,0.74)

Therefore the solution to [tex]\log_3(x+1)=\log_6(5-x)[/tex]  is the x-coordinate of the point of intersection of the two graphs which is x=1.25.

Answer:  a and d

Step-by-step explanation:

I got it right on edge

Answer: The m∠A is 55° and m∠B is 35°. Hope this helps

Step-by-step explanation:

Step 1: m∠A + m∠B = 90°

Step 2: m∠A + (m∠A − 20°) = 90°

Step 3: m∠A + (m∠A − 20°) = 90°

+20° = +20° Add 20° to both sides.

m∠A + m∠A = 110°

2(m∠A) = 110° Divide both sides by 2.

m∠A = 55°

Step 4: m∠A + m∠B = 90°

55° + m∠B = 90° Substitute 55° for m∠A.

m∠B = 35°

The measures of two complementary angles where one is 20° greater than the other, we set up equations based on the sum of their measures being 90°. Solving these equations, we find that the measure of angle A is 55° and the measure of angle B is 35°.

The measures of two complementary angles, where the measure of angle A is 20° greater than the measure of angle B. To find these measures, we can set up the following equations based on the properties of complementary angles:

Let m B be the measure of angle B.

Therefore, m A will be m B + 20° because it's given that angle A is 20° greater than angle B.

Since angles A and B are complementary, their measures must add up to 90°, hence m A + m B = 90°.

Substitute m A = m B + 20° into the equation m A + m B = 90° to get (m B + 20°) + m B = 90°.

Combine like terms to form 2m B + 20° = 90°.

Solve for m B by subtracting 20° from both sides to get 2m B = 70°.

Divide both sides by 2 to find m B = 35°.

Substitute m B = 35° into m A = m B + 20° to find m A = 35° + 20° = 55°.

Therefore, the measure of angle A is 55° and the measure of angle B is 35°.

Answer:

D. y=-2x-6

Step-by-step explanation:

First start with what we know....

y = -2x + 3 (Slope Intercept Form)

Because of this we can eliminate B.  

Parallel means that the lines wouldn't be touching which means they should have the same slope and the only one with the same slope is D.

For this case we have that an equation of a line of the slope-intersection form is given by:

[tex]y = mx + b[/tex]

Where:

m: It's the slope

b: It is the cutoff point with the y axis

They give us the following information:

[tex]m = -2\\b = 3[/tex]

Then the line is:

[tex]y = -2x + 3[/tex]

They ask us to find a parallel line. By definition, if two lines are parallel then they have the same slope. Thus, the line sought is of the form:

[tex]y = -2x + b[/tex]

We look for the cut point "b" substituting the point where the line passes: [tex](2,2)[/tex]

[tex]2 = -2 (2) + b\\2 = -4 + b\\2 + 4 = b\\b = 6[/tex]

Finally, the line is:

[tex]y = -2x + 6\\y + 2x = 6[/tex]

Answer:

Option A

Answer:

40

Step-by-step explanation:

6x7-8/4 =42 - 2 = 40

To solve the expression 6x7-8/4, follow PEMDAS rule: Multiply 6 by 7, divide 8 by 4, and subtract the result from the multiplication.

To solve the expression 6x7-8/4 using the order of operations, we follow the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication, and Division (from left to right), and Addition and Subtraction (from left to right). Firstly, we perform the multiplication: 6x7 = 42. Then, we divide 8 by 4, which equals 2. Finally, we subtract 2 from 42, which gives us the final answer of 40.

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Answer:

B. Pairs

Step-by-step explanation:

A P E X

Answer:

Triplets

Step-by-step explanation:

just did it on pex learning

Answer:

Option A asymptote.

Step-by-step explanation:

Asymptotes are lines to which the graph approaches very it can come very close to it but will never touch it. Asymptote are the limits of a graph.

IN the given graph of Hyperbola the two blue lines representing the Hyperbola are coming too close to the red lines but not touching it .

The two red lines are called the Asymptotoes of the Hyprbola.It is the limit of the Hyperbola.

Step-by-step explanation:

Answer:

It is an asymptote.

Step-by-step explanation:

I did it on A P E X

There would be 2 intercepts, the X and the Y.

To find the X intercept, set Y to 0 and solve for x.

To find the Y intercept, set X to 0 and solve for y.

x -2y = 2

x-2(0) = 2

x-0 = 2

x = 2

0 -2y = 2

-2y=2

y = 2/-2

y = -1

The x intercept is (2,0)

The y intercept is (0,-1)

Not sure what format you need the answer in, you may only need x = 2 and y = -1.

Answer:

$2.00

Step-by-step explanation:

2.50*.20=.5

2.50-.5= $2

The question involves calculating a 20% discount on a head of lettuce normally priced at $2.50. The discount amounts to $0.50, therefore the lettuce would cost $2.00 on Tuesdays.

The subject of this question is mathematics, specifically numerical problem solving involving discounts. Alan's Produce is having a 20% off sale on all produce on Tuesdays. If a head of lettuce normally costs $2.50, we need to calculate how much it would cost with the discount.

The discount can be calculated by multiplying the original price by the percentage reduction. So, $2.50 (the original price) times 20% (the discount) equals $0.50. This means the head of lettuce is $0.50 cheaper on Tuesdays.

Therefore, to find the discounted price, subtract this amount from the original price: $2.50 - $0.50 equals $2.00. So on Tuesdays, a head of lettuce at Alan's Produce would cost $2.00.

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Answer:

There are 95040 ways to chose the starting five players

The answer is d ⇒ Permutation; Ps - 95040

Step-by-step explanation:

* Lets explain the difference between permutations and combinations

- Both permutations and combinations are collections of objects

- Permutations are for lists (order matters)

- Combinations are for groups (order doesn't matter)

- A permutation is an ordered combination.

- Permutation is nPr, where n is the total number and r is the number

 of choices

# Example: chose the first three students from the group of 10 students

  n = 10 and r = 3,then 10P3 is 720

- Combinations is nCr, where n is the total number and r is the number

 of the choices

# Example: chose a group of three students from the group of 10 students

  n = 10 and r = 3,then 10C3 is 120

* Lets solve the problem

- We want to pick starting five players from a basketball team of

 twelve players

∵ We will pick the starting five

∴ The order is important

∴ We will use the permutations

∵ The total number of the players is 12

∵ The number of choices is 5

∴ n = 12 and r = 5

∵ The number of ways is nPr

∴ 12P5 = 95040

∴ There are 95040 ways to chose the starting five players

Answer is D

Step-by-step explanation:

Answer:

(cxd)(x) = 4x^3 + 18x^2 - 10x

Step-by-step explanation:

We have two functions:

c(x) = 4x – 2

d(x) = x2 + 5x

And we need to find (cxd)(x) which is the multiplication of both functions:

(cxd)(x) = (4x – 2)(x^2 + 5x) = 4x × x^2 + 20x^2 - 2x^2 -10x

= 4x^3 + 18x^2 - 10x

Then: (cxd)(x) = 4x^3 + 18x^2 - 10x

Answer: [tex](c*d)(x)=4x^3+18x^2-10x[/tex]

Step-by-step explanation:

You know that the function [tex]c(x)[/tex] and the function [tex]d(x)[/tex] are:

[tex]c(x) = 4x - 2\\\\d(x) = x^2 + 5x[/tex]

Then, in order to find [tex](c*d)(x)[/tex] you need to multiply the function [tex]c(x)[/tex] by the function [tex]d(x)[/tex]:

[tex](c*d)(x)=(4x - 2)(x^2 + 5x)[/tex]  

You must remember the Product of powers property, which states that:

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

Now you can apply Distributive property:

[tex](c*d)(x)=4x^3+20x^2-2x^2-10x[/tex]

Finally, add the like terms. Then:

[tex](c*d)(x)=4x^3+18x^2-10x[/tex]

Answer:

[tex]y = 3x + 5[/tex]

Step-by-step explanation:

The equation of a line in the pending intercept form has the following form:

[tex]y = mx + b[/tex]

Where m is the slope of the line and b is the intercept with the y axis.

If we know two points [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] that belong to the line then:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]b=y_1-mx_1[/tex]

In this case:

[tex](x_1, y_1)=(2,11)[/tex]

[tex](x_2, y_2)=(4,17)[/tex]

[tex]m=\frac{17-11}{4-2}[/tex]

[tex]m=\frac{6}{2}[/tex]

[tex]m=3[/tex]

[tex]b=11-3(2)[/tex]

[tex]b=11-6[/tex]

[tex]b=5[/tex]

Finally the equation is:

[tex]y = 3x + 5[/tex]

Answer:y=3x-5

Step-by-step explanation:

Answer:

b. it is not a function. it's not a function because I'm does not pass the horizontal lines test

Answer:

The correct option is B.

Step-by-step explanation:

Vertical line test: A vertical line intersects a function's graph at most once.

Horizontal line test: A horizontal line intersects a function's graph at most once.

If a graph passes the vertical line test, then it represents a function.

If a graph passes the horizontal line test, then its inverse is a function.

Check whether the given graph passes horizontal line test or not.

Let x-axis or y=0 be a horizontal line. The curve intersect x-axis at (-2,0) and (2,0).

Since the graph of the function intersect a horizontal line more than one time, therefore it does not passes the horizontal line test and inverse of the given function is not a function.

Hence the correct option is B.

Answer:

We know that in a scalene triangle all sides (and angles) are different.

Let

z = length of longest side

y =  length of second side

x =  length of shortest side

The perimeter of the scalene triangle is 60 cm

x + y + z = 60 cm

The length of the longest side is 4 times that of the shortest side.

z = 4*x

We are left with the following equations

x + y + z = 60

z = 4*x

We will test every affirmation

Number one

The value of x can equal 40.

FALSE. The longest side cannot be greater than the perimeter.

Number two

The longest side can equal 30 cm

z = 30 cm

x = 30/4 cm = 7.5 cm

y = [60 -30 - 7.5] cm  = 22.5 cm

This statement is TRUE

Number three

The shortest side can equal 7 cm

x = 7 cm

z = 4*7 cm = 28 cm

y = [60 -28 - 7] cm  = 25 cm

This statement is TRUE

Number four

The value of x can equal 25

x = 25 cm

z = 4*25 cm = 100 cm

The longest side cannot be greater than the perimeter.

This statement is FALSE

Number five

The shortest side can equal 5

x = 5 cm

z = 4*5 cm = 20 cm

y = [60 -20 - 5] cm  = 35 cm

This statement is FALSE. The second side (y) cannot be greater than the longest side (z).

Answer:

1/5,22%,0.3

Step-by-step explanation:

Please mark brainliest and have a great day!

Answer:

1/5,22%,0.3

Step-by-step explanation:

To put 22% 0.3 1/5 in order, convert all to decimal to determine the least

22% = 22/100

= 0.22

0.3 is already in decimal form

1/5 = 0.2

So, the least is 0.2

and the greatest is 0.3

Answer: 0.2, 0.22 and 0.3

Answer:

2401 • x^2

Step-by-step explanation:

7 √x7 √x • 7 √x • 7

=(7)^4  (√x)^4

=2401 • x^2

Answer: C 1/3

Step-by-step explanation:

The +1/3 is the y-intercept and the -2/9 is the slope.

It is the same as y=mx+b, just that f(x) means function of x and is usually referred to as y.

It can go into 100 feet 4 times. After you add four times you should have 97 and 4/8 or 97 and 1/2

Answer:  She can cut 49 pieces from the ball of string that’s is 100 feet.

Step-by-step explanation:  Given that Carla is cutting pieces of string that are exactly [tex]24\dfrac{3}{8}[/tex] inches long.

We are to find the number of pieces that she can cut from a ball of string with weight 100 feet.

We know that

1 feet = 12 inches.

So, 100 feet = 1200 inches.

Also, [tex]24\dfrac{3}{8}=\dfrac{195}{8}.[/tex]

Now, the number of pieces with length [tex]\dfrac{195}{8}[/tex] inches = 1.

So, the number of pieces with length 1 inch will be

[tex]\dfrac{1}{\frac{195}{8}}=\dfrac{8}{195}.[/tex]

Therefore, the number of pieces that can be cut from 1200 inches is given by

[tex]\dfrac{8}{195}\times1200=49.23.[/tex]

Thus, she can cut 49 pieces from the ball of string that’s is 100 feet.

recall your d = rt, distance = rate * time.

b = rate of the boat in still water

c = rate of the currrent

the distance going upstream is 8 miles, the distance going downstream is also the same 8 miles.

the boat took 1 hour going upstream, now, the boat is not going "b" mph fast, since it's going against the current, the current is eroding speed from, thus the boat going up is really going "b - c" fast.

likewise, when the boat goes downstream, is not going "b" fast either, is going faster because is going with the current and thus is really going "b + c" fast, and we know that trip back took 1/2 hour or 30 minutes.

[tex]\bf \begin{array}{lcccl} &\stackrel{miles}{distance}&\stackrel{mph}{rate}&\stackrel{hours}{time}\\ \cline{2-4}&\\ Upstream&8&b-c&1\\ Downstream&8&b+c&\frac{1}{2} \end{array}\qquad \begin{cases} 8=(b-c)(1)\\ 8+c=\boxed{b}\\ \cline{1-1} 8=(b+c)\left( \frac{1}{2} \right) \end{cases}[/tex]

[tex]\bf \stackrel{\textit{substituting in the 2nd equation}~\hfill }{8=\left(\boxed{8+c}+c \right)\cfrac{1}{2}\implies 8=(8+2c)\cfrac{1}{2}}\implies 16=8+2c \\\\\\ 8=2c\implies \cfrac{8}{2}=c\implies \blacktriangleright 4=c \blacktriangleleft \\\\\\ \stackrel{\textit{we know that }~\hfill }{8+c=b\implies 8+4=b\implies \blacktriangleright 12=b \blacktriangleleft}[/tex]

To find the speed of the boat in still water and the speed of the current, two equations are formed using the distances and times of the upstream and downstream trips. Solving the equations reveals that the speed of the boat is 12 miles/hour and the speed of the current is 4 miles/hour.

The problem describes a boat moving upstream and then downstream in a river with a constant current. To solve for the speed of the boat in still water and the speed of the current, we use some simple algebra based on the relative speeds of the boat and the current.

Let's denote:

When going upstream, the boat's effective speed is (b-c), and when going downstream, it is (b+c). We are told that the boat went 8 miles upstream in 1 hour and the return trip only took 30 minutes (or 0.5 hours).

Therefore, for the upstream trip:

Distance = Speed × Time

8 miles = (b - c) × 1 hour

8 = b - c ...... (equation 1)

For the downstream trip:

8 miles = (b + c) × 0.5 hours

16 = b + c ...... (equation 2)

Solve the two equations simultaneously:

Add equation 1 and equation 2:

8 + 16 = (b - c) + (b + c)

24 = 2b

b = 12 miles/hour

Substitute b into equation 1:

8 = 12 - c

c = 4 miles/hour

Therefore, the speed of the boat in still water is 12 miles/hour and the speed of the current is 4 miles/hour.

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