The length of a rectangle is 3 less than twice the width. If the perimeter of the rectangle is 78 inches, find the dimensions (length and width) of the rectangle.

[tex]~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{"a"~is~negative}{op ens~\cap}\qquad \stackrel{"a"~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x=3\\ y=-3 \end{cases}\implies y=a(x-3)^2 - 3\hspace{5em}\textit{we also know that} \begin{cases} x=2\\ y=-1 \end{cases} \\\\\\ -1=a(2-3)^2 -3\implies 2=a(-1)^2\implies 2=a~\hfill \boxed{y=2(x-3)^2 -3}[/tex]

Answer:

Step-by-step explanation:

Let's call the speed of the boat in the river x miles per hour. The boat travels 2 miles downstream in the same amount of time it takes to travel 2 miles upstream, so the speed of the boat relative to the water is the same in both directions. We can set up the following equation to represent this relationship:

21 - x = x + current speed

Solving for x, we find that the speed of the boat in the river is (21 - x)/2 = 10.5 miles per hour.

We can then use this value to find the current speed of the river. The boat travels 2.6 miles downstream in the same amount of time it takes to travel 2 miles upstream, so the speed of the boat relative to the ground is the same in both directions. We can set up the following equation to represent this relationship:

10.5 + current speed = 10.5 - current speed

Solving for the current speed, we find that it is (10.5 - 10.5)/2 = 0.

Therefore, the current speed of the river is 0 miles per hour.

The line parallel to line y= -x + 7  is y= -x + 9.

What is slope intercept form?

The intercept formula y = mx + b is used when you know the slope of the line you are examining, and the specified point is also the y-intercept (0, b). In this formula, b represents the y-value of the y-intercept.

Solution: We will use slope intercept formula y= mx + b

Given that the parallel line passes through (4,5), which means x=4 and y=5

Also, m = -1

Putting value of x and y in slope intercept form to find b, we get

y=mx + b

⇒ 5= (-1x4) + b

⇒ 5= -4 +b

⇒5 + 4= b

⇒b=9

Now putting value of b and m in slope intercept form, we get the parallel line

y= -x + 9

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Answer:

[tex]x=-7-\sqrt{11}, -7+\sqrt{11}[/tex]

Step-by-step explanation:

[tex](x+7)^2=11 \\ \\ (x+7)=\pm \sqrt{11} \\ \\ x=-7-\sqrt{11}, -7+\sqrt{11}[/tex]

Step-by-step explanation:

32^2-24^2=448

square root 448=21

the small and large parallel line is 21=21

24+9=33

33^2+21^2=1530

square root 1530=39

39-32=7

~7~=6.75

x=6.75

The elements of the required sets would be

(a) (P∩Q)∪R = {1, 2, 3, 4, 5, 7}

(b) (R∩P)'∪R = {2, 3, 4, 5, 6, 7, 8, 9, 10}

(c) Q'∩(P∪R) = {2, 3, 5, 7}

(d) (P∪Q∪R)'∩(P∩R) = ∅

What are sets?

A set is a mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

Given:

Let the universal set is (S) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

P = {Odd numbers}  = {1, 3, 5, 7, 9}

Q = {Prime Numbers} = {2, 3, 5, 7}

R = {1, 2, 3, 4, 5}

So by using the properties of sets we can find

P' = {2, 4, 6, 8, 10}

Q' = {1, 4, 6, 8, 9, 10}

R' = {6, 7, 8, 9, 10}

P∩Q = {3, 5, 7}

R'∪P' = {2, 4, 6, 7, 8, 9, 10}

P∪R = {1, 2, 3, 4, 5, 7, 9}

P∩R = {1, 3, 5}

P∪Q∪R = {1, 2, 3, 4, 5, 7, 9}

P'∩Q'∩R' = S - P∪Q∪R

                = {6, 8, 10}

Now we can find the elements of the required sets:

(a) (P∩Q)∪R = {1, 2, 3, 4, 5, 7}

(b) (R∩P)'∪R = {2, 3, 4, 5, 6, 7, 8, 9, 10}

(c) Q'∩(P∪R) = {2, 3, 5, 7}

(d) (P∪Q∪R)'∩(P∩R) = ∅

Hence, the elements of the required sets would be

(a) (P∩Q)∪R = {1, 2, 3, 4, 5, 7}

(b) (R∩P)'∪R = {2, 3, 4, 5, 6, 7, 8, 9, 10}

(c) Q'∩(P∪R) = {2, 3, 5, 7}

(d) (P∪Q∪R)'∩(P∩R) = ∅

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Answer:

10-x

Step-by-step explanation:

The greatest common factor between the given two numbers will be 20.

The greatest common factor is the largest positive number which can be divided into given numbers evenly.

Given that the two numbers are 1240 and 6660. First, factorize each number and find the common multiple of both numbers.

1240 = 2 x 2 x 2 x 5 x 31

6660 = 2 x 2 x 3 x 3 x 5 x 37

The common numbers are,

2 x 2 x 5

4 x 5

20

Therefore, the greatest common factor between the given two numbers will be 20.

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The first number is -4 and the second number is 7

A word problem is a mathematical exercise where significant background information on the problem is presented in ordinary language rather than in mathematical notation.

Let  first number = n

second number = n+11

2(n+n+11)= 6

expanding the bracket

2(2n+11)=6

4n+22 =6

collecting like terms

4n = -16

n=-4

The first number = n =-4

second number = n+11=-4+11=7

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Answer:

Step-by-step explanation:

2,400 consoles

The total area of the given figure is 68.56 units².

The area is the entire amount of space occupied by a flat (2-D) surface or an object's shape.

On a sheet of paper, draw a square using a pencil.

It has two dimensions.

The area of a shape on paper is the area that it occupies.

Consider your square as being composed of smaller unit squares.

So, the total area of the given figure:
We can divide the figure into separate rectangles and circles:

Area of rectangle: l*b

14*4

56 units²

Area of circle: πr² (D = 4, R = 4/2 = 2)

πr²

3.14(2)²

3.14*4

12.56 units²

The total area of the figure:

56 units² + 12.56 units² = 68.56 units²

Therefore, the total area of the given figure is 68.56 units².

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After solving, the value of f^(-1)(g(7)) is 6.

In the given question, we have to find the f^(-1)(g(7)).

The given table of functions f and g is:

x          f(x)                   y          g(x)  

-3          1                     -2          -5

0          4                     1            -4

1           5                     5            2

6         -6                     7           -6

To find the value of f^(-1)(g(7)) we put the value of g(7) from the table. As we can see in the table that at the value of g(x) at point 7 is -6.

Then we find the value of from the table of f(x).

f^(-1)(g(7)) = f^(-1)(-6)

The value of f^(-1)(-6) from the table f(x) is

f^(-1)(g(7)) = 6

Hence, the value of f^(-1)(g(7)) is 6.

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Answer:

[tex]\huge{\boxed{46 \text{ or } 67 \text{ or } 88 \text{ degrees}}}[/tex]

Step-by-step explanation:

An isosceles triangle is a triangle in which two of the sides and angles are equal.  Because the sum of all of the angles in a triangle is 180°, you may setup the following equation.

46 + x + z = 180°

Depending on the following scenario, your answer may be one of two options.  If the "third angle" refers to the angle that is not at the base of the triangle, then your equation is as follows:

z = x

46 + x + x = 180

46 + 2x = 180

2x = 134

x = 67

If the third angle is one of the base angles and x is a base angle, then x is also 46, because base angles are equal.

If the third angle is one of the base angles and x is the third angle, then your equation is as follows:

z = 46

46 + 46 + x = 180

92 + x = 180

x = 88

Hope it helps :) and let me know if you want me to elaborate.

Median won't change, however Mean will fall by 180.

By summing all the numbers in a data collection and dividing by the total number of values in the set, one may get the mean (average) of the data set. When a data collection is sorted from least to largest, the median represents the midpoint value.

African elephants total 555 in number.

Elephants had a 3800 kg average mass.

Elephants had a 3600 kg median mass.

Lola, the elephant that was the smallest, weighed 2700 kg.

Later, Lola became really ill and lost a lot of weight, reaching a mass of 1800 kg.

What effect will Lola's weight loss have on the mean and median as well as Currently, the median is 3600 and the lowest is 2700.

now 2700 becomes 1800

so Median remains the same as 3600

No change in Median

The mean mass of the elephants was  3800 kg

=> total weight = 5 x 3800 = 19000 kg

2700 kg becomes 1800 kg

Hence new total mass = 19000 - 2700 + 1800 = 181000 kg

New Mean = 18100/5   = 3620  kg

Mean reduced by 3800 - 3620  = 180 kg

No change in Median weight

Mean reduced by 180 kg

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Answer: 3.62 inches

Step-by-step explanation:

Diego's cousin is too short for this ride, we know how tall he is now and what height he needs to be.

This can be solved by subtracting his needed height from his current height.

39.37-35.75=3.62 inches

Answer:

Step-by-step explanation:

mid point = a+b/2

let b(x,y)

-2 = [7+x]/2                      3 = [-6+y]/2

-4 = 7 + x                          6 = -6 + y

-11 = x                                12 = y

Hence B is at (-11,12)

Answer:

6

Step-by-step explanation:

Using PEMDAS, -32 is solution of (4/5)² x (-25) - 4².

A rule that specifies the right procedure to follow while evaluating a mathematical equation is known as the order of operations. Parentheses, Exponents, Multiplication and Division (from Left to Right), Addition and Subtraction are the steps that we can remember in that order using PEMDAS (from left to right). We are provided with the proper steps for solving a mathematical expression by the PEMDAS rule. Operations are carried out in parenthesis first according to the PEMDAS rule. The next step is to conduct operations on exponents or powers. Then, depending on which occurs first, the operations of multiplication and division are performed from left to right. The final step is to conduct addition or subtraction from left to right, depending on which comes first.

Given

(4/5)² x (-25) - 4²

Using PEMDAS,

Squaring the exponents,

16/25 × -25 - 16

Multiplying,

-16 - 16

Subtracting,

- 32

Using PEMDAS, -32 is solution of (4/5)² x (-25) - 4².

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Answer:

Angle-Side-Angle or also known as ASA

Step-by-step explanation:

Answer:

  -1/2

Step-by-step explanation:

You want the slope of the line through points (8, -3) and (-6, 4) using the slope formula.

The formula for the slope of the line through points (x1, y1) and (x2, y2) is ...

  m = (y2 -y1)/(x2 -x1)

For the two given points, the slope is ...

  m = (4 -(-3))/(-6 -8) = 7/-14 = -1/2

The slope of the line is -1/2.

Answer:

1.3 is the common difference.

Step-by-step explanation:

The time it takes the trains 2.2 hours to be 396 miles apart.

Time is the duration of an event.

Since two trains leave the station at the same time, one heading west and the other east. The westbound train travels at 85 miles per hour. The eastbound train travels at 95 miles per hour.

Let

Also,

Since both trains move in opposite directions and leave at the same time, their distance apart after time t is D' = d + D

= vt + Vt

= 85t + 95t

= 180t

Since D' = 396 miles, we have that

396 = 180t

t = 396 m/180 mph

t = 2.2 hours

So, it takes the trains 2.2 hours to be 396 miles apart.

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With the help of AA similarity rule option C,D,E,F will be the answer which is C:-∠ adc is 52.8°

D:- ∠ dac is 26.7°

E:- dac/adc = 0.506

F:- cd/bc =1.8

What is AA similarity rule?

If two triangles have an equal number of corresponding sides and an equal number of corresponding angles, then they are comparable.

Similar figures are items that feature more than two figures that are the same form but different sizes.

When two triangles are comparable, all of their matching angle pairs and sides are congruent and proportionate. However, you do not necessarily need to have information about all of the sides and angles of two triangles in order to be certain that they are identical.

AA similarity rule

So, ∠ adc is 52.8°

and ∠ dac is 26.7°

So, the ratio of dac/adc = 0.506

And the ratio of side will also be equal as well which is 1.8

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The number of ways the 6 books can be arranged on the 4 shelves by Amanda is 360 ways

From the question, we have the following parameters that can be used in our computation:

Total number of books, n = 6

Total number of shelves, r = 4

The number of ways the books could be arranged on the shelves is calculated using the following permutation formula

Total = ⁿPᵣ

Where

n = 6 and r = 4

Substitute the known values in the above equation, so, we have the following representation

Total = ⁶P₄

Apply the permutation formula

ⁿPᵣ = n!/(n - r)!

So, we have the following representation

Total = 6!/2!

Evaluate the quotient

Total = 360

Hence, the number of ways is 360

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The distance of the student from the light pole is 21.4 feet.

A student sees a bird on top of a 15 foot high light pole. The angle of elevation to the top of the light pole is 35 degrees.  

The distance of the student from the pole can be calculated as follows:

The situation forms a right angle triangle.

using trigonometric ratios,

tan 35  = opposite / adjacent

tan 35 = 15 / x

cross multiply

x tan 35 = 15

divide both sides by tan 35

x = 15 / tan 35

x = 15 / 0.70020753821

x = 21.4222201012

x = 21.4 feet

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A table of values that models this situation is as follows;

Distance (miles)               Time (hour)

120                                          0

150                                          1

180                                          2

210                                          3

240                                          4

An equation for D in terms of t representing Elizabeth's distance from city A, t hours after leaving her house is d = 30t + 120.

Mathematically, a constant of proportionality (time) for Elizabeth's journey (distance) with respect to time can be represented with the following mathematical expression:

d = kt

Where:

Therefore, the required function that relates the distance (d) and speed (s) at t hours is given by;

d = 30t + 120

Next, we would develop a table of value;

At time (t) = 1 hour, we have;

Distance, d = 30(1) + 120

Distance, d = 150 miles.

At time (t) = 2 hours, we have;

Distance, d = 30(2) + 120

Distance, d = 180 miles.

At time (t) = 3 hours, we have;

Distance, d = 30(3) + 120

Distance, d = 210 miles.

At time (t) = 4 hours, we have;

Distance, d = 30(4) + 120

Distance, d = 240 miles.

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As ΔADE and ΔABC are similar because of their slope so correct option is A.

What are similar triangles?

Similar triangles square measure triangles that have identical form, however their sizes might vary. All equal triangles, squares of any facet lengths square measure samples of similar objects.

Main Body;

let co-ordinates of the points

A = (x₁ ,y₁ ) E =(x₂, y₂)

D =( x₃,y₃) C= (x₄ ,y₄)  B =( x₅,y₅)

Now, Slope of a line = Δy/Δx

.: Slope of line AD = y₃ -y₂/x₂-x1 .

                              = DE/AE

... Slope of line AB = y₅-y₄/x₄-x₁ = BC/ AC

Since ΔADE ~ ΔABC

Now , Slope of AD = slope of AB

=> DE/ AE = BC /AC

In option A

By property of Similar triangle -

DE /BC  = BC /АC

Hence, option A is correct answer.

All other options are incorrect

Since Sides of a similar triangle may not be equal.

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Answer:

3= 26

4= 11 because it is area of triangle

The inequality that shows how long the temperature in Round Rock  will remain colder than that in Austin is 72 - 3t > 65 - 2t and the time is less than 13 hours

From the question, we have the following parameters that can be used in our computation:

Austin

Initial temperature = 65 degrees Fahrenheit

Rate of decrement = 2 degrees Fahrenheit

Round Rock

Initial temperature = 72 degrees Fahrenheit

Rate of decrement = 3 degrees Fahrenheit

The above parameters mean that

Current temperature = Initial temperature + Rate of increment x Number of hours

So, we have

Austin: A(t) = 65 - 2t

Round Rock: B(t) = 72 - 3t

When Round Rock is colder than Austin, we have

B(t) > A(t)

Substitute the known values in the above equation, so, we have the following representation

72 - 3t > 65 - 2t

Evaluate the like terms

-t > -13

Divide both sides by -1

t < 13

Hence, the interval is less than 13 hours

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