[tex]\bf \stackrel{\textit{difference of squares}}{(6+3i)(6-3i)}\implies (6)^2-(3i)^2\implies \stackrel{\textit{recall }i^2=-1}{36-(3^2 i^2)}\implies 36-(9\cdot -1) \\\\\\ 36+9\implies 45[/tex]
Step-by-step explanation:
6*6 - 6*3i + 6*3i -3i*3i
36-18i+18i-9i^2
36-9(-1)= 36+9= 45
Mrs. Bell graded 6.5 students' tests in 0.5 hours. At this rate, how long will i
grade 30 students' papers? State your answer in hours and minutes
Answer:
2 hours 18 minutes (to the nearest minute)
Step-by-step explanation:
6.5 / 0.5 hours
--> Multiply the whole equation by 4.61538461538 so that 6.5 becomes 30
= 30 papers in 2.30769230769 hours = 02:18:28
Mrs. Bell will take 2 hrs 18 mins approximately.
What is the unitary method?The unitary approach is a strategy for problem-solving that involves first determining the value of a single unit, then multiplying that value to determine the required value.
Given that 6.5 students' papers can be checked in 0.5 hours
1 student's paper can be checked in 0.5/6.5 hours
30 students' paper can be checked in (0.5 /6.5)*30 hours = 2.307 hours
Therefore, Mrs. Bell will take 2 hrs 18 mins approximately.
To know more about the unitary method refer to :
https://brainly.com/question/27944694
#SPJ2
what substitution should be used to rewrite 6(x+5)^2+5(x+5)-4=0 as a quadratic equation
Step-by-step explanation:
6 (x+5 )^2 +5(x+5)-4=0
let x+5= y
6y^2 + 5y - 4=0
6y^2 + 8y - 3y-4 =0
2y (3y+4) - 1 (3y+4)=0
(2y-1)(3y+4)=0
2y-1=0 or 3y+4=0
y=1/3 or -4/3
Recall that x+5=y
When y=1/3
x+5=1/3
x= -14/3
when y= -4/3
x+5=-4/3
x = -11/3
Answer:
u=(x+5)
Step-by-step explanation:
I leave my family's vacation cabin at 8 a.m. and start driving home at a nice, safe 45 mph. Two hours later, my husband, who always drives as fast as the law will allow, leaves the cabin and starts driving home 65 mph. When can I expect him to pass me?
Answer:
2:30 pm
Step-by-step explanation:
In two hours (from 10am), she is 45 * 2 = 90 miles from vacation cabin.
From then on her average is still 45 mph and her husband's is 65 mph. So, husband catches her 20 miles (65-45=20) in every hour.
The husband has to catch up 90 miles. How long would it take if he is catching up at 20 miles an hour?
90/20 = 4.5 hours
So husband starts at 10am and 4.5 hours form that time is 2.30pm
y = –x + 4
x + 2y = –8
Answer:
(16,-12)
Step-by-step explanation:
To find where these lines intersect, we must first know their slope. If the slope is equal, then the lines don't intersect and they are parallel.
The best formula for knowing slope is slope-intercept form, y=mx+b whereas m is the slope and b is the y-intercept (where the line crosses x=0)
The first equation is already in slope-intersect form so we can substitute the section that is equal to y into the second equation where y is.
x + 2(-x + 4) = -8
x + -2x + 8 = -8
-x + 8 = -8
-x = - 16
x = 16
Now we can use this known variable to find y.
y = -(16) + 4
y = -16 + 4
y = -12
What is the equation of the line perpendicular to 2x – 3y = 13 that passes through the point (–6, 5)?
Answer:
[tex]y =- \frac{3}{2}x - 4[/tex]
Step-by-step explanation:
Given equation of line is:
2x-3y=13
We will convert the equation of line in point-slope form to find the slope of the line
Let
m_1 be the slope of the line
So,
2x-3y=13
-3y= -2x+13
Dividing both sides by -3
(-3y)/(-3)=(-2x)/(-3)+13/(-3)
y=(2/3)x-13/3
The co-efficient of x is the slope of the line.
So,
m_1=2/3
Let
m_2 be the slope of second line
As we know that product of slopes of two perpendicular line is -1
m_1 m_2= -1
2/3*m_2= -1
m_2= -1*3/2
m_2= -3/2
So m2 is the slope of the line perpendicular to given line.
The standard equation of a line is
y=mx+b
To find the equation of line through (-6,5), put the point and slope in the given form and solve for b
5= -3/2 (-6)+b
5=18/2+b
5=9+b
b=5-9
b= -4
Putting the values of slope and b, we get
[tex]y =- \frac{3}{2}x - 4[/tex]
Answer: [tex]y=-\frac{3}{2}x-4[/tex]
Step-by-step explanation:
The equation of the line in Slope-intercept form is:
[tex]y=mx+b[/tex]
Where "m" is the slope and "b" is the y-intercept.
To express the given equation of the line in this form, we need to solve for "y":
[tex]2x - 3y = 13\\\\-3y=-2x+13\\\\y=\frac{-2}{-3}x+\frac{13}{-3}\\\\y=\frac{2}{3}x-\frac{13}{3}[/tex]
We can identify that the slope of this line is:
[tex]m=\frac{2}{3}[/tex]
Since the slopes of perpendicular lines are negative reciprocals, then the slope of the other line is:
[tex]m=-\frac{3}{2}[/tex]
Now, we need to substitute the given point and the slope into [tex]y=mx+b[/tex] and solve for "b":
[tex]5=-\frac{3}{2}(-6)+b\\\\5=9+b\\\\5-9=b\\\\b=-4[/tex]
Substituting values, we get that the equation of this line is:
[tex]y=-\frac{3}{2}x-4[/tex]
Please help with this question!
At a fruit stand, apples cost 80 cents for x apples and oranges cost 75 cents for y oranges. Which of the following represents the cost, in dollars, of 5 apples and 6 oranges?
Answer:
($4.00/x)+($4.50/y)
Step-by-step explanation:
we know that
Apples cost 80 cents for x apples
so
One apple cost $0.80/x
Oranges cost 75 cents for y oranges
so
One orange cost $0.75/y
Find the cost of 5 apples and 6 oranges
Multiply the number of apples by the cost of one apple and multiply the number of oranges by the cost of one orange
so
5($0.80/x)+6($0.75/y)=($4.00/x)+($4.50/y)
Final answer:
The cost of 5 apples at 80 cents each is $4.00 and the cost of 6 oranges at 75 cents each is $4.50. The total cost for 5 apples and 6 oranges is therefore $8.50.
Explanation:
To calculate the cost, in dollars, of 5 apples and 6 oranges at a fruit stand where apples cost 80 cents for x apples and oranges cost 75 cents for y oranges, you can follow these steps:
Determine the cost per apple by dividing 80 cents by x. Since we do not have the value of x, assume x equals 1 for this explanation, making the cost 80 cents per apple.
Multiply the cost per apple by 5 to find the total cost for 5 apples.
Determine the cost per orange by dividing 75 cents by y. Assuming y equals 1 for this explanation, the cost is 75 cents per orange.
Multiply the cost per orange by 6 to find the total cost for 6 oranges.
Add the total cost for apples and oranges to find the combined cost, converting cents to dollars as necessary.
Therefore, the cost for 5 apples is 5 × 0.80 = $4.00, and the cost for 6 oranges is 6 × 0.75 = $4.50. Adding these together, the total cost for 5 apples and 6 oranges is $4.00 + $4.50 = $8.50.
A radar operator on a ship discovers a large sunken vessel lying fiat on the ocean
floor 200 m directly beneath the ship. The operator measures the angks of
depression of the front and back of the sunken ship to be 56° and 62°. How
long is the sunken ship?
The length of the sunken ship is 28.56m
To calculate the length of the sunken ship , we have to calculate the distance between the beneath of the ship to the front and to the back.
distance between the beneath of the ship to back of the sunken ship
Tan 62 = 200/x
x = 200/tan 62
x = 106.34m
distance between the beneath of the ship to front of the sunken ship
Tan 56 = 200/y
y = 200/tan 56
y = 134.90m
The length of the sunken ship = 134 .90 - 106.34
= 28.56m
What would y=
4.6
13.9
21.6
Answer:
13.9 is the right answer :)
Step-by-step explanation:
For this case we have to define trigonometric relations of rectangular triangles, that the tangent of an angle is given by the leg opposite the angle on the leg adjacent to the angle.
Then, according to the figure we have:
[tex]tg (60) = \frac {y} {8}\\y = 8 * tg (60)\\y = 8 * 1.73205081\\y = 13,85640648[/tex]
Rounding the value we have that [tex]y = 13.9[/tex]
ANswer:
Option B
Help. Find the value of x to the nearest tenth
I will assume that line 4 bisect line x exactly in half. This means that if we solve the last side of the triangle like seen in the pic below we will know what half of line x is
To find the length of the last side of the triangle (let's call this y) you must use Pythagorean theorem
[tex]a^{2} +b^{2}=c^{2}[/tex]
a and b are the legs (the sides that form a perpendicular/right angle)
c is the hypotenuse (the side opposite the right angle)
In this case...
a = 4
b = y
c = 6
^^^Plug these numbers/variables into the theorem
[tex]4^{2} +y^{2} =6^{2}[/tex]
solve for y
16 + [tex]y^{2}[/tex] = 36
[tex]y^{2}[/tex] = 20
x = √20
x ≈ 4.472
^^^This is only half of x so if you double it then it will get you the full length of x
4.472 * 2 = 8.944
8.9 (D)
Hope this helped!
~Just a girl in love with Shawn Mendes
A student needs at least 80% of all points on the tests in a class to get a B. There are three
100-point tests and a 250-point final. The student's test scores are 72%, 85%, and 78% for
the 100 point tests. What is the least score the student can make on the final and still get a B?
How many points are there total? (3*100)+250 = 550
So far, he has 72 + 85 + 78, or 235
He needs 0.80 * 550 for a B, or 440 points
440 - 235 = 205 points on the final to get a B
Percentage-wise, he needs 205/250, or 0.82
Can someone help me?
Is it possible that cos(A−B) = cos A−cos B? Why or why
Answer:
Distributive Property.
Step-by-step explanation:
Cos(A-B)= Cos A- Cos B
The variables in the parenthesis gets multiplied into the Cos
Cos A - Cos B
∠MPR is an acute angle and PQ is in the interior of ∠MPR. ∠QPR must be
A. Straight
B. Obtuse
C. Right
D.Acute
Answer:
∠QPR must be an acute angle
Step-by-step explanation:
Given : ∠MPR is an acute angle and PQ is in the interior of ∠MPR.
To Find :∠QPR must be?
Solution:
Acute Angle : An angle whose measure is less than 90°
Since ∠MPR is an acute angle
This means ∠MPR is less than 90°
Now PQ is in the interior of ∠MPR
So, referring the attached figure
∠QPR should be less than ∠MPR
So, the measure of ∠QPR should be less than ∠MPR
So, the measure of ∠QPR is also less than 90°
So, ∠QPR must be an acute angle
So, Option D is correct.
Answer: D. Acute
Step-by-step explanation:
We are given that : ∠MPR is an acute angle and PQ is in the interior of ∠MPR.
Definition :
Acute Angle : An angle having measure is less than 90°
.
Obtuse angle : An angle having measure is greater than 90° but less than 180
°.
Right angle : An angle whose measure exactly 90°
.
Straight angle : An angle whose measure exactly 180°
.
∵∠MPR is an acute angle ,
Then ∠MPR must be less than 90°
.
Also, PQ is in the interior of ∠MPR
.
⇒∠QPR should be less than ∠MPR
.
⇒ the measure of ∠QPR is also less than 90°
⇒∠QPR must be an acute angle . [By definition of Acute angle.]
Hence, the correct answer is option D.
Helpppppppp plssssssssss
your correct it is 240
lengthxwidthxheight
8x6x5=240
The students in a class were asked how many siblings they have. The data obtained is represented in the dot plot. The number of students who have no siblings is . The number of students who have three or more siblings is .
Answer:
0 Siblings: 4 Students
3 or more siblings: 5 students
Step-by-step explanation:
The numbers at the bottom represents how many siblings they have and the dots represent the number of studens so using this we know:
0 Siblings: 4 Students
1 Sibling: 3 Students
2 Siblings: 3 Students
3 Siblings: 1 Student
4 Siblings: 0 Students
5 Siblings: 2 Students
6 Siblings: 1 Student
7 Siblings: 1 Student
3 or more: 1 + 0 + 2 + 1 + 1 = 5
3 or more siblings: 5 students
Answer:
0 Siblings: 4 Students3 or more siblings: 5 students
Step-by-step explanation:
HELPP ME PLEASE ASPAAAAAAP PLEASEEEEE
Answer:
4Step-by-step explanation:
1, 2 and 3 are interior angles
4 is the exterior angle
What percent is represented by the shaded area? ( sorry if the picture is sideways)
Answer:
40%
Step-by-step explanation:
2 out of 5 of the rectangles are shaded. With a ratio of 2/5 this represented as a percent is 40%
Answer:
40%.
Step-by-step explanation:
2 out of 5 rectangles are shaded.
As a percentage that is 2/5 * 100 = 40%.
What is the distance between (6, 2) and (-3, -2)?
9
3
5
Answer:
Option A is correct.
Step-by-step explanation:
Distance between two points can be calculated from formula
[tex]d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 }[/tex]
Here x₁ = 6, y₁ = 2,x₂ = -3 and y₂ = -2
Putting the values in the formula
[tex]d = \sqrt{(-3-(6))^2+(-2-(2))^2}\\d = \sqrt{(-3-6)^2+(-2-2)^2} \\d = \sqrt{(-9)^2+(-4)^2} \\d = \sqrt{81+16}\\d = \sqrt{97}\\d= 9.85[/tex]
So, the distance between points (6, 2) and (-3, -2) is 9.85 or ≈ 9.
So, Option A is correct.
ANSWER
[tex] \sqrt{ 97 }[/tex]
EXPLANATION
The formula for calculating the distance between two points is
[tex]d = \sqrt{ {(x_2-x_1)}^{2} +(y_2-y_1)^{2} } .[/tex]
We use this formula to find the distance between (6,2) and (-3,-2)
We plug in the points to get,
[tex]d = \sqrt{ {( - 3-6)}^{2} +( - 2-2)^{2} } .[/tex]
[tex]d = \sqrt{ {( - 9)}^{2} +( - 4)^{2} } .[/tex]
[tex]d = \sqrt{ 81+16}[/tex]
[tex]d = \sqrt{ 97 }[/tex]
The distance between the two given points is
[tex] \sqrt{ 97 }[/tex]
Write [tex]\frac{4}{50}[/tex] as a percentage
Answer:
8%
Step-by-step explanation:
[tex]4 \div 50 \times 100 \: = \: 8[/tex]
Hello There!
[tex]\frac{4}{50}[/tex] as a percentage is 8%
You can first reduce this fraction by dividing both the numerator and denominator by the Greatest Common Factor of 4 and 50 using 2.
4 ÷ 2 ≈ 2
50 ÷ 2 ≈ 25
We now have a fraction [tex]\frac{2}{25}[/tex]
We now divide 2 by 25 and we get a quotient of 0.08
Finally we multiply 0.08 by 100 and we get 8 so 4/50 as a percentage is 8%
Which statement about this figure is true?
It has rotational symmetry with an angle of rotation of 30°.
It has no reflectional symmetry.
It has no rotational symmetry.
It has reflectional symmetry with 6 lines of symmetry.
Answer:
b and d
Step-by-step explanation:
The figure has 12 peaks. A full circle has 360 degrees.
(360 degrees)/12 = 30 degrees.
b. It has rotational symmetry with an angle of rotation of 30 deg.
If you draw a line between each pair of opposite peaks, that line is an axis of reflectional symmetry. Since there are 12 peaks, there are 6 such lines.
d. It has reflectional symmetry with 6 lines of symmetry.
Answer:
It has rotational symmetry with an angle of rotation of 30°.
Step-by-step explanation:
A shape is said to possess rotational symmetry, when it still looks the same after some rotation.
We can see, that the given shape looks the same from every angle and every side.
So, the correct answer is: It has rotational symmetry with an angle of rotation of 30°.
30 degrees because it has 12 angles and [tex]360/12=30[/tex] degrees.
NEED HELP 50 POINTS!! WILL MARK BRAINLIEST ANSWER. Find the lengths of all the sides and the measures of the angles.
Answer:
Part 1) ∠ABD=23.9°
Part 2) ∠ADB=119.1°
Part 3) AB=506.7 ft
Part 4) ∠BDE=60.9°
Part 5) ∠BED=77.1°
Part 6) DE=239.6 ft
Part 7) BE=312.8 ft
Part 8) ∠BEC=102.9°
Part 9) ∠EBC=36.1°
Step-by-step explanation:
Let
A-----> Zebra house
B ----> Entrance
C ----> Tiger house
D ---> Giraffe house
E ----> Hippo house
see the attached figure with letters to better understand the problem
step 1
In the triangle ABD
Find the measure of angle ABD
Applying the law of sines
sin(37°)/349=sin(ABD)/235
sin(ABD)=235*sin(37°)/349
sin(ABD)=0.4052
∠ABD=arcsin(0.4052)=23.9°
step 2
In the triangle ABD
Find the measure of angle ADB
Remember that the sum of the internal angles of a triangle must be equal to 180 degrees
so
∠ADB+23.9°+37°=180°
∠ADB+23.9°+37°=180°-60.9°
∠ADB=119.1°
step 3
In the triangle ABD
Find the measure of side AB
Applying the law of sines
sin(37°)/349=sin(119.1°)/AB
AB=349*sin(119.1°)/sin(37°)
AB=506.7 ft
step 4
In the triangle BDE
Find the measure of angle BDE
we have
∠BDE+∠ADB=180° ----> by supplementary angles
∠ADB=119.1°
substitute
∠BDE+119.1°=180°
∠BDE=180°-119.1°
∠BDE=60.9°
step 5
In the triangle BDE
Find the measure of angle BED
Remember that the sum of the internal angles of a triangle must be equal to 180 degrees
so
∠BED+60.9°+42°=180°
∠BED=180°-102.9°
∠BED=77.1°
step 6
In the triangle BDE
Find the measure of side DE
Applying the law of sines
sin(77.1°)/349=sin(42°)/DE
DE=349*sin(42°)/sin(77.1°)
DE=239.6 ft
step 7
In the triangle BDE
Find the measure of side BE
Applying the law of sines
sin(77.1°)/349=sin(60.9°)/DE
BE=349*sin(60.9°)/sin(77.1°)
BE=312.8 ft
step 8
In the triangle BEC
Find the measure of angle BEC
we have
∠BEC+∠BED=180° ----> by supplementary angles
∠BED=77.1°
substitute
∠BEC+77.1°=180°
∠BEC=180°-77.1°
∠BEC=102.9°
step 9
In the triangle BEC
Find the measure of angle EBC
Remember that the sum of the internal angles of a triangle must be equal to 180 degrees
so
∠EBC+102.9°+41°=180°
∠EBC=180°-143.9°
∠EBC=36.1°
What number belongs to the solution of the inequality
3x>96
[tex]\text{Hey there!}[/tex]
[tex]\text{3x}>\text{96}[/tex]
[tex]\text{You have to DIVIDE by 3 on BOTH OF YOUR SIDES.}[/tex]
[tex]\dfrac{3\text{x}}{3}>\dfrac{96}{3}[/tex]
[tex]\text{Cancel out: }\dfrac{3\text{x}}{3}\text{ because it equals to 1}[/tex]
[tex]\text{Keep: }\dfrac{96}{3}\text{ because it helps us solve for our answer, to compare}[/tex] [tex]\text{both numbers.}[/tex]
[tex]\text{96}\div3\text{ = 32}[/tex]
[tex]\boxed{\boxed{\bf{Answer:x > 32}}}\checkmark[/tex]
[tex]\text{It's a}[/tex] [tex]\text{(o)(p)(e)(n)(e)(d) circle shaded to the right side of the number line}[/tex]
[tex]\text{Good luck on your assignment and enjoy your day!}[/tex]
~[tex]\frak{LoveYourselfFirst:)}[/tex]
Answer:
x>32
Step-by-step explanation:
Divide by 3 from both sides of equation.'
3x/3>96/3
Simplify, to find the answer.
96/3=32
x>32 is the correct answer.
I hope this helps you, and have a wonderful day!
-11 - (-5)
————
2 x 3
Is an example of
A.) an algebraic expression
B.) A numerical equation
C) A numerical equation
D.) An algebraic equation
Answer:
A.) an algebraic expression
Step-by-step explanation:
Note that there are only integers (whole numbers) and the usage of algebraic operations. Also note that there is no "equal" sign (which would make it an equation), making the problem a expression.
~
Are the graphs of the lines in the pair parallel? Explain.
y = 2/3x– 17
4x – 6y = –6
Answer:
Yes, they are parallel.
Step-by-step explanation:
Parallel lines have the same slope. We must find the slopes of the two lines.
When the equation of a line is written in the slope-intercept form,
y = mx + b,
the slope is m.
The first line has equation
[tex] y = \dfrac{2}{3}x - 17 [/tex]
It is already written in the slope-intercept form. Comparing y = 2/3x - 17 with y = mx + b, you see that m = 2/3. The slope of the first line is 2/3.
Now we solve the second equation for y to obtain the slope-intercept form of that equation.
4x - 6y = -6
Subtract 4x from both sides.
-6y = -4x - 6
Divide both sides by -6.
[tex] \dfrac{-6}{-6}y = \dfrac{-4}{-6}x + \dfrac{-6}{-6} [/tex]
[tex] y = \dfrac{2}{3}x + 1 [/tex]
We now compare this form of the second equation with y = mx + b, and we see that m = 2/3.
Both equations have the same slope, 2/3, so the lines are parallel.
The Bishop family celebrated a birthday by dining out at a local restaurant. Their
18 percent tip at the table for the server. What is the total amount that the family v
if necessary.
O
$17.54
$107.20
$114.99
$116.94
Answer:
$114.99
Step-by-step explanation:
rt4by5by4by5v4fcefrcekjj3ivc24ino3rmf44g4g
gv5y43t4r3wgy54bt3bt 4hnht6bgrvf
sorry if there is random numbers and letters i am glitchy
Answer:
97.45 • 0.18 = 17.54. 97.45 + 17.54 = 114.99. I think this is right.
Step-by-step explanation:
From the top of a 100 foot high pole, an observer measures the angle of depression of a car on the road as 28 degrees. Find, to the nearest hundredth of a foot, the distance from the car to the base of the pole.
This is a quick sketch of the problem
the answer is about 235.31 ft
you have to use this equation to get the length of the other side, which is roughly 213 ft:
[tex]100 \div \sin(28) [/tex]
then it's just Pythagorean Theorem to get the end result:
[tex] {100}^{2} + {213}^{2} = {c}^{2} [/tex]
To find the distance from the car to the base of the pole, use the tangent function. Plug in the values and solve for the adjacent side.
Explanation:To find the distance from the car to the base of the pole, we can use trigonometry. Since the observer is looking down at an angle of depression, we can use the tangent function. Tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the pole (100 ft) and the adjacent side is the distance from the car to the base of the pole.
So, using the formula tan(angle) = opposite/adjacent, we can plug in the values and solve for the adjacent side:
tan(28 degrees) = 100 ft/adjacent
adjacent = 100 ft / tan(28 degrees)
Using a calculator, we find that the distance from the car to the base of the pole is approximately 169.38 ft.
Learn more about Trigonometry here:https://brainly.com/question/11016599
#SPJ2
Which system of equations is represented by the graph?
A) y = x
y = -x-10, over x-4
B) y = −x
y = x+10, over x-4
C) y = x
y = -x+10 divided by the quantity x-4
D) y = −x
y = -x+10 divided by the quantity x-4
Answer:
[tex]\large\boxed{C)\ \left\{\begin{array}{ccc}y=x\\\\y=\dfrac{-x+10}{x-4}\end{array}\right}[/tex]
Step-by-step explanation:
We have the points
(-2 , -2) → x = -2 and y = -2 → x = y
(5, 5) → x = 5 and y = 5 → x = y
The equation of a line is [tex]y=x[/tex]
The hyperbola has the vertical asymptote x = a and the horizontal asymptote y = -1.
Therefore the second equation is:
[tex]y=\dfrac{a}{x-4}-1=\dfrac{a}{x-4}-\dfrac{x-4}{x-4}=\dfrac{a-(x-4)}{x-4}=\dfrac{a-x+4}{x-4}[/tex]
Where a > 0.
The corresponding equation in the solutions to choose is:
[tex]y=\dfrac{-x+10}{x-4}[/tex]
If 2x+b = y + 2b what is b^2 in terms of x and y
Answer:
b² = 2bx - 2by
Step-by-step explanation:
2x + b = y + 2b
b = 2x - y
b² = (2x - y)²
b² = (2x - y)(2x - y)
b² = 4x² - 2xy - 2xy + y²
b² = 4x² - 4xy + y²
Samantha swam upstream for some distance in one hour. She then swam downstream the same river for the same distance in only 12 minutes. If the river flows at 4 mph, how fast can Samantha swim in still water?
recall your d = rt, distance = rate * time.
s = Samantha's speed rate
one thing to bear in mind is that, when Samantha is going upstream, she's not really going "s" mph fast, because she's going against the current, and the current's rate is 4 mph and subtracting speed from her, then she's really going "s - 4" fast.
Likewise, when she's going downstream because she's going with the current, the current is adding speed to her, so she's going "s + 4" fast.
Now, let's say the distance she covered both ways was the same "d" miles.
Let's recall that since there are 60 minutes in 1 hour, 12 minutes is 12/60 = 1/5 of an hour.
[tex]\bf \begin{array}{lcccl} &\stackrel{miles}{distance}&\stackrel{mph}{rate}&\stackrel{hours}{time}\\ \cline{2-4}&\\ Upstream&d&s-4&1\\ Downstream&d&s+4&\frac{1}{5} \end{array}\qquad \implies \begin{cases} d=(s-4)(1)\\ d=(s+4)\left( \frac{1}{5} \right) \end{cases} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf \stackrel{\textit{from 1st equation}}{\boxed{d}=s-4}\qquad \qquad \stackrel{\textit{substituting on the 2nd equation}~\hfill }{\boxed{s-4}=\cfrac{s+4}{5}\implies 5s-20=s+4} \\\\\\ 4s-20=4\implies 4s=24\implies s=\cfrac{24}{4}\implies s=6[/tex]
Answer:
6 mph
Step-by-step explanation:
Let the the total distance is D and the speed of Samantha's speed in still water is x ( in mph ),
Here, speed of stream = 4 mph,
Thus, the speed in upstream = (x - 4) mph,
Speed in downstream = (x + 4) mph,
We know that,
[tex]Speed =\frac{Distance}{Time}\implies Time =\frac{Distance}{Speed}[/tex]
∴ Time taken in upstream = [tex]\frac{D}{x-4}[/tex]
And, time taken in downstream = [tex]\frac{D}{x+4}[/tex]
According to the question,
[tex]\frac{D}{x-4}=1\implies D = x-4-----(1)[/tex]
[tex]\frac{D}{x+4}=\frac{12}{60}\implies \frac{D}{x+4}=\frac{1}{5}\implies 5D=x+4----(2)[/tex]
Equation (1) - 5 equation (2),
0 = x + 4 - 5x + 20
0 = -4x + 24
4x = 24
⇒ x = 6 mph.
Hence, Samantha swims with the speed 6 mph.
What is the equation of the given circle?
Answer:
A
Step-by-step explanation:
The equation of a circle is:
(x - h)² + (y - k)² = r²
where (h, k) is the center and r is the radius.
Here, the center is (2, 1) and the radius is 1.
(x - 2)² + (y - 1)² = 1²
The equation of the given circle is Option(A) [tex](x-2)^{2} + (y - 1)^{2} = 1[/tex] .
What is equation of circle ?The standard equation of any circle is given as -
[tex](x-h)^{2} + (y - k)^{2} = r^{2}[/tex]
where (h,k) is the coordinate of the center of the given circle and r is the length of radius of the circle.
How to form the equation of the given circle ?In the diagram given aside, we can see that the circle has its center at (2,1) and also the radius of the circle measures 1 units.
Thus, we have h = 2 , k = 1 and r = 1 in the standard representation.
The equation of the circle is -
⇒ [tex](x-2)^{2} + (y - 1)^{2} = 1^{2}[/tex]
∴ [tex](x-2)^{2} + (y - 1)^{2} = 1[/tex]
Therefore, the equation of the given circle is Option (A) [tex](x-2)^{2} + (y - 1)^{2} = 1[/tex] .
To learn more about equation of circle, refer -
https://brainly.com/question/4287512
#SPJ2
What is the distance between –3 and 2 on the number line?
Answer: 5
Step-by-step explanation: The distance between two numbers can be found by finding the absolute value of the difference of the numbers. Put more simply, subtract -3 - 2 to get -5, which has an absolute value of 5. Or, you can do 2 - -3 to get 5, which has the same absolute value of 5.
The distance between -3 and 2 on the number line is 5 units.
What is the distance?It is the measurement done between two points on a surface.
Distance=speed*time
How to calculate distance?If we plot -3 and 2 on the number we find that -3 will be on the left side of 0 and 2 will be on the right side of 0. To reach 2 from -3 we have to first go to-2 then -1, then 0 and then 0 to 1 and 1 to 2.
Hence we have to go to 5 units. That's why the distance between -3 and 2 is 5 units.
Learn more about distance at https://brainly.com/question/2854969
#SPJ2