Answer:
[tex]x=\frac{-2}{3} \pm \frac{\sqrt{10}}{3}[/tex]
Step-by-step explanation:
Compare [tex]ax^2+bx+c[/tex] to [tex]3x^2+4x-2[/tex].
We have [tex]a=3,b=4,c=-2[/tex].
The quadratic formula is for solving equations of the form [tex]ax^2+bx+c=0[/tex] and is [tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex].
So we are going to plug in our values in that formula to find our solutions,x.
If you want to notice it in parts you can.
Example I might break it into these parts and then put it in:
Part 1: Evaluate [tex]b^2-4ac[/tex]
Part 2: Evaluate [tex]-b[/tex]
Part 3: Evaluate [tex]2a[/tex]
------Let's do these parts.
Part 1: [tex]b^2-4ac=(4)^2-4(3)(-2)=16-12(-2)=16+24=40[/tex].
This part 1 is important in determining the kinds of solutions you have. It is called the discriminant. If it is positive, you have two real solutions. If it is negative, you have no real solutions (both of the solutions are complex). If it is 0, you have one real solution.
Part 2: [tex]-b=-4[/tex] since [tex]b=4[/tex].
Part 3: [tex]2a=2(3)=6[/tex].
Let's plug this in:
[tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]
or in terms of our parts:
[tex]x=\frac{\text{Part 2} \pm \sqrt{\text{Part 1}}}{\text{Part 3}}[/tex]
[tex]x=\frac{-4 \pm \sqrt{40}}{6}[/tex]
40 itself is not a perfect square but it does contain a factor that is. That factor is 4.
So we are going to rewrite 40 as [tex]4 \cdot 10[/tex].
[tex]x=\frac{-4 \pm \sqrt{4 \cdot 10}}{6}[/tex]
[tex]x=\frac{-4 \pm \sqrt{4} \cdot \sqrt{10}}{6}[/tex]
[tex]x=\frac{-4 \pm 2\cdot \sqrt{10}}{6}[/tex]
I'm going to go ahead and separate the fraction like so:
[tex]x=\frac{-4}{6} \pm \frac{2 \cdot \sqrt{10}}{6}[/tex]
Now I'm going to reduce both fractions:
[tex]x=\frac{-2}{3} \pm \frac{1 \cdot \sqrt{10}}{3}[/tex]
[tex]x=\frac{-2}{3} \pm \frac{\sqrt{10}}{3}[/tex]
Final answer:
Using the quadratic formula with a=3, b=4, and c=-2, the solution set for the quadratic equation 3x^2 + 4x - 2 = 0 is x = (-4 + 2√(10)) / 6 and x = (-4 - 2√(10)) / 6.
Explanation:
To find the solution set of the quadratic equation 3x2 + 4x - 2 = 0, we can use the quadratic formula:
x = (-b ± √(b2 - 4ac)) / (2a)
Here, comparing with the standard form ax2 + bx + c = 0, we identify a = 3, b = 4, and c = -2. Substituting these values into the quadratic formula gives us:
x = (-(4) ± √((4)2 - 4(3)(-2))) / (2(3))
x = (-4 ± √(16 + 24)) / 6
x = (-4 ± √(40)) / 6
x = (-4 ± 2√(10)) / 6
Which simplifies to two solutions:
x = (-4 + 2√(10)) / 6
x = (-4 - 2√(10)) / 6
These are the two solutions to the given quadratic equation.
How many pairs of parallel faces are shown?
Final answer:
There are three pairs of parallel faces in a cube, as each face has one parallel counterpart. An icosahedron has no parallel faces. When standing between two parallel mirrors, an infinite series of images is theoretically created, but only a finite number can be seen due to practical limits.
Explanation:
To determine the number of pairs of parallel faces in a geometric shape, we have to consider the characteristics of the shape provided. In the case of a cube, which seems to be the most closely related shape mentioned in the details provided, there are three pairs of parallel faces. Each face of a cube is parallel to one other face that is opposite to it. For instance, the top face of the cube is parallel to the bottom face, one of the side faces is parallel to the face directly opposite to it, and the same applies to the front and back faces.
If the student's question instead pertains to an icosahedron, which is mentioned among the details, an icosahedron does not have any parallel faces as it is made up of equilateral triangles, and each face is inclined to the others.
For the discussion question regarding images formed between two parallel mirrors, an infinite number of images would be formed. This is because the mirrors reflect not only the object but also the images in the other mirror, thus creating a series of reflections that seem to go on indefinitely. However, due to practical limitations such as the quality of the mirror and ambient light, only a finite number of these images can actually be observed.
A rectangular prism has a length of 5 meters, a width of 6 meters, and a height of 3 meters.
What is the volume of the prism?
Enter your answer in the box.
Answer:
V=90m³
Step-by-step explanation:
If a rectangular prism has a length of 5 meters, a width of 6 meters, and a height of 3 meters, the volume of the prism is 90m³.
Formula: V=whl
V=whl=6·3·5=90m³
Lines s and t are perpendicular. If the slope of line s is -5, what is the slope of line ?
A: -1/5
B: 1/5
C: -5
D: 5
Answer:
B: 1/5
Step-by-step explanation:
If the lines are perpendicular, they have negative reciprocal slopes
s has a slope of -5
t must have a slope of - (1/ -5)
= 1/5
Line PQ passes through the points P(-5,-13) and Q(5,17) what is the equation of line PQ in standard form
Answer:
[tex]3x - y = - 2[/tex]
Step-by-step explanation:
The given line PQ passes through the points P(-5,-13) and Q(5,17) .
Find the slope using the formula:
[tex]m = \frac{y_2-y_1}{x_2-x_1} [/tex]
Plug in the points to get:
[tex]m = \frac{17 - - 13}{5 - - 5} [/tex]
[tex]m = \frac{30}{10} [/tex]
[tex]m = 3[/tex]
Use the point-slope formula :
[tex]y-y_1 = m(x - x_1)[/tex]
Substitute the point (5,17)
[tex]y - 17 = 3(x - 5)[/tex]
Expand:
[tex]y - 17 = 3x - 15[/tex]
The standard form is when the equation is put in the form
[tex]ax + by = c[/tex]
Regroup the terms to get:
[tex]3x - y = - 17 + 15[/tex]
The standard form is
[tex]3x - y = - 2[/tex]
Louisa states that the solution to the equation 1/4x - 3 = 3/8x + 4 is x = 56. She verifies her solution using the steps below.
Equation: 1/4x - 3 = 3/8x + 4
Step 1: 1/4(56) - 3 = 3/8(56) + 4
Step 2: 14 - 3 = 21 + 4
Step 3: 11 = 25
Which statement most accurately describes Louisa’s error?
A. Louisa made an error when determining the original solution of x = 56.
B. Louisa made an error when substituting the solution in for x.
C. Louisa made an error when multiplying 56 by 1/4 and by 3/8
D. Louisa made an error when adding or subtracting.
Answer:
A. Louisa made an error when determining the original solution of x = 56.
Step-by-step explanation:
1/4x - 3 = 3/8x + 4
To verify x=56 is a solution, substitute x=56 into the equation
1/4(56) - 3 = 3/8(56) + 4
Multiply
14 -3 = 21 +4
Combine like terms
11 = 25
The steps are correct.
x=56 must not be a solution to the equation
Answer:
option A
Step-by-step explanation:
Given that Louisa states that the solution to the equation 1/4x - 3 = 3/8x + 4 is x = 56.
She did the following steps
Step 1 is right because she substituted x=14 to check whether right side = left side
Step 2: Multiplication is done correctly
Step 3: Is incorrect because 11 can never be equal to 25 in Mathematics.
Hence the mistake she did is not in substitution, multiplication or addition/subtraction. But in determining the original solution
Hence A) Louisa made an error when determining the original solution of x = 56.
Need help with 13 and 14
Answer:
13. A
14. C.
Step-by-step explanation:
A relation is a function if there is only one y assigned to each x.
That is if you have a set of points, there should be no repeated x value.
So looking at {(0,0),(-2,4),(-2,-4),(-3,7)} this is not a function because you have two x's that are the same. A.
The slope-intercept form of a linear equation is y=mx+b where m is the slope and b is the y-intercept.
The y-intercept is where the graph goes through the y-axis at. It goes through at -1 so b=-1.
The slope is rise/run. So starting from the y-intercept (0,-1) we need to find another point to count the rise & run to... How about (3,1)? That works. You can do the counting if you want. You could also use the slope formula.
To use the slope formula, I just like to line the points up vertically and subtract vertically then put 2nd difference over the 1st difference.
(0,-1)
-(3,1)
--------
-3 -2
So the slope is -2/-3 or just 2/3.
So the equation is y=2/3 x-1
C.
HELP PLS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
X is left to right. Since the translation is X + 1 , you are adding 1 to the X direction, which moves the shape 1 place to the right.
Y is up and down, the translation is y - 2, which means the shape moves 2 places down.
Looking at one point on the original image, lets use Point C. It is located at X = 2 and Y = 1. Now add 1 to the X and subtract 2 from the y:
The new location for C needs to be X = 2+1 = 3 and Y = 1-2 = -1
The image that has C at (3,-1) is C.
Find the value of X in the picture please
Answer:
The measure of the arc x is 130°
Step-by-step explanation:
we know that
The semi-inscribed angle is half that of the arc it comprises
so
65°=(1/2)[arc x]
solve for x
arc x=(2)(65°)=130°
Huntsville’s population grows from 25,000 to 28,000. What is the percent increase in Huntsville’s population?
How do you solve 3,000/ 25,000 without a calculator to still get 12%?
Answer:
[tex]\%increase = 12\%[/tex]
Step-by-step explanation:
The formula to calculate the percentage of increase is:
[tex]\% = \frac{x_f-x_i}{x_i}*100\%[/tex]
Where
[tex]x_i[/tex] is the initial amount and [tex]x_f[/tex] is the final amount
So:
[tex]\% = \frac{28000-25000}{25000}*100\%[/tex]
[tex]\% = \frac{3000}{25000}*100\%[/tex]
You can write it as:
[tex]\% = \frac{3*1000}{25*1000}*100\%[/tex]
This is:
[tex]\% = \frac{3}{25}*100\%[/tex]
[tex]\%increase = 12\%[/tex]
solve 5n-7p+3n=25p for n
Answer:
n=4p
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
Help pls I don't really understand
Y = 6
13 + [tex]\frac{6}{y}[/tex] means that we are adding 13 and 1 together because 6 divided by 6 is 1.
13 + 1 = 14
What is the ratio of letters of the first name to the last name of Justin Morris? (4
You are a school photographer taking individual and class pictures for 2 classes of 21 students each. On average, each individual picture takes 3 minutes and a class picture takes 10 minutes. About how long should it take you to take all of the pictures?
Answer:
It will take 143 minutes to take the pictures
The volume of a cylinder is 980pi in3. The radius is 7 in. What is its height?
A) 10 in
B) 140 in
C) 70 in
D) 20 in
Answer:
D) 20 in
Step-by-step explanation:
The volume of a cylinder is given by
V = pi r^2 h where r is the radius and h is the height
Substituting what we know
980 pi = pi (7)^2 h
Divide each side by pi
980 pi /pi= pi 49 h/pi
980 = 49h
Divide each side by 49
980/49 = 49h/49
20 =h
Answer:
h ≈ 20 inches
Step-by-step explanation:
Given volume 980 pi inches cubed and radius 7 inches.
Using the height formula for right cylinder
h = V/π r^2
h = 3078.78/π7^2
h ≈ 20 inches
the rectangle has an area of 24 square centimeters. find the length a of the rectangle
Answer : The length of the rectangle (a) is, 8 cm
Step-by-step explanation :
As we are given that,
Area of rectangle = [tex]24cm^2[/tex]
Length of rectangle = a
Breadth of rectangle = a - 5
As we know that,
Area of rectangle = Length × Breadth
Now put all the given values in this formula, we get the value of 'a'.
[tex]24cm^2=(a)\times (a-5)[/tex]
[tex]24cm^2=a^2-5a[/tex]
[tex]a^2-5a-24=0[/tex]
By the solving the term 'a', we get the value of 'a'.
a = 8
Thus,
Length of rectangle = a = 8 cm
Breadth of rectangle = a - 5 = 8 - 5 = 3 cm
Therefore, the length of the rectangle (a) is, 8 cm
The length a of the rectangle is 8 cm
Further explanationTo solve the above questions, we need to recall some of the formulas as follows:
Area of Square = (Length of Side)²
Perimeter of Square = 4 × (Length of Side)
Area of Rectangle = Length × Width
Perimeter of Rectangle = 2 × ( Length + Width )
Area of Rhombus = ½ × ( Diagonal₁ + Diagonal₂ )
Perimeter of Rhombus = 4 × ( Length of Side )
Area of Kite = ½ × ( Diagonal₁ + Diagonal₂ )
Perimeter of Kite = 2 × ( Length of Side₁ + Length of Side₂ )
Let us now tackle the problem !
Given:
Area of Rectangle = A = 24 cm²
Length of Rectangle = L = a cm
Width of Rectangle = W = (a - 5) cm
Unknown:
Length of Rectangle = a = ?
Solution:
This problem is about Area of Rectangle.
[tex]\text{Area of Rectangle} = \text{Length} \times \text{Width}[/tex]
[tex]A = L \times W[/tex]
[tex]24 = a \times (a - 5)[/tex]
[tex]24 = a^2 - 5a[/tex]
[tex]a^2 - 5a - 24 = 0[/tex]
[tex](a - 8)(a +3) = 0[/tex]
[tex](a - 8) = 0[/tex]
[tex]a = \boxed {8 ~ \text{cm}}[/tex]
Learn moreThe perimeter of a polygon : https://brainly.com/question/6361596The perimeter of a rectangle : https://brainly.com/question/7619923The perimeter of a triangle : https://brainly.com/question/2299951Answer detailsGrade: College
Subject: Mathematics
Chapter: Two Dimensional Figures
Keywords: Perimeter, Area , Square , Rectangle , Side , Length , Width
What is the value of x in x + 2 = 5?
Answer:
x = 3Step-by-step explanation:
x + 2 = 5 subtract 2 from both sides
x + 2 - 2 = 5 - 2
x = 3
Check:
3 + 2 = 5 CORRECT :)
PLEASE HELP ASAP:
At Lincoln High School, approximately 7 percent of enrolled juniors and 5 percent of enrolled seniors were inducted into the National Honor Society last year. If there were 562 juniors and 602 seniors enrolled at Lincoln High School last year, which of the following is closest to the total number of juniors and seniors at Lincoln High School last year who were inducted into the National Honor Society?
Answer:
39.34 juniors and 30.1 seniors
Answer:
The closest number of juniors and seniors who were inducted is 69.
Step-by-step explanation:
At Lincoln High School, approximately 7% of enrolled juniors and 5% of enrolled seniors were inducted into the National Honor Society last year.
There were 562 juniors and 602 seniors enrolled at Lincoln High School last year.
We will calculate the number of students as:
[tex]0.07(562)+0.05(602)[/tex]
= [tex]39.34+30.1[/tex]
= 69.44
Therefore, the closest number of juniors and seniors who were inducted is 69.
You hike uphill at a rate of 200 feet per minute. Your friend hikes downhill on a same trail at a rate of 250 feet per minute. How long will it be until you meet?
Answer:
After x/450 minutes
Step-by-step explanation:
The speed uphill is 200 ft/min
The speed down hill is 250 ft/min
Lets take the hiking distance top to down hill is x ft
Remember speed=distance/time
Relative speed =200+250=450ft/min
Time to meet=distance/relative speed
Time to meet=x/450 min
Final answer:
To find out how long it will take until you meet, we need to determine the distances each hiker covers and then divide it by their respective rates.
Explanation:
To find out how long it will take until you meet, we need to determine the distances each hiker covers and then divide it by their respective rates.
The first hiker covers a certain distance while hiking uphill, while the second hiker covers a certain distance while hiking downhill. When they meet, their combined distances will add up to the total distance of the trail.
Let's say it takes x minutes for you to meet. The first hiker covers a distance of 200x feet in x minutes, while the second hiker covers a distance of 250x feet in the same x minutes. When they meet, their combined distances will equal the length of the trail, so we can set up the equation: 200x + 250x = total distance. Solving for x will give us the time it takes for you to meet.
pls help solve this problem
Answer:
166 mm²
Step-by-step explanation:
The area is given by ...
A = (1/2)Pa . . . . . where P is the perimeter and "a" is the apothem
One side of a 6-sided figure is shown as 8 mm, so the perimeter is ...
P = 6·(8 mm) = 48 mm
Filling in the apothem value, we have ...
A = (1/2)(48 mm)(4√3 mm) = 96√3 mm² ≈ 166 mm²
The area of the hexagon is about 166 mm².
solve the quadratic equation
1)x²+16-48=0
Answer:
x = 12
x = 4
Step-by-step explanation:
Answer:
X=12
X=4
Step-by-step explanation:
Step-1 : Multiply the coefficient of the first term by the constant 1 • 48 = 48
Step-2 : Find two factors of 48 whose sum equals the coefficient of the middle term, which is -16 .
-48 + -1 = -49
-24 + -2 = -26
-16 + -3 = -19
-12 + -4 = -16 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -12 and -4
x2 - 12x - 4x - 48
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-12)
Add up the last 2 terms, pulling out common factors :
4 • (x-12)
Step-5 : Add up the four terms of step 4 :
(x-4) • (x-12)
Which is the desired factorization
write y + 1 = -2x - 3 in standard form. A. -2x-y = 4 B. x + 1/2y = - 2 C. y = -2x-4 D. 2x + y = -4
Answer:
D. 2x + y = -4
Step-by-step explanation:
Standard form for the equation of a line is Ax + By =C where A is a positive integer and B and C are integers
y + 1 = -2x - 3
Add 2x to each side
2x+y + 1 = -2x+2x - 3
2x+y +1 = -3
Subtract 1 from each side
2x+y +1-1 = -3-1
2x+y = -4
Answer:
y=-2x-4
Step-by-step explanation:
There are many standard form of any equations. One of them is called the
y intercept form. The standard form is
y=mx+c
where m is the slope and c is the y intercept
Let us see our given equation
y+1= -2x - 3
subtracting 1 from both sides we get
y = -2x - 3 - 1
y= -2x - 4
Here if we compare it with the standard equation we get
m = -2 and c = -4
For a certain weekend, the weatherman predicts that it will rain with a 40% probability on Saturday and a 50% probability on Sunday. Assuming these probabilities are independent, what is the probability that it rains over the weekend (that is, on at least one of the days)? Express your answer as a percentage.
Answer:
70%
Step-by-step explanation:
The probability it rains on at least one of the days is:
P = P(Sat & not Sun) + P(Sun & not Sat) + P(Sat & Sun)
P = (0.40)(1−0.50) + (0.50)(1−0.40) + (0.40)(0.50)
P = 0.20 + 0.30 + 0.20
P = 0.70
Another way to look at it is 1 − the probability that it doesn't rain on either day.
P = 1 − P(not Sat & not Sun)
P = 1 − (1−0.40)(1−0.50)
P = 1 − 0.30
P = 0.70
There is a 70% probability that it will rain on at least one day.
The probability that it will rain on at least one day over the weekend is 70%, based on independent probabilities given for Saturday and Sunday.
Explanation:The subject of your question is probability, a branch of mathematics that studies randomness and the likelihood of events occurring. To calculate the probability that it will rain over the weekend, i.e., on at least one of the days (Saturday or Sunday), we first need to find the probability that it won't rain on either day.
The probability it doesn't rain on Saturday is 60% (100% - 40%), and the probability it doesn't rain on Sunday is 50% (100% - 50%). Since these events are independent, we multiply these probabilities to find the joint probability. Therefore, the probability it doesn't rain on either day is 30% (0.6 × 0.5 = 0.3).
Now, this is the case which we don’t want (no rain), so the case we want (that it rains at least one day) is the complementary event of this. Thus, we subtract this from the total probability, which is 1, yielding a 70% (1 - 0.3) probability that it will rain on at least one of the days over the weekend.
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An object is thrown upward at a speed of 58 feet per second by a machine from a height of 7 feet off the ground. The height h of the object after t seconds can be found using the equation h=−16t^2+58t+7
a.When will the height be 17 feet? ______
b. When will the object reach the ground? ______
Answer:
First part:
Set h(t) = 17and solve for t.
-16t²+ 58t + 7= 17
-16t² + 58t - 10 = 0
Solve this quadratic equation for t. You should get 2 positive solutions. The lower value is the time to reach 17 on the way up, and the higher value is the time to reach 17 again, on the way down.
Second part:
Set h(t) = 0 and solve the resulting quadratic equation for t. You should get a negative solution (which you can discard), and a positive solution. The latter is your answer.
The time required to reach 17 feet and the ground by the ball is required.
The time taken to reach 17 feet is 0.181 s.
To reach the ground the time taken is 3.74 s.
The equation is
[tex]h=-16t^2+58t+7[/tex]
[tex]h=17[/tex]
[tex]17=-16t^2+58t+7\\\Rightarrow -16t^2+58t-10=0\\\Rightarrow t=\frac{-58\pm \sqrt{58^2-4\left(-16\right)\left(-10\right)}}{2\left(-16\right)}\\\Rightarrow t=0.181,3.44[/tex]
The time taken reach a height of 17 feet while going up is 0.181 s.
On the ground [tex]h=0[/tex]
[tex]0=-16t^2+58t+7\\\Rightarrow t=\frac{-58\pm \sqrt{58^2-4\left(-16\right)\times 7}}{2\left(-16\right)}\\\Rightarrow t=-0.12,3.74[/tex]
The time taken to reach the ground is 3.74 s.
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The value of China's exports of automobiles and parts (in billions of dollars) is approximately f(x)=1.8208e.^3387x, where x = 0 corresponds to 1998.
In what year did/will the exports reach $8 billion?
Give your answer as the year, with at least one decimal place
Answer:
This was occur sometimes in year 2002.4
Step-by-step explanation:
* Lets explain how to solve the problem
- The value of China's exports of automobiles and parts
(in billions of dollars) is approximately f(x) = 1.8208 e^(0.3387 x)
# You must pay attention about the function is already calculated in
billions dollars so you will not multiply the value of f(x) by 10^9 to
change it to billions
∵ The value of x = 0 at 1998
- Remember that e^(0) = 1
∴ f(0) = 1.8208 e^(0) = 1.8208 billion dollars
- You need to calculate the year that the export reaches 8 billion
∵ f(x) = 1.8208 e^(0.3387 x)
∵ f(x) = 8 billion
∴ 8 = 1.8208 e^(0.3387 x)
- Divide both sides by 1.8208
∴ 4.393673111 = e^(0.3387 x)
- Insert ㏑ in both sides
∴ ㏑(4.393673111) = ㏑[e^(0.3387 x)]
- Remember ㏑(e^n) = n ㏑(e), ㏑(e) = 1, then ㏑(e^n) = n
∴ ㏑(4.393673111) = 0.3387 x
- Divide both sides by 0.3387
∴ x = ㏑(4.393673111) ÷ 0.3387
∴ x = 4.37 ≅ 4.4
- Lets add the number of years to 1998
∴ The year is 1998 + 4.4 = 2002.4
∴ This was occur sometimes in year 2002.4
The value of China's exports of automobiles and parts reached $8 billion in approximately the year 2001.8. The mathematical calculation is based on the exponential function representing the export values over time with the base year 1998.
Explanation:The student has asked to determine the year in which China's exports of automobiles and parts reached $8 billion.
To solve for the year when exports reach $8 billion, we use the given exponential function for the value of China's exports of automobiles and parts, which is f(x) = 1.8208e0.3387x, where x = 0 corresponds to the year 1998. We need to solve the equation f(x) = 8 for x.
Firstly, set f(x) = 8:
Next, divide both sides of the equation by 1.8208 to isolate the exponential part:
Then, take the natural logarithm of both sides to solve for x:
Finally, add the result to the base year 1998 to get the approximate year:
Therefore, the exports reached $8 billion in approximately 2001.8, or the year 2001 when considering just the full year.
Points A(-2,4) , B(1,3), C(4,1) and D form a parallelogram.What are the coordinates of D?
Answer:
The coordinates of vertex D are (1,2).
Step-by-step explanation:
Let the coordinates of D are (a,b).
It is given that ABCD is a parallelogram and the vertices of the parallelogram are A(-2,4), B(1,3) and C(4,1).
According to the property of parallelogram, the diagonals of the parallelogram bisect each other.
AC and BD are diagonals of the parallelogram ABCD.
Midpoint formula:
[tex]Midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]
Using midpoint formula, the midpoint of AC is
[tex]Midpoint_{AC}=(\frac{-2+4}{2},\frac{4+1}{2})=(1,\frac{5}{2})[/tex]
Using midpoint formula, the midpoint of BD is
[tex]Midpoint_{BD}=(\frac{1+a}{2},\frac{3+b}{2})[/tex]
Midpoint of both diagonals is the intersection point of the diagonals.
[tex]Midpoint_{AC}=Midpoint_{BD}[/tex]
[tex](1,\frac{5}{2})=(\frac{1+a}{2},\frac{3+b}{2})[/tex]
On comparing both the sides we get
[tex]1=\frac{1+a}{2}[/tex]
[tex]2=1+a[/tex]
[tex]2-1=a[/tex]
[tex]1=a[/tex]
The value of a is 1.
[tex]\frac{5}{2}=\frac{3+b}{2}[/tex]
[tex]5=3+b[/tex]
[tex]5-3=b[/tex]
[tex]2=b[/tex]
The value of b is 2.
Therefore the coordinates of vertex D are (1,2).
The coordinates of point D in the parallelogram formed by points A(-2,4), B(1,3), and C(4,1) are (7,2), calculated based on the property of parallelograms, where opposite sides are congruent.
Explanation:In order to find the coordinates of point D forming a parallelogram with points A(-2,4), B(1,3), and C(4,1), we can use the property of a parallelogram that opposite sides are equal. This means the vector from A to B is equivalent to the vector from D to C. The vector AB is calculated through subtraction of the coordinates: B minus A (1-(-2), 3-4) = (3, -1).
Applying the same vector (movement) to point C to find point D (since D to C equals B to A), we get D as C plus the vector (4+3, 1-(-1)) = (7,2). So, the coordinates of point D in the parallelogram are (7,2).
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3. The width and length of a rectangle are
consecutive integers. If the perimeter of
the rectangle is 142 inches, find the width
and length of the rectangle.
Answer:
35 and 36
Step-by-step explanation:
If the smaller dimension is x, then the larger dimension is x + 1. Therefore:
2x + 2(x + 1) = 142
2x + 2x + 2 = 142
4x = 140
x = 35
One dimension is 35, and the other dimension is 36.
Answer:
The dimensions of the rectangle are 35 inches by 36 inches.
Step-by-step explanation:
If the length and width are consecutive integers and L=n, then W=n+1 assuming the width is larger.
We are given the perimeter is 142 inches so: 2L+2W=142.
Substituting L=n and W=n+1 we have: 2(n)+2(n+1)=142.
Let's solve it:
2(n)+2(n+1)=142
Distribute:
2n+2n+2=142
Combine like terms:
4n+2=142
Subtract 2 on both sides:
4n=140
Divide both sides by 4:
n=140/4
n=35
Since L=n, then the length is 35 inches.
Since W=n+1, then the width is 36 inches.
The dimensions of the rectangle are 35 inches by 36 inches.
If g(x)=x^2-4 find g(5)
Answer:
21Step-by-step explanation:
[tex]g(x)=x^2-4\\\\g(5)-\text{put}\ x=5\ \text{to the equation of the function}\ g(x):\\\\g(5)=5^2-4=25-4=21[/tex]
The value of the function g(x) = x² - 4 at g(5) is 21. Option C is correct.
What is an expression?Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication and division.
The expression is solved as:-
g(x) = x² - 4
To calculate the value g(5) put x=5 in the expression.
g(5) = ( 5 )² - 4
g(5) = 25 - 4
g(5) = 21
Therefore, the value of the function g(x) = x² - 4 at g(5) is 21. Option C is correct.
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The length of the sides of one triangle are 2/3 the length of the side of a similar triangle. If a side of the lower triangle is 36 millimeters, what is the measure of the matching side of the smaller triangle?
Answer:
24
Step-by-step explanation:
36= three part in a length
so, one part=36/3=12
smaller triangle's length=2/3=12×2=24
Answer:
24 millimeters
Step-by-step explanation:
We have 2 triangles with proportional sides. The smaller side has a length of 2/3 of the similar triangle:
[tex]L_1=\frac{2L_2}{3}[/tex]
[tex]L_1=\frac{2*36}{3}[/tex]
[tex]L_1=24 mm[/tex]
24 millimeters
I need help please.
Answer:
[tex]\frac{7\sqrt{5} }{8}[/tex]
Step-by-step explanation:
Using the rules of radicals
[tex]\sqrt{\frac{a}{b} }[/tex] = [tex]\frac{\sqrt{a} }{\sqrt{b} }[/tex]
[tex]\sqrt{a}[/tex] × [tex]\sqrt{b}[/tex] ⇔ [tex]\sqrt{ab}[/tex]
Given
[tex]\sqrt{\frac{245}{64} }[/tex] = [tex]\frac{\sqrt{245} }{\sqrt{64} }[/tex]
Simplifying the radicals
[tex]\sqrt{245}[/tex]
= [tex]\sqrt{5(7)(7)}[/tex]
= [tex]\sqrt{49(5)}[/tex]
= [tex]\sqrt{49}[/tex] × [tex]\sqrt{5}[/tex] = 7[tex]\sqrt{5}[/tex]
and
[tex]\sqrt{64}[/tex] = 8
The simplified radical is
[tex]\frac{7\sqrt{5} }{8}[/tex]
Please help I’m timed!!
Which expression represents a rational number?
5/9 + (square root of 18)
Pie+ (square root of 16)
2/7 + (square root of 121)
3/10 + (square root of 11)
Answer:
5/9, 2/7, 3/10 are all rational numbers
Step-by-step explanation:
Remember that all rational numbers are able to be written as a fraction or ratio of two integers.
Answer:
[tex]\frac{2}{7}+\sqrt{121}[/tex]
Step-by-step explanation:
Since, a real number is called rational number if it can be expressed in the form of [tex]\frac{p}{q}[/tex],
Where, p and q are integers,
S.t. q ≠ 0,
If the number is not a rational number then it is irrational,
Now, the sum or difference of two rational numbers is a rational number,
While, the sum or difference of a rational number and an irrational number is an irrational number.
∵ √18, √11 and [tex]\pi[/tex] are irrational numbers,
Also, [tex]\frac{5}{9}[/tex], [tex]\sqrt{16}[/tex] and [tex]\frac{3}{10}[/tex] are rational number,
[tex]\implies \frac{5}{9}+\sqrt{18},\pi+\sqrt{16}, \frac{3}{10}+\sqrt{11}\text{ are irrational numbers}[/tex]
Now,
[tex]\frac{2}{7}\text{ and }\sqrt{121}\text{ are rational numbers}[/tex]
Hence,
[tex]\frac{2}{7}+\sqrt{121}\text{ is rational number}[/tex]