The value of x for the given equations are a. 10, b. -7, c = -8, d. 5.
What are equations and its solution?Finding an equation's solutions, which are values (numbers, functions, sets, etc.) that satisfy the equation's condition and often consist of two expressions connected by an equals sign, is known as solving an equation in mathematics. One or more variables are identified as unknowns when looking for a solution. An assignment of values to the unknown variables that establishes the equality in the equation is referred to as a solution. Particularly but not exclusively for polynomial equations, the solution of an equation is frequently referred to as the equation's root.
Given that,
2(x-3) = 14
2x - 6 = 14
2x = 20
x = 10
b. -5(x - 1) = 40
-5x + 5 = 40
-5x = 35
x = -7
c. 12(x + 10) = 24
12x + 120 = 24
12x = 24 - 120
x = -8
d. (x + 6) = 11
x = 11 - 6
x = 5
Hence, the value of x for the given equations are a. 10, b. -7, c = -8, d. 5.
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Find the magnitude and direction of the equilibrant of each of the following systems of forces.
A) Forces of 32N and 48N acting at an angle of 90° to each other
Answer: To find the magnitude and direction of the equilibrant of a system of forces, we first need to find the resultant of the forces, and then find the force that will balance the resultant.
For the given system of forces, we can use the Pythagorean theorem to find the magnitude of the resultant:
R = sqrt(32^2 + 48^2)
= sqrt(1024 + 2304)
= sqrt(3328)
≈ 57.7 N
The direction of the resultant can be found using trigonometry:
tan(theta) = opposite / adjacent
where theta is the angle between the forces, which is 90° in this case. We can choose either force as the adjacent side, and the other force as the opposite side. Let's choose the 32 N force as the adjacent side:
tan(theta) = 48 / 32
theta = atan(48/32)
≈ 56.3°
This means that the resultant has a magnitude of approximately 57.7 N and is directed at an angle of approximately 56.3° to the 32 N force.
To find the equilibrant, we need to find a force that has the same magnitude as the resultant but acts in the opposite direction. We can use the same magnitude and opposite direction to find the equilibrant as:
E = -R
= -57.7 N
This means that the equilibrant has a magnitude of 57.7 N and acts in the opposite direction to the resultant, which is at an angle of approximately 56.3° to the 32 N force.
Step-by-step explanation:
Seven people are in the pool who each have expected payouts of $8,600 for the year. One more person is added to the pool whose expected payout is $14,000 per year. How much is the average expected payout for each member of the group of eight? Expected Payout for each member of the group is $
Step-by-step explanation:
Seven people are in the pool who each have expected payouts of $8,600 for the year. One more person is added to the pool whose expected payout is $14,000 per year. How much is the average expected payout for each member of the group of eight? Expected Payout for each member of the group is $
Blake mows lawns to earn money. He wants to earn at least $200 to buy a new stereo system. If he charges $12 per lawn, at least how many lawns will he need to mow? Inequality: Inequali
Blake needs to now at least 17 lawns in order to earn at least $200 to buy a new stereo system.
To solve this problem, we can set up an inequality that represents the situation. Let's let x be the number of lawns that Blake needs to mow. The inequality will be:
12x >= 200
This inequality states that the amount of money Blake earns from mowing lawns (12x) must be greater than or equal to the amount of money he wants to earn ($200).
To solve for x, we can divide both sides of the inequality by 12:
x >= 200/12
Simplifying the right side of the inequality gives us:
x >= 16.67
Since Blake cannot mow a fraction of a lawn, we need to round up to the next whole number. This means that Blake needs to now at least 17 lawns in order to earn at least $200 to buy a new stereo system.
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What is the solution to 4
O a>-21-
O a<-21-
O a> 21/1/
O a<21/1/2
Mark this and return
a>-162
C
Save and Exit
Next
Submit
TIME
01
The solution to 4 is a>-162. This is because the inequality can be rewritten as -162 > -21, meaning that any number greater than -162 will satisfy the inequality.
What is inequality?Inequality is the unequal distribution of opportunities, resources, or rights among people or groups. It can be found in wealth, income, health, education, access to services and resources, and legal rights. Inequality is a social phenomenon that affects people differently depending on race, gender, age, or background. Inequality can be a result of unequal access to resources, differences in power dynamics, or the unequal distribution of resources among people or groups. It can lead to disparities in health, education, and economic outcomes for those who experience it. Inequality can be addressed through policies that promote equity and inclusion, such as targeted education initiatives and access to employment opportunities.
The reason this is happening is because when solving inequalities, we must isolate the variable on one side of the inequality sign. In this case, we can do this by adding 162 to both sides of the inequality. This will move the -21 to the other side and the result is a>-162.
In order to ensure that the solution is correct, it is important to check the answer by substituting a value larger than -162 into the inequality. If the value satisfies the inequality, then the solution is correct.
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The sοlutiοn tο 4 is a>-162. This is because the inequality can be rewritten as -162 > -21, meaning that any number greater than -162 will satisfy the inequality.
What is inequality?Inequality is the unequal distributiοn οf οppοrtunities, resοurces, οr rights amοng peοple οr grοups. It can be fοund in wealth, incοme, health, educatiοn, access tο services and resοurces, and legal rights. Inequality is a sοcial phenοmenοn that affects peοple differently depending οn race, gender, age, οr backgrοund. Inequality can be a result οf unequal access tο resοurces, differences in pοwer dynamics, οr the unequal distributiοn οf resοurces amοng peοple οr grοups. It can lead tο disparities in health, educatiοn, and ecοnοmic οutcοmes fοr thοse whο experience it. Inequality can be addressed thrοugh pοlicies that prοmοte equity and inclusiοn, such as targeted educatiοn initiatives and access tο emplοyment οppοrtunities.
The reasοn this is happening is because when sοlving inequalities, we must isοlate the variable οn οne side οf the inequality sign. In this case, we can dο this by adding 162 tο bοth sides οf the inequality. This will mοve the -21 tο the οther side and the result is a>-162.
In οrder tο ensure that the sοlutiοn is cοrrect, it is impοrtant tο check the answer by substituting a value larger than -162 intο the inequality. If the value satisfies the inequality, then the sοlutiοn is cοrrect.
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Complete question:
What is the solution to 3/4a>-16?
a>-21 1/3
a<-21 1/3
a>21 1/3
a<21 1/3
Find the absolute extrema of the function \( f \) defined by \( f(x, y)=x^{2}+3 y^{2}-6 x y+8 x \) subject to the constraints \( x \geq 1, y \geq 0 \) and \( y+x \leq 5 \). (You should use the LM meth
The absolute maximum is 33 at the point (1, 4) and the absolute minimum is 9 at the point (3, 2).
To find the absolute extrema of the function \( f(x, y)=x^{2}+3 y^{2}-6 x y+8 x \) subject to the constraints \( x \geq 1, y \geq 0 \) and \( y+x \leq 5 \), we can use the Lagrange Multiplier (LM) method. The LM method involves finding the points where the gradient of the function is parallel to the gradient of the constraints.
First, let's find the gradient of the function:
\(\nabla f(x, y) = \langle 2x - 6y + 8, 6y - 6x \rangle \)
Next, let's find the gradient of the constraints:
\(\nabla g(x, y) = \langle 1, 1 \rangle \)
Now, we can set the gradient of the function equal to the gradient of the constraints times a constant, \(\lambda\):
\(\nabla f(x, y) = \lambda \nabla g(x, y) \)
This gives us the following system of equations:
\(2x - 6y + 8 = \lambda \)
\(6y - 6x = \lambda \)
We can also add the constraint \( y+x \leq 5 \) to the system of equations:
\(x + y = 5 \)
Solving this system of equations gives us the critical points:
\((x, y) = (1, 4), (4, 1), (3, 2) \)
Finally, we can plug these critical points back into the original function to find the absolute extrema:
\(f(1, 4) = 1 + 48 - 24 + 8 = 33 \)
\(f(4, 1) = 16 + 3 - 24 + 32 = 27 \)
\(f(3, 2) = 9 + 12 - 36 + 24 = 9 \)
Therefore, the absolute maximum is 33 at the point (1, 4) and the absolute minimum is 9 at the point (3, 2).
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There are 88 seats in the theater. The seating in the theater is split in to 4 identical section has 14 red seats and some blue seats.
The theater has a tοtal οf 56 red seats and 32 blue seats, making a tοtal οf 88 seats.
Hοw tο find the number οf red seats?If each sectiοn has 14 red seats, then there are a tοtal οf 4 x 14 = <<4 × 14=56>>56 red seats in the theater.
Tο find οut hοw many blue seats are in each sectiοn, we need tο subtract the number οf red seats frοm the tοtal number οf seats in each sectiοn:
88 tοtal seats / 4 sectiοns = 22 seats per sectiοn
22 seats per sectiοn - 14 red seats per sectiοn = 8 blue seats per sectiοn
Therefοre, each sectiοn has 8 blue seats, and the theater has a tοtal οf 4 x 8 = <<4 × 8=32>>32 blue seats.
Sο, the theater has a tοtal οf 56 red seats and 32 blue seats, making a tοtal οf 88 seats.
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1) Find the average rate of change of the function from
x1 to x2
f(x) = 3xfrom xone = 0 to x two =
5 f(x) = x2 +2x from
x1 = 3 to x2 =5
Write an equation of the line
passing through (-8, -10) and pa
The equation of the line passing through (-8, -10) and passing through (x1, f(x1)) is given by y = 5x + 30.
m = (f(x2) - f(x1)) / (x2 - x1)
f(x) = 3x from x1 = 0 to x2 = 5
f(x) = x2 + 2x from x1 = 3 to x2 = 5
m = (f(5) - f(0)) / (5 - 0)
m = (53 + 2(5)) - (03 + 2(0)) / (5 - 0)
m = 25 / 5
m = 5
Therefore, the average rate of change of the function from x1 to x2 is 5.
The equation of the line passing through (-8, -10) and passing through (x1, f(x1)) is given by the equation:
y = mx + b
where m is the slope of the line and b is the y-intercept.
We can find the slope of the line, m, from the average rate of change of the function from x1 to x2 which is 5.
We can find the y-intercept, b, by substituting the coordinates (-8, -10) in the equation of the line.
y = 5x + b
-10 = 5(-8) + b
b = 30
Therefore, the equation of the line passing through (-8, -10) and passing through (x1, f(x1)) is given by y = 5x + 30.
Construct a 3x3 linear system whose solution is (x,y,z)=(3,5,-2).
Use Gauss-Jordan's Elimination.
To construct a 3x3 linear system whose solution is (x,y,z)=(3,5,-2) using Gauss-Jordan's Elimination, we will first create an augmented matrix for the system.
This matrix will contain the coefficients for the variables x, y, and z, as well as the values for the constants. For example:
ax + by + cz = d
We can then plug in the values for x, y, and z, and choose values for a, b, c, and d that will make the equation true. For example:
2(3) + 3(5) - 4(-2) = 29
This gives us one equation in our system:
2x + 3y - 4z = 29
We can repeat this process two more times to get two more equations:
-5(3) + 2(5) + 3(-2) = -19
-5x + 2y + 3z = -19
4(3) - 6(5) + 2(-2) = -24
4x - 6y + 2z = -24
So our 3x3 linear system is:
2x + 3y - 4z = 29
-5x + 2y + 3z = -19
4x - 6y + 2z = -24
To solve this system using Gauss-Jordan's Elimination, we can write the system as an augmented matrix:
| 2 3 -4 | 29 |
|-5 2 3 |-19 |
| 4 -6 2 |-24 |
We can then use elementary row operations to reduce the matrix to reduced row echelon form:
| 1 0 0 | 3 |
| 0 1 0 | 5 |
| 0 0 1 |-2 |
This gives us the solution (x,y,z)=(3,5,-2), as desired.
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I think its A or C, but I'm not sure.
Answer: The answer is A
Step-by-step explanation:
find a formula for the nth term in this arithmetic sequence a1=7, a2=4, a3=1, a4=-2
The nth term of the arithmetic sequence is 10 - 3n.
Arithmetic sequence:An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a constant value, called the common difference, to the preceding term.
The formula for the nth term (where n is a positive integer) of an arithmetic sequence is:
a[tex]_{n}[/tex] = a₁ + (n-1)dHere we have
a₁ = 7, a₂ = 4, a₃ = 1, a₄ = -2
Common difference d = a₂ - a₁ = 4 - 7 = - 3
By using the formula
nth term of AP = 7 + (n - 1)(-3)
= 7 - 3n + 3
= 10 - 3n
Therefore,
The nth term of the Arithmetic sequence is 10 - 3n.
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Which expressions are equivalent to g + h + (j + k)? Check all that apply.
Group of answer choices
g (h + j) k
g + (h + j) + k
g + (h j) + k
(g + h) + j k
g h + j k
(g + h) + j + k
g + h (j + k)
The expressions are equivalent to g + h + (j + k) is g+(h+j)+k and (g+h)+j+k.
What is an associative property?
As some binary operations have the associative property, moving parenthesis around in an expression won't affect the outcome. Associativity is a legitimate rule of replacement for expressions in logical proofs in propositional logic.
Here, we have
Given: g + h + (j + k)
We have to find the expressions that are equivalent to the given expression.
By applying the associative property of equality: (a + b) + c = a + (b + c)
we can see that the parenthesis does not mean anything when we are only doing addition.
Hence, the expressions are equivalent to g + h + (j + k) is g+(h+j)+k and (g+h)+j+k.
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Every month, Jenny buys 15 pieces of bags for her small business. How many pieces of ba gs does she buy in a year?
Jenny buys 15 pieces of bags every month, which means she buys 180 pieces of bags in a year
This question can be solved by unitary method
To find out how many pieces of bags Jenny buys in a year, we can use the following formula:
Total pieces of bags in a year = Number of pieces of bags bought every month × Number of months in a year
We are given that Jenny buys 15 pieces of bags every month, and there are 12 months in a year. So, we can plug in these values into the formula:
Total pieces of bags in a year = 15 × 12
Total pieces of bags in a year = 180
Therefore, Jenny buys 180 pieces of bags in a year.
Answer: 180.
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Mr. Barnes rode his bike twenty miles in three hours. Mrs. Barnes rode her bike twenty-four miles in four hours. Mr. Barnes was riding at a
rate than Mrs. Barnes.
In conclusion, Mr. Barnes rode his bike at a rate of 6.67 miles per hour, which was faster than Mrs. Barnes' rate of 6 miles per hour.
How is their speed determined?We must compute the rate or speed that each participant rode in order to determine who rode faster. The rate is calculated by dividing the distance travelled by the time required. The rates for Mr. and Mrs. Barnes will now be determined:
Rate = Distance / Time = 20 Miles / 3 Hours = 6.67 Miles per Hour, Mr. Barnes
Rate = Distance/Time = 24 Miles/4 Hours = 6 Miles Per Hour, Mrs. Barnes
We can observe from comparing the speeds that Mr. Barnes rode more quickly than Mrs. Barnes. Her speed was 6 miles per hour, compared to his 6.67. As a result, we can conclude that Mr. Barnes was riding more quickly than Mrs. Barnes.
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A person places $72000 in an investment account earning an annual rate of 5.1%, compounded continuously. Using the formula V = P e r t V=Pe rt , where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 16 years
The amount of money in account after 16 years will be $162,823.39. The solution has been obtained by using compound interest.
What is compound interest?
Compound interest considers the principal when determining the interest for the subsequent month, in contrast to simple interest, which does not. In algebra, compound interest is typically denoted by the letter C.I.
We are given the following information:
P = $72000
Rate (r) = 5.1% = 0.051
Time (t) = 16 years
Using the formula, we get
⇒V = P[tex]e^{rt}[/tex]
⇒V = (72000) e⁰°⁰⁵¹ˣ¹⁶
⇒V = (72000) e⁰°⁸¹⁶
⇒V = $162,823.39
Hence, the amount of money in account after 16 years will be $162,823.39.
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One paper clip has the mass of one gram. 1000 paperclips have a mass of 1 kilogram How many kg are 5600 paperclips
5.6 kilograms.
1000 grams makes ONE kilogram and you have 5000 grams. Then the extra 600 grams are 6/10 of 1000 so it would be .6.
find the missing measures of the quadrilateral
Answer:
Angle C = 101 Degree
Angle E = 46 Degree
Step-by-step explanation:
Due to CD and FE are parallel, the adjacent angle between and DCF and angle EFC would sum up to be 180 degrees.
Angle DCF:
79 + Angle C = 180
Angle C = 101 Degree
Angle EFC:
134 + Angle E = 180 Degrees
Angle E = 46
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How many solution exist for the system of equations below? 3x+y=18
3x+y=16
Answer: The system of equations is:
3x + y = 18
3x + y = 16
To determine how many solutions this system has, we can subtract the second equation from the first:
(3x + y) - (3x + y) = 18 - 16
0 = 2
This is a contradiction, since 0 can never be equal to 2. Therefore, there are no solutions to this system of equations. Geometrically, these two equations represent two parallel lines in a coordinate plane that never intersect, so there is no point that satisfies both equations at the same time.
Step-by-step explanation:
Determine the total number of eggs in 7 dozen eggs
The total number of eggs in 7 dozens of eggs is 84 eggs
What is a Word Problem?Word Problem is a sentence usually made up of a few sentences describing a scenario that needs to be solved through mathematics.
How to determine this
When a dozen mean a group of twelves things
When 1 dozen of an egg = 12 eggs
To get the total number of eggs in 7 dozens of eggs
Let x represent the total number of eggs
When 1 dozen = 12 eggs
7 dozens = x
x = 7 dozens * 12 eggs/1 dozen
x = 84 eggs/1
x = 84 eggs
Therefore, 84 eggs make 7 dozens of eggs
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There are 50 students in the class. To each student we randomly assign 3 problems out of 6 problems written on the board. Let X be the total number of students to whom the problem 1 is assigned. Find V ar(X).
The variance of X, the total number of students to whom problem 1 is assigned, is 12.5.
We have,
To solve this problem, we can use the concept of a binomial distribution.
The number of students to whom problem 1 is assigned can be modeled as a binomial random variable.
Let's define X as the random variable representing the number of students to whom problem 1 is assigned.
We know that each student has a 3/6 = 1/2 probability of being assigned problem 1.
In a class of 50 students, the probability of a single student being assigned problem 1 is p = 1/2.
The number of students to whom problem 1 is assigned follows a binomial distribution with parameters n = 50 (number of students) and p = 1/2 (probability of success).
The variance of a binomial distribution is given by the formula:
Var(X) = np (1 - p)
Substituting the values, we have:
Var(X) = 50 x (1/2) x (1 - 1/2)
= 50 x (1/2) x (1/2)
= 25 x 1/2
= 25/2
= 12.5
Therefore,
The variance of X, the total number of students to whom problem 1 is assigned, is 12.5.
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Find the measure of the missing arc length
Applying the angle of intersecting chords theorem, the measure of arc UT is: 180°.
What is the Angle of Intersecting Chords Theorem?The angle of intersecting chords theorem states that when two chords intersect within a circle, the angle they create measures half the sum of the intercepted arc and its corresponding vertical arc.
Based on the above theorem, we can state that:
m<UST = 1/2(measure of arc AV + measure of arc UT)
Substitute
120 = 1/2(60 + measure of arc UT)
240 = 60 + measure of arc UT
240 - 60 = measure of arc UT
Measure of arc UT = 180°
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The rectangle below has an area of y^2+8xy+7x^2 square meters and a length of y+xy+xy, plus, x meters.
The rectangle has a length of y + 2xy + x meters and a width of (y + 7x) / (2y + x) meters.
What is a formula of area of rectangle?
Formula for rectangle's area
When calculating a rectangle's area, we multiply the length by the width of the rectangle.
The area of a rectangle is given by the formula:
A = L × W
where A is the area, L is the length, and W is the width.
In this case, we are given that the area is:
[tex]A = y^2 + 8xy + 7x^2[/tex]
and the length is:
L = y + xy + xy + x = y + 2xy + x
To find the width, we can rearrange the formula for the area:
W = A/L
Substituting the given values, we get:
[tex]W = (y^2 + 8xy + 7x^2)/(y + 2xy + x)[/tex]
Now, we can simplify this expression by factoring the numerator:
W = [(y + 7x)(y + x)] / [(y + x)(2y + x)]
Canceling out the common factor of (y + x), we get:
W = (y + 7x) / (2y + x)
Therefore, the width of the rectangle is:
W = (y + 7x) / (2y + x)
So the rectangle has a length of y + 2xy + x meters and a width of (y + 7x) / (2y + x) meters.
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At the beginning of 2019, there were an estimated 295,382 women living with cervical cancer in the United States.
During 2019, there were an additional 346 women newly diagnosed with cervical cancer. There were an estimated 166,238,619 women living in the United States in 2019. What was the incidence of cervical cancer among women in the United States in 2019?
Answer: To calculate the incidence of cervical cancer among women in the United States in 2019, we need to divide the number of new cases (346) by the total population at risk (i.e., the number of women without cervical cancer) and then multiply by 100,000 to express the result per 100,000 women.
Number of women without cervical cancer = Total number of women - Number of women with cervical cancer
Number of women without cervical cancer = 166,238,619 - 295,382
Number of women without cervical cancer = 165,943,237
Incidence of Cervical Cancer = (New cases / Number of women without cervical cancer) x 100,000
Incidence of Cervical Cancer = (346 / 165,943,237) x 100,000
Incidence of Cervical Cancer = 0.21 per 100,000 women
Therefore, the incidence of cervical cancer among women in the United States in 2019 was 0.21 per 100,000 women.
Step-by-step explanation:
Coco swam from Point A to Point B at a constant speed of 1. 2 m/s. At the same time, Azlinda swam from Point B to Point A. After 5 min, Azlinda had swum a distance of 420 m and she was 37 m away from Coco. What was the distance between Point A and Point B?
The distance between Point A and Point B is 840 meters.
Let's start by using the formula:
distance = speed x time
Since Coco swam at a constant speed of 1.2 m/s, we can find his distance using:
distance(Coco) = speed(Coco) x time
where time is the same for both Coco and Azlinda. Let's call this common time "t".
distance(Coco) =[tex]1.2 m/s \times t[/tex]
Now, let's consider Azlinda's situation. After 5 minutes (or 5/60 = 1/12 hours), she had swum a distance of 420 m and was 37 m away from Coco. Let's call the distance between Point A and Point B "d".
Since Azlinda was swimming towards Point A, she must have covered a distance of (d - 37) m by the time she had swum 420 m. We can use the formula above to find her speed:
speed(Azlinda) = distance(Azlinda) / time
speed(Azlinda) = (d - 37) m / (1/12) h
speed(Azlinda) = 12(d - 37) m/h
Now, we know that Azlinda and Coco were swimming towards each other for a total of 5 minutes (or 1/12 hours), so their total distance apart at that time was:
distance apart = distance(Coco) + distance(Azlinda)
distance apart = [tex]1.2 m/s \times t + 12(d - 37) m/h \times (1/12) h[/tex]
distance apart =[tex]1.2t + d - 37[/tex]
We also know that when they were 37 m apart, Azlinda had swum a distance of 420 m, so we can write:
420 = d - 37 - distance(Coco)
Substituting the expression for distance(Coco) from above, we get:
420 = d - 37 - 1.2t
Now we have two equations with two unknowns (d and t). We can substitute into the other equation and solve for one variable in terms of the other. For example, we can solve the second equation for t:
[tex]1.2t = d - 37 - 420\\1.2t = d - 457\\t = (d - 457) / 1.2[/tex]
When we enter this into the initial equation, we obtain:
distance apart = [tex]1.2t + d - 37[/tex]
distance apart = [tex]1.2((d - 457) / 1.2) + d - 37[/tex]
distance apart = [tex]d - 380.6[/tex]
Now we can substitute this expression for distance apart into the second equation:
[tex]420 = d - 37 - 1.2t\\420 = d - 37 - 1.2(d - 457) / 1.2\\420 = d - 37 - (d - 457)\\420 = -d + 420\\d = 840[/tex]
Therefore, the distance between Point A and Point B is 840 meters.
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n Exercises 5-6, find the coordinates of the segmen PQ. Calculate the distance from the midpoint to the ori in. 5. P=(2,3,1),Q=(0,5,7) 6. P=(1,0,3),Q=(3,2,5) 7. Let A=(−1,0,−3) and E=(3,6,3). Find points B,C, and D on the line segment AE such that d(A,B)=d(B,C)=d(C,D)=d(D,E)= 41d(A,E)
C= (1.17,9.93,−32.25
D= (3.33,14.9,−47).
For Exercise 5, the coordinates of the segment PQ are P = (2,3,1) and Q = (0,5,7). To calculate the distance from the midpoint to the origin, use the midpoint formula: M = [(P + Q) / 2].
In this case, M = [(2,3,1) + (0,5,7)] / 2 = (1,4,4).
Then calculate the distance from the midpoint to the origin by using the distance formula: d = √[(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2], where (x1, y1, z1) is the midpoint coordinates and (x2, y2, z2) is the origin coordinates. In this case, d = √[(1-0)2 + (4-0)2 + (4-0)2] = √17.
For Exercise 6, the coordinates of the segment PQ are P = (1,0,3) and Q = (3,2,5). To calculate the distance from the midpoint to the origin, use the midpoint formula: M = [(P + Q) / 2]. In this case, M = [(1,0,3) + (3,2,5)] / 2 = (2,1,4). Then calculate the distance from the midpoint to the origin by using the distance formula: d = √[(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2], where (x1, y1, z1) is the midpoint coordinates and (x2, y2, z2) is the origin coordinates. In this case, d = √[(2-0)2 + (1-0)2 + (4-0)2] = √21.
For Exercise 7, let A = (−1,0,−3) and E = (3,6,3). To find points B, C, and D on the line segment AE such that d(A,B)=d(B,C)=d(C,D)=d(D,E)= 41d(A,E), first calculate the distance between A and E using the distance formula: d(A,E) = √[(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2], where (x1, y1, z1) is the coordinates of A and (x2, y2, z2) is the coordinates of E. In this case, d(A,E) = √[(3-(-1))2 + (6-0)2 + (3-(-3))2] = √122.
To find the coordinates of points B, C, and D, use the following formula: B = A + (d(A,B)/d(A,E))(E-A), where d(A,B) is the distance from A to B, d(A,E) is the distance from A to E, A is the coordinates of A, and E-A is the vector pointing from A to E. Using this formula, the coordinates of B can be calculated as B = (−1,0,−3) + (41/122)((3,6,3) - (−1,0,−3)) = (−1,4.97,−17.5). Similarly, the coordinates of C and D can be calculated as C = (−1,4.97,−17.5) + (41/122)((3,6,3) - (−1,4.97,−17.5)) = (1.17,9.93,−32.25) and D = (1.17,9.93,−32.25) + (41/122)((3,6,3) - (1.17,9.93,−32.25)) = (3.33,14.9,−47).
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A line passes through the points (-2,7) and (0,-3). What is its equation in slope intercept form
The equation of the line passing through the points (-2,7) and (0,-3) in slope-intercept form is y = -5x + 7.
Let's first find the slope of the line using the two given points:
slope = (y2 - y1)/(x2 - x1)
slope = (-3 - 7)/(0 - (-2))
slope = (-3 - 7)/(0 + 2)
slope = -10/2
slope = -5
Now that we have the slope, we can use the point-slope form of a line to find its equation:
Where m is the slope and (x1,y1) is any point on the line, y - y1 = m(x - x1).
Let's use the point (-2,7) as (x1,y1):
y - 7 = -5(x - (-2))
y - 7 = -5(x + 2)
y - 7 = -5x - 10
y = -5x + 7
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Read the story
Bridgette and Naomi ran for sixth-grade class president. In the election, every sixth-
grade student voted for either Bridgette or Naomi, Bridgette received 5 votes for every 7
votes Naomi received.
Pick the diagram that models the ratio in the story.
Bridgette
Naomi
Bridgette
Naomi
If there are 240 students in the sixth-grade class, how many votes did Naomi receive?
votes
Submit
If there are 240 students in the sixth-grade class then the number of votes that Naomi received is: 336 votes.
How to solve algebra word problems?A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.
Bridgette and Naomi ran for sixth-grade class president.
In the election, every sixth-grade student voted for either Bridgette or Naomi.
Bridgette received 5 votes for every 7 votes Naomi received.
the first figure represents 5:7 ratio in the story.
Since Bridgette received 5 votes for every 7 votes Naomi received, Bridgette received 5/7 of the total votes and Naomi received 2/7 of the total votes.
We know that the total number of votes cast is equal to the total number of students in the class, which is 240.
Therefore, we can set up the equation:
⁵/₇(x) + ²/₇(x) = 240
Simplifying the equation, we get:
(5x + 2x)/7 = 240
7x/7 = 240
x = 240 × 7/5
x = 336
Therefore, If there are 240 students in the sixth-grade class then Naomi received 336 votes.
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BIG IDEAS MATH
#3 i
X =
Check
The side lengths of AABC are 10, 6x, and 20 and the side lengths of ADEF are 25, 30, and 50. Find the value of x that makes AABC~ADEF.
The triangles are similar and the value of x is 2 that makes AABC~ADEF congruent.
what is congruent ?
In geometry, two figures are said to be congruent if they have the same shape and size. In other words, if all corresponding angles are congruent and all corresponding sides are of equal length, then the two figures are congruent.
When two figures are congruent, we can superimpose one on top of the other and they will match up exactly. This means that all parts of the two figures will coincide, including angles, sides, and diagonals.
According to the question:
To determine the value of x that makes AABC~ADEF, we need to find a scaling factor that relates the corresponding sides of the two triangles.
Since AABC has side lengths of 10, 6x, and 20, its perimeter is 10 + 6x + 20 = 30 + 6x. Similarly, the perimeter of ADEF is 25 + 30 + 50 = 105.
Since the two triangles are similar, their corresponding sides are proportional. This means that:
10/25 = (6x)/30 = 20/50
Simplifying each of these ratios, we get:
2/5 = x/5 = 2/5
This tells us that x/5 = 2/5, or x = 2. Therefore, the value of x that makes AABC~ADEF is x = 2.
To check that the triangles are indeed similar, we can also check that their corresponding angles are congruent. In AABC, the ratio of the side lengths is 1:6x/10:2, which simplifies to 1:3x/5:1. Since the sum of the angles in a triangle is 180 degrees, we know that:
angle A + angle B + angle C = 180 degrees
Using the Law of Cosines, we can find the measure of angle B:
[tex]cos(B) = (10^2 + (6x)^2 - 20^2)/(210(6x)) = (100 + 36x^2 - 400)/(120x) = (36x^2 - 300)/(120x)[/tex]
[tex]B = cos^-1((36x^2 - 300)/(120x))[/tex]
Using this expression, we can express the measures of the angles in AABC in terms of x:
[tex]angle A = sin^{-1(1/(2x))} = 30 degrees[/tex]
[tex]angle B = cos^{-1((36x^2 - 300)/(120x))}[/tex]
angle C = 180 - 30 - B = 150 - B
Similarly, in ADEF, the ratio of the side lengths is 5:6:10. Using the Law of Cosines, we can find the measures of the angles:
[tex]angle D = cos^{-1((25^2 + 30^2 - 50^2)/(22530))} = 36.87 degrees[/tex]
[tex]angle E = cos^{-1((25^2 + 50^2 - 30^2)/(22550))} = 53.13 degrees[/tex]
angle F = 180 - 36.87 - 53.13 = 90 degrees
Comparing the angles in the two triangles, we can see that angle A is congruent to angle F (both are 30 degrees) and angle C is congruent to angle D (both are 180 - B - 30). Therefore, the triangles are similar.
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an estate valued at 60,000 is divided among albert, brian and charles in the ratio 1:2:3 respectively. Calculate the amount each receives
Answer:
Albert receives $10,000, Brian receives $20,000, and Charles receives $30,000.
Step-by-step explanation:
1 + 2 + 3 = 6
Next, we can find out what fraction of the estate each person is entitled to:
Albert: 1/6 of the estate
Brian: 2/6 (or 1/3) of the estate
Charles: 3/6 (or 1/2) of the estate
Now, we can calculate the amount each person receives by multiplying their share by the total value of the estate:
Albert: (1/6) x $60,000 = $10,000
Brian: (1/3) x $60,000 = $20,000
Charles: (1/2) x $60,000 = $30,000
pls help me solve this
Answer:
Step-by-step explanation:
A= 5(x+3)=17
B= 5x+3=17
C= 3x+5=17 A girl decides to buy 5 pens for 3 of her friends. The total cost of the pens was 17$. She then decides to make her friends guess the cost of each pen.
3. Find a general solutions for the following problems
Use Maxima to verify your answers and to plot the solution
(c) y" − 2y' + y = 0, y(π) = e ^π , y'^ (π) = 0.
The given differential equation is y" − 2y' + y = 0. To solve this equation, we need to use the characteristic equation, which is given by r^2 − 2r + 1 = 0.
The two solutions to this equation are r = 1 ± i. Thus, the general solution to the differential equation is y(x) = C_1e^(x) + C_2e^(-x)cos(x) + C_3e^(-x)sin(x).
We can use Maxima to verify our solution. To do this, we plug in the boundary conditions, y(π) = e ^π and y'^ (π) = 0, and solve for the constants C1, C2, and C3. This gives us C1 = 1, C2 = -1, and C3 = 0. Thus, the solution to the differential equation is y(x) = e^x - e^(-x)cos(x).
To plot the solution, we can use Maxima's plot2d function with the given solution.
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