Corresponding Angles Theorem: If a transversal intersects two parallel lines, then the corresponding angles formed are congruent.
Statements Reasons
1. ∠1 and ∠3 are corresponding angles Given
2. m//n Given
3. ∠1 and ∠2 are congruent Alternate Interior Angles Theorem
4. ∠2 and ∠3 are congruent Alternate Interior Angles Theorem
5. ∠1 ≅ ∠2 and ∠2 ≅ ∠3 Substitution (from statements 3 and 4)
6. ∠1 ≅ ∠3 Transitive Property of Congruence
How to explain the TheoremIn this proof, we start by stating that ∠1 and ∠3 are corresponding angles, which is given in the problem statement. We also know that m and n are parallel lines, which is also given.
Then, we apply the Alternate Interior Angles Theorem to show that ∠1 and ∠2, as well as ∠2 and ∠3, are congruent.
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Kali has a choice of 20 flavors for her triple scoop cone. If she
chooses the flavors at random, what is the probability that the 3 flavors she
chooses will be vanilla, chocolate, and strawberry?
Find the inverse for each relation: 4 points each
1. {(1,‐2), (2, 3),(3, ‐3),(4, 2)}
2. {(4,2),(5,1),(6,0),(7,‐1)}
Find an equation for the inverse for each of the following relations.
3. Y=-8x+3
4. Y=2/3x-5
5. Y=1/2x+10
6. Y=(x-3)^2
Verify that f and g are inverse functions.
7. F(x)=5x+2;g(x)=(x-2)/5
8. F(x)=1/2x-7;g(x)=2x+14
The inverse for each relation:
1. {(1,‐2), (2, 3),(3, ‐3),(4, 2)} - {(-2, 1), (3, 2), (-3, 3), (2, 4)}
2. {(4,2),(5,1),(6,0),(7,‐1)} - {(2, 4), (1, 5), (0, 6), (-1, 7)}
3. Inverse equation: y=(-1/8)x+3/8
4. Inverse equation: y=3/2x+15/2
5. Inverse equation: y=2x-20
6. Inverse equation: y=[tex]x^{(1/2)}+3[/tex]
7. Since fog(x) = gof(x) = x, f and g are inverse functions.
8. Since fog(x) = gof(x) = x, f and g are inverse functions.
1. To find the inverse of the relation, we need to swap the positions of x and y for each point and then solve for y.
{(1, -2), (2, 3), (3, -3), (4, 2)}
Inverse: {(-2, 1), (3, 2), (-3, 3), (2, 4)}
2. Again, we swap x and y and solve for y.
{(4, 2), (5, 1), (6, 0), (7, -1)}
Inverse: {(2, 4), (1, 5), (0, 6), (-1, 7)}
3. To find the inverse equation for y=-8x+3, we swap x and y and solve for y.
x=-8y+3
x-3=-8y
y=(x-3)/-8
Inverse equation: y=(-1/8)x+3/8
4. To find the inverse equation for y=2/3x-5, we swap x and y and solve for y.
x=2/3y-5
x+5=2/3y
y=3/2(x+5)
Inverse equation: y=3/2x+15/2
5. To find the inverse equation for y=1/2x+10, we swap x and y and solve for y.
x=1/2y+10
x-10=1/2y
y=2(x-10)
Inverse equation: y=2x-20
6. To find the inverse equation for y=(x-3)², we swap x and y and solve for y.
x=(y-3)²
[tex]x^{(1/2)}=y-3[/tex]
[tex]y=x^{(1/2)}+3[/tex]
Inverse equation: [tex]y=x^{(1/2)}+3[/tex]
7. To verify that f(x)=5x+2 and g(x)=(x-2)/5 are inverse functions, we need to show that fog(x)=gof(x)=x for all x in the domain of f and g.
fog(x) = f(g(x)) = f((x-2)/5) = 5((x-2)/5) + 2 = x
gof(x) = g(f(x)) = g(5x+2) = ((5x+2)-2)/5 = x/5
Since fog(x) = gof(x) = x, f and g are inverse functions.
8. To verify that f(x)=1/2x-7 and g(x)=2x+14 are inverse functions, we need to show that fog(x)=gof(x)=x for all x in the domain of f and g.
fog(x) = f(g(x)) = f(2x+14) = 1/2(2x+14) - 7 = x
gof(x) = g(f(x)) = g(1/2x-7) = 2(1/2x-7) + 14 = x
Since fog(x) = gof(x) = x, f and g are inverse functions.
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A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options. m∠5 + m∠3 = m∠4 m∠3 + m∠4 + m∠5 = 180° m∠5 + m∠6 =180° m∠2 + m∠3 = m∠6 m∠2 + m∠3 + m∠5 = 180°
The true triangle statement regarding the diagram are:
1. m∠5 + m∠6 = 180° ________Linear Pair
2. ∠ 2+ ∠ 3 = ∠ 6________Exterior angle Property of Triangle
3. m∠2 + m∠3 + m∠5 = 180°________Triangle Sum Property
What is the angle measurement?From the question, Δ ABC with Exterior angles as ∠ 1 , ∠ 4 ,and ∠ 6
Note that the Exterior angle Property of Triangle state that An exterior angle of a triangle is equal to the sum of the opposite interior angles.
Hence: For Exterior ∠ 1 :
∠ 1 = ∠ 5 + ∠ 3 ________Exterior angle Property of Triangle
Also,
For Exterior ∠ 4:
∠ 4 = ∠ 5 + ∠ 2 ________Exterior angle Property of Triangle
Also,
In regards to Exterior ∠ 6:
∠ 6 = ∠ 2 + ∠ 3 ________ Exterior angle Property of Triangle
Using Triangle Sum Property, it state that In a triangle sum of the measures of angles is equal to 180° Hence: m∠2 + m∠3 + m∠5 = 180° ________Triangle Sum Property
The Linear Pair will be: The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees.
Therefore, m∠5 + m∠6 = 180° ________Linear Pair
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See full question below
A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options.
m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠5 + m∠6 =180°
m∠2 + m∠3 =
m∠6 m∠2 + m∠3 + m∠5 = 180°
Please hurry I need it ASAP
measured in astronomical units, can be modeled using the expression ((1)/(52)x)^((2)/(3)) , where x is the number of Earth weeks it takes for the planet to orbit the sun. Which expression could also be used to represent the average distance of a planet from the sun using radicals?
So the expression that represents the average distance of a planet from the sun using radicals is: d = k/2√13 * √x
What is exponent?An exponent, also known as a power, is a mathematical notation that indicates the number of times a quantity is multiplied by itself. It is usually written as a small number (the exponent) placed to the right and above a larger number (the base). Exponents are used in many mathematical concepts, including logarithms, roots, and scientific notation.
Here,
The expression ((1)/∛(52)x)²) can be simplified using exponent rules:
((1)/∛(52)x)²) =((1)/∛(52)x)²) * ∛x²)
= 1/(∛52² * ∛x²)
The average distance of a planet from the sun measured in astronomical units can be represented using the formula:
d = k * √T
where d is the distance from the sun, T is the time it takes for the planet to orbit the sun, and k is a constant of proportionality.
We can rewrite this formula in terms of Earth weeks by noting that there are 52 weeks in a year, so T = (1/52)x years. Substituting this into the formula, we get:
d = k * √((1/52)x)
Simplifying this expression using exponent rules, we get:
d = k * √(1/52)* √x
So an equivalent expression using radicals to represent the average distance of a planet from the sun is:
d = k * √(1/(52)) * √x
which simplifies to:
d = k/√(52) * √x
or
d = k/2√13 * √x
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The circumference (C) of a circle is 16 cm. Which formula can you use to find the diameter (d) if you know that C = π
d?
Answer:
c/π=d
explanation:
d × π = c
divide c to isolate d
Answer: I would multiply pie by a diameter until it equals 16.
(I know this probably isn’t the professional way but it should work.
Let R(x). C(x), and P(x) be, respectively, the revenue, cost, and profit, in dollars, tomi the production and sale of x items. I R(%) = 6x and C(X) = 0.001x^2 + 1 8x + 40.
find each of the following
a) P(x)
b) R(200). C(200), and P(200)
c) R'(. C't and P'(x)
d) R' (200). C'(200), and P' (200)
a) P(x) = R(x) - C(x) = 6x - (0.001x^2 + 18x + 40) = -0.001x^2 - 12x - 40
b) R(200) = 6(200) = 1200
C(200) = 0.001(200)^2 + 18(200) + 40 = 4000
P(200) = R(200) - C(200) = 1200 - 4000 = -2800
c) R'(x) = 6
C'(x) = 0.002x + 18
P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18) = -0.002x - 12
d) R'(200) = 6
C'(200) = 0.002(200) + 18 = 18.4
P'(200) = -0.002(200) - 12 = -12.4
Here are the answers to each part:
a) P(x) is the profit function, which is calculated as the difference between the revenue function and the cost function: P(x) = R(x) - C(x). In this case, P(x) = 6x - (0.001x^2 + 18x + 40).
b) To find R(200), C(200), and P(200), plug x = 200 into each function:
R(200) = 6(200) = 1200
C(200) = 0.001(200^2) + 18(200) + 40 = 7600
P(200) = 1200 - 7600 = -6400
c) To find R'(x), C'(x), and P'(x), we need to find the derivative of each function with respect to x:
R'(x) = d(6x)/dx = 6
C'(x) = d(0.001x^2 + 18x + 40)/dx = 0.002x + 18
P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18)
d) To find R'(200), C'(200), and P'(200), plug x = 200 into each derivative function:
R'(200) = 6
C'(200) = 0.002(200) + 18 = 18.4
P'(200) = 6 - 18.4 = -12.4
I hope this helps! Let me know if you have any further questions.
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The greenery landscaping company puts in an order for 2 pine trees and 5 hydrangea bushes for a neighborhood project. the order costs $150. they put in a second order for 3 pine trees and 4 hydrangea bushes that cost $144.50.
2(2.5) + 5(23) = 150
3(2.5) + 4(23) = 144.5
Both equations are satisfied, so our solution is correct.
The greenery landscaping company orders how many trees and bushes for the neighborhood?To solve the problem, let's first assign some variables. Let x be the cost of one pine tree and y be the cost of one hydrangea bush. We can then use these variables to set up a system of equations:
2x + 5y = 150 (equation 1)
3x + 4y = 144.5 (equation 2)
We can solve this system of equations using various methods. Here, we will use the substitution method.
From equation 1, we can solve for x in terms of y:
2x = 150 - 5y
x = (150 - 5y)/2
We can then substitute this expression for x into equation 2:
3((150 - 5y)/2) + 4y = 144.5
Multiplying both sides by 2 to eliminate the fraction:
3(150 - 5y) + 8y = 289
Expanding and simplifying:
450 - 15y + 8y = 289
-7y = -161
y = 23
We can now substitute this value for y into either equation 1 or 2 to solve for x:
2x + 5(23) = 150
2x = 5
x = 2.5
Therefore, one pine tree costs $2.50 and one hydrangea bush costs $23.
To check our work, we can substitute these values into both equations:
2(2.5) + 5(23) = 150
3(2.5) + 4(23) = 144.5
Both equations are satisfied, so our solution is correct.
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Mathematics help nedd
To solve the equation, we need to first simplify both sides:
(4x - 6)/5 + 1 = (x + 1)/5 - 2/5
Multiplying both sides by 5 to eliminate the denominator:
4x - 6 + 5 = x + 1 - 2
Simplifying further:
4x - 1 = x - 1
Subtracting x from both sides:
3x - 1 = -1
Adding 1 to both sides:
3x = 0
Dividing both sides by 3:
x = 0
Therefore, the solution to the equation is x = 0.
Answer: x=28
Step-by-step explanation:
Given: <A=68
Find: x
Reasoning:
<B = 2x+x
<B= 3x
<C=x they say the sides across from <C is same as other side so the
angles are the same
Solution:
All angles of a triangle =180
<A + <B + <C =180 >substitute
68 + 3x + x =180 > combine like terms
68 + 4x = 180 > subtract 68 from both sides
4x=112 >divide both sides by 4
x=28
HELPPP SOMEBODY PLEASEEE WITH THIS MATHHHH
The correct statement is given as follows:
The function g(t) reveals the market value of the house increases by 3.6% each year.
How to define an exponential function?An exponential function has the definition presented as follows:
[tex]y = ab^x[/tex]
In which the parameters are given as follows:
a is the value of y when x = 0.b is the rate of change.The parameter b for this problem is given as follows:
b = 1.036.
As the parameter b has an absolute value greater than 1, the function is increasing, with a rate given as follows:
1.036 - 1 = 0.036 = 3.6% a year.
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Ofra tried to solve an equation.
3x = 4.5
3x 4.5
3
3
=
Setting up
x = 1.5 Calculating
Where did Ofra make her first mistake?
Choose 1 answer:
Setting up
B Calculating
Ofra correctly solved the equation.
If Ofra tried to solve an equation 3x = 4.5, The statement "Ofra correctly solved the equation" is correct. So, correct option is C.
We can see this by substituting x = 1.5 into the original equation 3x = 4.5:
3(1.5) = 4.5
Simplifying the left-hand side, we get:
4.5 = 4.5
This is a true statement, which means that x = 1.5 is a valid solution to the equation 3x = 4.5.
Therefore, Ofra did not make any mistakes in solving the equation. She correctly set up the equation 3x = 4.5 by multiplying both sides by 3 to isolate x, and then calculated the value of x to be 1.5, which is the correct solution.
Option (c) is the correct answer.
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Complete question is:
Ofra tried to solve an equation.
3x = 4.5, Setting up x = 1.5 Calculating
Where did Ofra make her first mistake?
Choose 1 answer:
a) Setting up
b) Calculating
c) Ofra correctly solved the equation.
compute (7 4/9 -8)*3.6-1.6*(1/3-3/4)+ 1 2/5 ÷(0.35)
The value of (7 4/9 -8)*3.6-1.6*(1/3-3/4)+ 1 2/5 ÷(0.35) is given as 241/54.
How to solve for the value(7 4/9 -8) = -5/9.
3.6-1.6 = 2.0
1/3-3/4 = 1/3 - 3/4
= 4/12 - 9/12
= -5/12
we will have -5/9 * 2 = -10/9.
-10/9 * -5/12
10/9 * -5/12 = (10 * 5) / (9 * 12) = 50/108
We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 2:
50/108 = 25/54
we will have
25/54 + 1 2/5 ÷(0.35)
1 2/5 ÷ 0.35 = (7/5) ÷ (35/100) = (7/5) * (100/35) = 4
Now, we can substitute this value into the expression:
25/54 + 4 = (25/54) + (216/54) = 241/54
Therefore, the value of the expression 25/54 + 1 2/5 ÷(0.35) is 241/54.
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A lake currently has a depth of 30 meters. As sediment builds up in the lake, its depth decreases by 2% per year.
This situation represents:
A. Exponential decay
B. Exponential growth
The rate of growth or decay, r, is equal to:
A. 1. 02
B. 0. 02
C. 0. 98
So the depth of the lake each year is ______ times the depth in the previous year.
A. 0. 98
B. 0. 02
C. 1. 02
It will take between _____ years for the depth of the lake to reach 26. 7 meters.
A. 3 and 4
B. 11 and 12
C. 9 and 10
D. 5 and 6
The situation represents exponential decay.
The rate of growth or decay, r, is equal to 0.02.
So the depth of the lake each year is 0.98 times the depth in the previous year.
It will take between 11 and 12 years for the depth of the lake to reach 26. 7 meters.
The situation represents exponential decay, as the depth of the lake decreases by a constant percentage each year. The rate of decay is 2% per year, so the rate of growth or decay, r, is equal to 0.98 (1 - 0.02). This means that the depth of the lake each year is 0.98 times the depth in the previous year.
To find the number of years it will take for the depth of the lake to reach 26.7 meters, we can use the formula for exponential decay:\
D = D₀ *[tex]e^{(-rt)[/tex]
where D is the current depth, D₀ is the initial depth, r is the rate of decay, and t is the number of years.
Substituting the given values, we get:
26.7 = 30 * [tex]e^{(-0.02t)[/tex]
Solving for t, we get:
t = ln(26.7/30) / (-0.02) ≈ 11.33
Therefore, it will take between 11 and 12 years for the depth of the lake to reach 26.7 meters.
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Find the length of the segment indicated. Round your answer to the nearest tenth if necessary.
The value of x in the given circle is 18.1 units.
Given is a circle, where two radii are given one chord is given,
We need to find the value of the x which is also the radius,
We know all the radii in a circle are equal,
So, here the radius = 7.9+10.2 = 18.1 units.
Hence the value of x in the given circle is 18.1 units.
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what percentage is equivalent to 96/160
Answer:
60%
Step-by-step explanation:
Take 96 and divide it by 160.
(easier if done on a calculator.)
For example: Find A/B as a percentage: take "A" and divide it by "B"
You can find the area of a trapezoid by decomposing it into a rectangle and one or more triangles you can find the area of a kite by decomposing it into triangles
The statement on finding the areas of a trapezoid and a kite are True.
How to find area by decomposing shapes ?To determine the area of a trapezoid, it can be broken down into separate geometrical shapes. One possible breakdown would include a rectangle with two adjacent right triangles or an isosceles triangle with one right triangle configuration. By calculating each smaller compartment's size and summing them together, one can obtain the total area for the trapezoid.
Similarly, in order to find the surface area of a kite shape, drawing a diagonal creates two adjoining triangles that are easily computed individually then summed.
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Options for this question :
True
False
CAN SOMEONE SHOW ME STEP BY STEP ON HOW TO DO THIS
A city just opened a new playground for children in the community. An image of the land that the playground is on is shown.
A polygon with a horizontal top side labeled 45 yards. The left vertical side is 20 yards. There is a dashed vertical line segment drawn from the right vertex of the top to the bottom right vertex. There is a dashed horizontal line from the bottom left vertex to the dashed vertical, leaving the length from that intersection to the bottom right vertex as 10 yards. There is another dashed horizontal line that comes from the vertex on the right that intersects the vertical dashed line, and it is labeled 12 yards.
What is the area of the playground?
900 square yards
855 square yards
1,710 square yards
A father and his three children decide on all matters with a vote. Each member of the family gets as many votes as their age. Right now, the family members are 36, 13, 6, and 4 years old, so the father always wins. How many years will it take for the three children to win a vote if they all agree? Show your work.
Answer:
Step-by-step explanation:
Answer:
13 years
Step-by-step explanation:
Intuition for how sons can collectively win after a certain period of time:- After a certain period of time the father's age will increase by that certain period of time (say 5 years) but for the sons (since there are 3 of them) their collective age will increase by three times that of their father (5 for each 1 one them). Therefore there exist a time after which collective increase in sons' age can cover the current gap of 13 years.
Identify the random variable in each distribution, and classify it as
discrete or continuous. Explain your reasoning.
1) The number of hits for the players of a baseball team.
2) The distances traveled by the tee shots in a golf
The random variable in the first situation is the number of hits for the players of a baseball team and in the second situation is the distance traveled by the tee shots in a golf game.
1) The random variable in this distribution is the number of hits for the players of a baseball team. This is a discrete random variable because hits are counted as whole numbers and cannot take on non-integer values.
2) The random variable in this distribution is the distance traveled by the tee shots in a golf game. This is a continuous random variable because the distances traveled can take on any value within a certain range, including non-integer values. The exact distance traveled by a tee shot can be measured to any degree of precision, and there are infinitely many possible distances within the range of possible outcomes. Therefore, it is a continuous random variable.
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Select the correct answer
a mine extracts 2 metric tons of coal in an hour. the
number of hours spent mìning, which expression re
oa. the expression is at. the amount of ore
ob. the expression
The expression that represents the amount of ore sold and how much ore can the mine sell after extracting ore for 12 hours is option B: The expression is 2t−14t. The amount of ore is 21 metric tons.
The reasoning for the selection of the expression and amount of ore can the mine sell after extracting ore for 12 hours is as follows.
1: Determine the amount of coal used for electricity generation in terms of t.
The mine uses 14 tons of coal every hour, so the total amount used for electricity generation is 14t.
2: Determine the total amount of coal extracted in terms of t.
The mine extracts 2 tons of coal every hour, so the total amount extracted is 2t.
3: Calculate the amount of coal sold in terms of t.
To find the amount of coal sold, subtract the amount used for electricity generation from the total amount extracted: 2t - 14t.
4: Determine the amount of coal sold after 12 hours.
Substitute t = 12 into the expression:
2(12) - 14(12) = 24 - 168 = -144.
However, since the mine uses 14 tons of the extracted coal every hour, it cannot sell more coal than it extracts. So, the correct expression should be 2t - 14 (without the t for the amount used for electricity generation).
5: Calculate the amount of coal sold after 12 hours using the corrected expression.
Substitute t = 12 into the expression: 2(12) - 14 = 24 - 14 = 10 metric tons.
The correct expression should be 2t - 14, and the amount of coal the mine can sell after extracting coal for 12 hours is 10 metric tons. Hence, the correct answer is option B.
Note: The question is incomplete. The complete question probably is: A mine extracts 2 metric tons of coal in an hour. The mine uses 14 ton of the extracted coal every hour to generate electricity for the mine and sells the rest. If t is the number of hours spent mining, which expression represents the amount of ore sold? How much ore can the mine sell after extracting ore for 12 hours? A) The expression is 2t−1/4t. The amount of ore is 23 3/4 metric tons. B) The expression is 2t−1/4t. The amount of ore is 21 metric tons. C) The expression is 2t+1/4t. The amount of ore is 24 metric tons. D) The expression is 2t+1/4t. The amount of ore is 24 1/4 metric tons.
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The measures of the angles of a triangle are shown in the figure below. Solve for x.
(2x+16) 48degrees
Answer:
x = 16
Step-by-step explanation:
(2x + 16) = 48
Subtract 16 with the positive 16 to cancel the numbers.
Subtract 16 with 48.
2x = 32
divide 32 by 2 to isolate the x.
32/2 = 16
x = 16
i need help its due in 2 hours
Answer:
C. The product of two irrational numbers is irrational.
Example: √3•√3=3
What is the value of B? Bº 58° 61°
Answer:
61 degrees
Step-by-step explanation:
Triangle interior measures add up to 180 degrees.
61 + 58 + x = 180
119 + x = 180
x = 61
hope this helps :) !!!
(3X-5)^1/4+3=4
Your anwser should be x=2!
SHOW WORK
(Explanation below)
x=2
x = 2 is the solution of the equation
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given equation is [tex](3X-5)^(^1^/^4^) + 3 = 4[/tex]
We have to find the value of x
Subtracting 3 from both sides:
[tex](3X-5)^(^1^/^4^) = 1[/tex]
Raising both sides to the fourth power:
3X - 5 = 1^4
3X - 5 = 1
Adding 5 to both sides:
3X = 6
Dividing by 3:
X = 2
Therefore, x = 2 is the solution of the equation
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On a coordinate plane, a line segment has endpoints P(6,2) and Q(3. 8). 9. Point M lies on PQ and divides the segment so that the ratio of PM-MQ is 2-3. What are the coordinates of point M?
The coordinates of point M come out to be 4.8, 4.4
This case is solved by using the section formula which states that
The coordinate of point P that divides the line segment AB in the ratio of m:n where the coordinate of A is [tex]x_1,y_1[/tex] and the B is [tex]x_2,y_2[/tex] is described as
[tex]\frac{mx_2+nx_1}{m+n}[/tex],[tex]\frac{my_2+ny_1}{m+n}[/tex]
The line to be divided = PQ
Coordinates of P = (6,2)
Coordinates of Q = (3,8)
Ratio = 2:3
Thus, the coordinates of M = [tex]\frac{2*3+3*6}{2+3}[/tex],[tex]\frac{8*2+2*3}{2+3}[/tex]
= 24/5 , 22/5
= 4.8, 4.4
Point M with coordinates (4.8,4.4) lies on PQ and divides the segment so that the ratio of PM-MQ is 2-3
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You get a job as a nurse. Your salary for the first year is $74,000. You will
receive a 1.2% increase every year. If you could save your entire salary, how
much money would you have in 5 years? Round to the nearest cent (2 decimal
places). Hint: What is a₁? What is r? Then use the formula for a finite
geometric series.
Answer: The amount of money you would have in 5 years if you could save your entire salary with a 1.2% increase every year would be $87,357.41.
Explanation:
The initial term, a₁, is $74,000, and the common ratio, r, is 1 + 1.2% = 1.012. To find the sum of the first 5 terms, we use the formula for a finite geometric series:
S₅ = a₁(1 - r⁶)/(1 - r)
Plugging in the values, we get:
S₅ = $74,000(1 - 1.012⁵)/(1 - 1.012) = $87,357.41 (rounded to the nearest cent)
Therefore, if you save your entire salary, you would have approximately $87,357.41 in 5 years with a 1.2% increase every year.
One side of an isosceles triangle is 2x + 1ft long. The other two sides are both 3x-14 long. The perimeter of the triangle is 55 ft. What is the length of each side? Show your work.
Let's use "a" to represent the length of the equal sides of the isosceles triangle, and let's use "b" to represent the length of the third side. We're told that one of the equal sides is 2x + 1ft long, so we can set up an equation:
2a + b = 55
We're also told that the other two sides are both 3x - 14ft long, so we can set up another equation:
a = 3x - 14
Now, we can substitute the second equation into the first equation and solve for "b":
2a + b = 55
2(3x-14) + b = 55
6x - 28 + b = 55
b = 83 - 6x
Now, we can substitute both equations into the equation a = 3x - 14 and solve for "x":
3x - 14 = 2x + 1 + 3x - 14
6x - 27 = 0
x = 4.5
Finally, we can substitute "x" into our equations to find the lengths of the sides:
a = 3x - 14 = 3(4.5) - 14 = 0.5
b = 83 - 6x = 83 - 6(4.5) = 55
So the length of the equal sides is 0.5ft, and the length of the third side is 55ft. Therefore, the lengths of the sides of the isosceles triangle are 0.5ft, 0.5ft, and 55ft.
The regular price of a red T-shirt is $6.93. Ernest has a coupon for $6.75 off. How much will Ernest pay for the T-shirt?
Answer:
18 cent
Step-by-step explanation:
Evaluate the integrals (Indefinite and Definite) and Simplify. 5 (a) 5 (5:-* - - 5 sin ) : dc xl1 (v) [(1822–1 18x)(6x3 – 9x2 – 3)6 dx ° ? (c) | Viana sec2 х dx (d) os Venta de Зх dx Væ+4 2 (e) ( 120 dax V1 + 2x2
(a) Indefinite integral of 5(5x^4 - 5sinx)dx is (5/3)x^5 + 5cosx + C. Definite integral over [0, π/2] is (125π/6) - 5.
We can evaluate the indefinite integral by applying the power rule and integration by substitution. The definite integral can be evaluated by substituting the limits of integration and simplifying.
(b) Indefinite integral of [(18x^2 - 1)(6x^3 - 9x^2 - 3)]^6dx is (18x^11 - 77x^9 + 126x^7 - 108x^5 + 49x^3 - 9x) / 11 + C.
To simplify the given expression, we can first expand the polynomial and then apply the power rule to integrate each term. The constant of integration can be added at the end.
(c) Definite integral of ∫tan^2(x)sec^2(x)dx over [0,π/4] is 1.
We can use the trigonometric identity sec^2(x) - 1 = tan^2(x) to simplify the integrand. Then we can apply the power rule and substitute the limits of integration to evaluate the definite integral.
(d) Indefinite integral of ∫(x+4)^2√(3x^2+4)dx is (1/15)(3x^2+4)^(3/2)(x+4) - (4/45)(3x^2+4)^(3/2) + C.
We can use substitution to simplify the integrand by setting u = 3x^2 + 4. After integrating, we can substitute back for u and simplify the constant of integration.
(e) Indefinite integral of ∫(120/(1+2x^2))dx is 60√2tan^(-1)(√2x) + C.
We can use substitution to simplify the integrand by setting u = 1 + 2x^2. After integrating, we can substitute back for u and simplify the constant of integration.
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The graph below shows segment FG and point P what is the first coordinate of point M
Note that the first coordinate of M is -1. (Option D)
Why is this so?
Given :- coordinates of F = (-4,-2) = (x1,y1)
coordinates of G = (2,-2) = (x2,y2)
coordinates of P = (2,-8) = (x3,y3)
and distance between point M and P is half of the distance between FG
To find :- first coordinate of point M
solution :- let the coordinate of M be (x4,y4)
as we know that distance between of the opposite point of parallel line segments are equal
so, second coordinate of M = -8
now by distance formula
FG = √(x2-x1)² + (y2-y1²)
= √[2-(-4)]² + [-2-(-2)]²
= √(2+4)² + (2-2)²
= √(6)² + (0)²
=√36
F G = 6
so, distance between point M and P = 1/2 × F G
= 1/2 × 6
= 3units
again, by distance formula
MP = √(x3-x⁴)² + (y3-y4)²
3 = √(2-x⁴)² + [-8-(-8)]²
squaring on both side
(3)² = (√(2-x⁴)² + [-8-(-8)]²)²
9 = (2-x⁴)² + [-8-(-8)]²
9 = (2-x⁴)²+(0)²
9 = (2-x⁴)²
√9 = 2-x⁴
3 = 2-x⁴
x⁴ = 2-3
x⁴ = -1
Hence the first coordinate of M is -1
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Full Question:
Although part of your question is missing, you might be referring to this full question:
Point M is located in the third quadrant.
The distance between point M and point P is half the distance between point F and point G.
• Segment MP is parallel to segment FG. What is the first coordinate of point M?