Answers
The measures of the two missing sides are:
c = 5√6 b = 10√2 How to find the lengths of b and c?Here we can see a right triangle where we know the measures of the interior angles and one of the sides. And we want to find the lengths of the other two sides.
We can use trigonometric relations like.
sin(a) = (opposite cathetus)/hypotenuse.
Here we know that b is the hypotenuse, and the known side is the opposite cathetus of the 30° angle, then we can write:
sin(30°) = 5√2/b
Solving this for b, we will get:
b = 5√2*(1/sin(30°))
b = 5√2*(2) = 10√2
And the value of c is given by:
sin(60°) = c/10√2
10√2*sin(60°) = c
10√2*(√3/2) = c
5*√(2*3) = c
5√6 = c
These are the two missing lengths.
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Related Questions
Extended sheet of paper in Europe is 21.0 CM wide and 29.7 cm long which has a greater area the standard sheet of paper with using stage or the standard sheet of paper in Europe explain and show your reasoning
Answers
the extended sheet in Europe has a greater area than the standard letter size paper. The difference in area between the two is about 20 square cm, it can make a difference in certain applications.
What is area of rectangle ?area of rectangle = length * breath
The standard sheet of paper commonly used in the United States and Canada is known as "Letter" size paper, which measures 8.5 inches by 11 inches (21.59 cm x 27.94 cm). To compare the area of this paper to the extended sheet of paper used in Europe, we need to convert the dimensions to centimeters.
So, the area of the standard letter size paper in centimeters would be:
21.59 cm x 27.94 cm = 603.78 square cm
On the other hand, the extended sheet of paper in Europe measures 21.0 cm by 29.7 cm. Therefore, its area would be:
21.0 cm x 29.7 cm = 623.70 square cm
the extended sheet of paper in Europe has a greater area than the standard letter size paper used in the United States and Canada. The difference in area between the two is about 20 square cm, it can make a difference in certain applications.
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75% of the people in a small city have car insurance with the company Geico. This unusually high amount is due to the amount of advertising withing the city and social media ads. If 10 people are selected at random from this city, what is the probability that at least 8 of them have car insurance with Geico?
Answers
Answer:
Step-by-step explanation:
This is a binomial distribution problem, where each person either has car insurance with Geico or doesn't have.
Let's denote the probability of a person having car insurance with Geico as p = 0.75, and the probability of not having car insurance with Geico as q = 0.25.
The probability of getting exactly k people out of n people who have car insurance with Geico is given by the binomial distribution formula:
P(k) = (n choose k) * p^k * q^(n-k)
where (n choose k) is the number of ways to choose k items out of n items, which is calculated by the formula:
(n choose k) = n! / (k! * (n-k)!)
where n! denotes the factorial of n.
We need to find the probability of getting at least 8 people out of 10 who have car insurance with Geico, which is the same as the probability of getting 8, 9, or 10 people who have car insurance with Geico. So, we can calculate this probability by adding the probabilities of these three cases:
P(at least 8) = P(8) + P(9) + P(10)
P(8) = (10 choose 8) * 0.75^8 * 0.25^2 = 0.002
P(9) = (10 choose 9) * 0.75^9 * 0.25^1 = 0.054
P(10) = (10 choose 10) * 0.75^10 * 0.25^0 = 0.056
Therefore, the probability of at least 8 people out of 10 having car insurance with Geico is:
P(at least 8) = P(8) + P(9) + P(10) = 0.002 + 0.054 + 0.056 = 0.112
So, the probability that at least 8 people out of 10 have car insurance with Geico is 0.112 or approximately 11.2%.
3. On a state math exam the scores were normally distributed with a mean of 72 and a standard deviation of 8. Use bell curve with standard deviations to help you complete the problem.
b. What percentage of students will score less than 72?
C. What percentage of students will score above 80?
Answers
The mean is 72, which is the center of the distribution, 50% of the students will score less than 72. The area to the right of z = 1, so approximately 15.87% of students will score above 80.
a) Since the mean is 72, and we want to know the percentage of students who scored less than 72, we need to find the area under the normal distribution curve to the left of the z-score that corresponds to 72. Since the standard deviation is 8, the z-score is:
z = (72 - 72) / 8 = 0
Using a standard normal distribution table or a calculator, we can find that the area to the left of z = 0 is 0.5. Therefore, 50% of students scored less than 72.
b) To find the percentage of students who scored above 80, we need to find the area under the normal distribution curve to the right of the z-score that corresponds to 80. The z-score is:
z = (80 - 72) / 8 = 1
Using a standard normal distribution table or a calculator, we can find that the area to the right of z = 1 is 0.1587. Therefore, approximately 15.87% of students scored above 80.
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The line plot above shows the amount of olive oil, in ounces, used in 13 different pizza recipes. Isabella wants to make one pizza from each of the recipes. Will she have enough olive oil to make the pizzas if she buys a 16-ounce bottle of olive oil? Explain or show your reasoning.
Answers
By adding all the numbers in the line plot, we conclude that she will have enough olive oil.
Will she have enough olive oil for the 13 recipes?To check this, we just need to add the 13 numbers and see if it is more than 16.
Each point above the line plot means how many times that number will apear in our sum, then the sum will be:
(2*0 + 4*(4/8) + 3*(6/8) + 3*1 + (1 + 2/8))
Now we need to solve that:
(2*0 + 4*(4/8) + 3*(6/8) + 3*1 + (1 + 2/8))
= 2 + 9/4 + 3 + 1 + 1/4
= 2 + 3 + 1 + 9/4 + 1/4
= 6 + 10/4
= 6 + 8/4 + 2/4
= 6 + 2 + 1/2
= 8 + 1/2
That is the amount of ounces of oil that she needs for the 13 recipes, it is less than 16 ounces, so she will have enough oil.
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Zoe's ball pool is a cylinder with height 20 cm and diameter 90 cm.
Calculate:
(a) The radius of the pool
(b) The circumference of the pool
(c) The area of the base of the pool
(d) The volume of the pool
Answers
Solution :
Zoe's ball pool is a cylinder with height 20 cm and diameter 90 cm.
Height = 20 cm
Diameter = 90 cm
Calculate :
(a) The Radius of the pool.
Answer :
We are given with Diameter of pool 90 cm.
We know that,
Radius = Diameter/2
Radius = 90/2 = 45 cm.
(b) The circumference of the pool .
Answer :
Circumference = 2 πr
[tex]2 \times \dfrac{22}{7} \times 45 \\ \\ \dfrac{ 44 \times 45}{7} \\ \\ \frac{1980}{7} \\ \\ 282.8 \: cm[/tex]
(c) The area of the base of the pool.
Answer :
Area of base = πr²
[tex] \dfrac{22}{7} \times {(45)}^{2} \\ \\ \frac{22}{7} \times 2025 \\ \\ 22 \times 289.28 \\ \\ 578.56 \: {cm}^{2} \\ [/tex]
(d) The volume of the pool
Answer :
Volume of cylinder = πr²h
[tex] \dfrac{22}{7} \times {(45)}^{2} \times 20 \\ \\ \dfrac{22}{7} \times 2025 \times 2 \\ \\ \dfrac{22}{7} \times 4050 \\ \\ \frac{89100}{7} \\ \\ 12728.57 \: {cm}^{3} [/tex]
The circumference of a circle is 81.64 miles. What is the circle's radius?
Use 3.14 for л.
Answers
The radius of the circle with given circumference is 13.
What is circumference?
In mathematics, the circumference of any shape determines the path or boundary that surrounds it. In other words, the perimeter, also referred to as the circumference, helps determine how lengthy the outline of a shape is.
We are given that the circumference of a circle is 81.64 miles.
We know that circumference of a circle is given by 2πr.
So, using this we get
⇒ C = 2πr
⇒ 81.64 = 2 * 3.14 * r
⇒ 81.64 = 6.28 * r
⇒ r = 13
Hence, the radius of the circle with given circumference is 13.
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Patrick had earned 125 points so far this year in math class. After the most recent assignment, Patrick now has 328 points. What was the percentage increase in points? Round your answer to the nearest
Answers
Answer: 162%.
Step-by-step explanation:
To find the percentage increase in points, we need to calculate the difference between the two values, divide it by the original value, and then multiply by 100 to get the percentage.
The difference between Patrick's old score and his new score is:
328 - 125 = 203
The percentage increase is:
203/125 x 100% = 162.4%
Rounded to the nearest whole number, the percentage increase is 162%.
Kayla already has 42.50 in gift card at store the hats Kayla sells have cash value x dollars the store pays additional 30% when hats are sold for gift card rather than for cash after selling hats for gift card Kayla has 140.26 in gift card at store
Answers
Step-by-step explanation:
7 is the answer to the question
From a hot-air balloon, Brody measures a 39-degree angle of depression to a landmark that’s 532 feet away, measuring horizontally. What’s the balloon’s vertical distance above the ground? Round your answer to the nearest hundredth of a foot if necessary.
Answers
Using a trigonometric relation we can see that the balloon’s vertical distance above the ground is 430.8ft
What’s the balloon’s vertical distance above the ground?We can see this as a right triangle, such that we know one angle of 39°, and the adjacent cathetus of that angle has a measure of 532 feet, then we can use a trigonometric relation to find the opposite cathetus, which is the height.
tan(a) = (opposite cathetus)/(adjacent cathetus)
Then we can write:
tan(39°) = H/532ft
532ft*tan(39°) = 430.8ft
That is the vertical height.
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what are the x- and y- coordinates of point P on the directed line segment from A to B such that P is 2/3 the length of the line segment from A to B? x= [m/m+n](x2 -x1) +x1 y = [m/m+n](y2 - y1) +y1
Answers
The coordinates of the point P on the directed line segment from A to B such that P is 2/3 the length of the line segment from A to B are (-1, 2).
How to obtain the coordinates of point P?The coordinates of point P are obtained applying the proportions in the context of the problem.
P is 2/3 the length of the line segment from A to B, hence the equation is given as follows:
P - A = 2/3(B - A)
Then the x-coordinate of P is obtained considering the x-coordinates of A and B as follows:
x - 9 = 2/3(-6 - 9)
x - 9 = -10
x = -1.
The y-coordinate of P is obtained considering the y-coordinates of A and B as follows:
y + 8 = 2/3(7 + 8)
y + 8 = 10
y = 2.
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At the gift shop, they sell small greeting cards and large greeting cards. The cost of a
small greeting card is $2 and the cost of a large greeting card is $5.45. How much
would it cost to get 3 small greeting cards and 2 large greeting cards? How much
would it cost to get x small greeting cards and y large greeting cards?
Answers
Answer:
cpiom t
Step-by-step explanation:dede
State the principle of mathematical induction
Answers
The principle of mathematical induction is a method of proof used in mathematics to prove that a statement is true for all natural numbers.
It is based on the idea that if the statement is true for one number, then it can be used to prove that it is true for the next number. Mathematical induction can be expressed mathematically as follows:
Let P(n) be a statement involving an integer n
Base Case: P(m) is true for some m
Induction Hypothesis: Assume P(k) is true for some k>m.
Induction Step: Show that P(k+1) is true.
Therefore, P(n) is true for all n>m
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Lauren’s age is half of Joe’s age. Emma is four years old than Joe. The sum of Lauren, Emma, and Joe’ age is 54. How old is Joe?
Answers
Joe = x
Lauren = y
Emma = a
y= x/2
Therefore, 2y = x......1
a = x+4
But, since x = 2y,
Then, a = 2y+4........2
y + x + a = 54
y + 2y + 2y+4 = 54
5y + 4 = 54
5y = 54 - 4
5y = 50
y = 10
Subs y=10 into x=2y
x=2(10)
x=20
Therefore, Joe is 20 years old.
Louise deposits $250 into a new savings account.
The account earns 6% simple interest per year.
No money is added or removed from the savings account for 3 years.
What is the total amount of money in her savings account at the end of the 3 years?
Answers
Answer:
Step-by-step explanation:
297.754
If Marie were to paint her living room alone, it would take 3 hours. Her sister Fran could do the job in 6 hours. How many hours would it take them working together? Express your answer as a fraction reduced to lowest terms, if needed.
Answers
The time it takes them to paint the living room together is 2 hours. Expressing this as a fraction reduced to lowest terms 2 hours is 2/1.
What is time working together?Time working together refers to the amount of time that two or more people need to work together to complete a certain task. It is often used in situations where people are working collaboratively or cooperatively to achieve a common goal.
According to question:Let's denote the time it takes Marie to paint the living room alone by M and the time it takes Fran to paint the living room alone by F.
From the problem, we know that:
M = 3 hours
F = 6 hours
To determine the time it takes them to paint the living room together, we can use the following formula:
1 / (time working together) = 1 / M + 1 / F
Substituting the values we have:
1 / (time working together) = 1 / 3 + 1 / 6
Simplifying the right-hand side of the equation:
1 / (time working together) = 2 / 6 + 1 / 6
1 / (time working together) = 3 / 6
1 / (time working together) = 1 / 2
Therefore, the time it takes them to paint the living room together is 2 hours. Expressing this as a fraction reduced to lowest terms:
2 hours = 2/1
So the answer is: 1 / (time working together) = 1 / 2, or equivalently, the time working together is 2/1.
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Quadrilateral has a vertex (5,2), coordinates after translation (x,y)- (x-1, y+3), followed by a dilation by 2
Answers
As a result, after translation and dilation, the initial quadrilateral's apex (5, 2) changes into the vertex (8, 10).
A quadrilateral form is what?One quadrilateral is a circular shape with four sides. These geometric shapes are quadrilaterals: produced by Raphael. a quadrilateral form. There is only one pair of parallel edges to the form, and there are no right angles.
To find the coordinates of the vertices of the quadrilateral after translation and dilation, we can follow these steps:
Translate the vertex (5, 2) by subtracting 1 from the x-coordinate and adding 3 to the y-coordinate to get the new coordinates: (5-1, 2+3) = (4, 5).
Dilate the translated vertex by a factor of 2 with respect to the origin (0, 0) to get the final coordinates: (24, 25) = (8, 10).
Therefore, the vertex (5, 2) of the original quadrilateral is transformed into the vertex (8, 10) after translation and dilation.
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The complete question is: After translating a quadrilateral with a vertex at (5,2) to a new position given by (x-1, y+3) and then dilating it by a factor of 2, what are the coordinates of the new vertices of the quadrilateral?
Answer:
(8,10)
Step-by-step explanation:
Which table of values represents the linear function y=4x+1
Answers
The values represents the linear function are:
x y
0 1
1 5
2 9
3 13
4 17
What is linear function ?
A linear function is a mathematical function that can be represented by a straight line on a graph. It has the form of:
y = mx + b
where m is the slope of the line, which determines how steeply the line rises or falls, and b is the y-intercept, which is the point where the line crosses the y-axis.
According to the question:
The table of values that represents the linear function y=4x+1 is:
x y
0 1
1 5
2 9
3 13
4 17
To generate these values, we can substitute different values of x into the equation y=4x+1 and solve for y.
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Q) which table of values represents the linear function y=4x+1 ?
K
Add the fractions. Simplify if possible.
2
9x
+
10
9x
Answers
The sum of the given fractions is (2x + 10)/x.
What is fraction ?
A fraction is a way of representing a part of a whole or a ratio between two quantities. It is written in the form of a numerator and a denominator separated by a horizontal line. The numerator represents the number of equal parts under consideration, while the denominator represents the total number of equal parts that make up the whole. For example, the fraction 3/4 represents three equal parts out of a total of four equal parts. Fractions can be added, subtracted, multiplied, and divided to perform various mathematical operations.
According to the question:
The given fractions can be added if they have a common denominator. In this case, the common denominator is 9x.
So, we need to write the fractions with 9x as the denominator:
2/1 * (9x/9x) = 18x/9x
10/1 * (9/9x) = 90/9x
Now, we can add these two fractions:
18x/9x + 90/9x = (18x + 90)/9x = 9(2x + 10)/9x = 2x + 10/ x
Therefore, the sum of the given fractions is (2x + 10)/x.
Q: Find K by Adding the fractions and simplify if possible k=(2*(9x))+(10*(9x))
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The scale of a map is 1: 1 000 000 . What is the real distance between two points given the following distances between them on the map??
35 mm
PLEASE HELP URGENTLY
Answers
Answer: 35 000 000mm
Step-by-step explanation:
A cone has a volume of 300 in³ and a diameter of 10 in. What is the height and slant height of the cone?
Answers
The height and slant height of the cone is 7.64 inches and 9.38 inches respectively.
What is volume of cone ?
The volume of a cone is the amount of space occupied by the cone and is given by the formula:
[tex]Volume of a cone = (1/3) * pi * r^2 * h[/tex]
where pi is the mathematical constant approximately equal to 3.14, r is the radius of the circular base of the cone, and h is the height of the cone.
The formula for the volume of a cone can be derived by using calculus or by dividing the cone into a series of infinitesimally thin circular disks, calculating the volume of each disk, and summing up the volumes to obtain the total volume of the cone
According to the question:
To solve this problem, we need to use the formulas for the volume and surface area of a cone:
[tex]Volume of a cone = (1/3) * pi * r^2 * h[/tex]
Surface area of a cone = [tex]pi * r * (r + \sqrt{h^2 + r^2})[/tex]
where r is the radius of the base, h is the height, and pi is a mathematical constant approximately equal to 3.14.
First, we need to find the radius of the cone. The diameter is 10 inches, so the radius is half of that, or 5 inches.
The volume of the cone is given as 300 cubic inches. We can plug in the values we know and solve for the height:
[tex]300 = (1/3) * pi * 5^2 * h[/tex]
[tex]h = 300 / ((1/3) * pi * 5^2)[/tex]
[tex]h \approx 7.64 inches[/tex]
So the height of the cone is approximately 7.64 inches.
Next, we can use the Pythagorean theorem to find the slant height of the cone.
[tex]slant height^2 = radius^2 + height^2[/tex]
[tex]slant height^2 = 5^2 + 7.64^2[/tex]
[tex]slant height \approx 9.38 inches[/tex]
So the slant height of the cone is approximately 9.38 inches.
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Last year, a person wrote 120 checks. Let the random variable x represent the number of checks he wrote in one day, and assume that it has a Poisson distribution. What is the mean number of checks written per day? What is the standard deviation? What is the variance?
Answers
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following mass probability function:
[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]
The parameters are listed and explained:
x is the number of successes that we want to find the probability of.e = 2.71828 is the Euler number[tex]\mu[/tex] is the mean in the given interval or range of values of the input parameter.An year is composed by 365 days, hence the daily mean of the number of checks written is given as follows:
120/365 = 0.3288 checks.
The variance has the same value of the mean for the Poisson distribution, in units squared, while the standard deviation is the square root of the variance, hence:
sqrt(0.3288) = 0.5734 checks.
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The two top concert tours in 2016 were concert A and concert B. Based on average ticket prices, it cost a total of $1707 to purchase six tickets for concert A and six tickets for concert B. Three tickets for concert B cost a total of $687. How much did an average ticket cost for each tour?
Answers
The average ticket cost for each concert is given as follows:
Concert A: $188.83.Concert B: $95.67.How to obtain the ticket costs?The ticket costs are obtained by a system of equations, for which the variables are given as follows:
Variable a: cost for Concert A.Variable b: cost for Concert B.It cost a total of $1707 to purchase six tickets for concert A and six tickets for concert B, hence:
6a + 6b = 1707
a + b = 284.5.
Three tickets for concert B cost a total of $687, hence the cost for concert B is of:
3b = 687
b = 287/3
b = $95.67.
Replacing into the first equation, the cost for concert A is given as follows;
a = 284.5 - 95.67
a = $188.83.
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graphing a Quadratic function given in factored from
Answers
Answer:
y
Step-by-step explanation:
Eliana enjoys listening to podcasts on a variety of topics, and she has many saved on her phone. She plays some on a cross-country road trip, in a random order. Here are the topics she has listened to so far:
cooking, mystery, sci-fi, mystery, history, mystery, cooking, sci-fi, mystery, sci-fi, cooking
Based on the data, what is the probability that the next podcast Eliana listens to will be a cooking podcast?
Write your answer as a fraction or whole number.
Answers
After answering the provided question, we can conclude that As a probability result, the answer is 3/11.
What is probability?Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, with 0 indicating an unlikely event and 1 indicating an unavoidable event. Because there are two equally likely outcomes, switching a fair coin and coin flips has a probability of 0.5 or 50%. (Either heads or tails). Probability theory, a branch of mathematics, is concerned with the investigation of random events rather than their properties. It is used in a variety of fields, including statistics, finance, science, and engineering.
To calculate the likelihood that the next podcast Eliana listens to will be a cooking podcast, multiply the number of cooking podcasts she has listened to by the total number of podcasts she has listened to.
We can see from the list of topics that Eliana has listened to three cooking podcasts out of a total of 11 podcasts. As a result, the likelihood that Eliana's next podcast will be a cooking podcast is:
3/11
As a result, the answer is 3/11.
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Which decimal is equivalent to
4/15
Please!
Answers
Answer:
0.266666
Step-by-step explanation:
1) Use the algorithm method.
0 . 2 6 6 6 6 6
____________________________
15 | 4 .
3 . 0
__________
1 . 0 0
9 0
_________
1 0 0
90
___________
1 0 0
9 0
___________
1 0 0
9 0
______
1 0 0
9 0
___
10
2) therefore, 4/15 ≈0.266666.
0.266666
A baseball player tosses a ball straight up into the air. The function y= -16x squared + 30x +5 models the motion of the ball, where x is the time in seconds and y is the height of the ball in feet. Write an equation you can solve to find out when the ball is at a height of 15 feet.
Answers
Therefore, at two separate moments, x = 0.28 and x = 1.17, the ball stands at a height of 15 feet. (rounded to two decimal places).
What sort of equation would that be?The meaning of an equation in algebra is a mathematical assertion that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two expressions 3x + 5 and 14, which are split by the 'equal' symbol.
To find out when the ball is at a height of 15 feet, we need to solve the equation:
-16x² + 30x + 5 = 15
First, we can simplify the equation by subtracting 15 from both sides:
-16x² + 30x - 10 = 0
Now we can divide both sides by -2 to make the coefficient of the x² term positive:
8x² - 15x + 5 = 0
This is a quadratic equation in standard form, where a = 8, b = -15, and c = 5. Using the quadratic method, we can find x:
x = (-b ± √(b² - 4ac)) / 2a
Plugging in the values, we get:
x = (-(-15) ± √((-15)² - 4(8)(5))) / 2(8)
x = (15 ± √(225 - 160)) / 16
x = (15 ± √65) / 16
Therefore, the ball is at a height of 15 feet at two different times:
x ≈ 0.28 seconds and x ≈ 1.17 seconds (rounded to two decimal places).
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Which function is best represented by the graph?
Answers
Answer:This would be option letter C
Step-by-step explanation:The diagram is a gram of external geofram so the position Is fFX(4) ^N
Answer:
D
Step-by-step explanation:
f(x)= -x^2+4
Draw graph of -x^2 and shift it 4 units above
What ordered pair contains the y-intercept of y= -2x^ -4x -5
(-1,3)
(0,-5)
(-5,0)
(0,-2)
Answers
Answer:
To find the y-intercept, we can set x = 0 in the equation y= -2x^2 -4x -5, which gives us:
y = -2(0)^2 - 4(0) - 5 = -5
So the y-intercept is (0, -5). Therefore, the correct answer is (0,-5).
A family with a spending budget of $22,300 receives an increase in wages of 3% in a year in which inflation was 5.6%. Find the net gain or loss in their purchasing power
Answers
The family actually lost purchasing power due to the combination of the wage increase and inflation, as their budget did not keep pace with rising prices.
To find the net gain or loss in the family's purchasing powerWe need to calculate the inflation-adjusted increase in their wages.
First, we can find the amount of the wage increase by multiplying the original budget by the percentage increase:
Wage increase = $22,300 x 0.03 = $669
Next, we need to adjust for inflation. We can do this by finding the inflation rate as a decimal (5.6% = 0.056) and subtracting it from 1 to get the inflation-adjustment factor:
Inflation-adjustment factor = 1 - 0.056 = 0.944
Finally, we can calculate the inflation-adjusted wage increase by multiplying the wage increase by the inflation-adjustment factor:
Inflation-adjusted wage increase = $669 x 0.944 = $631.08
Therefore, the family's net gain in purchasing power is:
Net gain = Inflation-adjusted wage increase - Original budget
Net gain = $631.08 - $22,300
Net gain = -$21,668.92
So the family actually lost purchasing power due to the combination of the wage increase and inflation, as their budget did not keep pace with rising prices.
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Write the solution in the interval notation x=1
Answers
Answer:
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Answer:
x < 2 O R x > 1 ⇔ ( − ∞ , ∞ )
Explanation:
x < 2 means x can take any value less than two and interval notation, this means ( − ∞ , 2 ) , meaning that all numbers between − ∞ and 1 are included and as − ∞ and 2 are not included we have use small brackets. This forms one set of numbers, say P . x > 1 means x can take any value greater than one and interval notation tis means ( 1 , ∞ ) , meaning that all numbers between 1 and ∞ are included, but not 13 and ∞ . This forms another set of numbers, say Q . Hence x < 2 O R x > 1 represents the union of two sets P and Q i.e P ∪ Q or in other words ( − ∞ , 2 ) ∪ ( 1 , ∞ ) . Observe that P ∪ Q includes all the numbers from − ∞ to ∞ and hence x < 2 O R x > 1 ⇔ ( − ∞ , ∞ )