Step-by-step explanation:
To find the probability of an advertising firm selected at random having under $1 million or $1 million to $20 million in income after taxes, we need to add the number of firms in each of these two categories and divide the result by the total number of firms.
Number of firms with under $1 million in income after taxes = 95
Number of firms with $1 million to $20 million in income after taxes = 74
Total number of firms = 95 + 74 + 31 = 200
Therefore, the probability of an advertising firm selected at random having under $1 million or $1 million to $20 million in income after taxes is:
(95 + 74) / 200 = 0.845
So the probability is 0.845 or 84.5%.
Given: trap. SPQR with SP || QR ; MN is the median of the trap. m∠QRS = 120 ; m∠QPS = 135 ; SP = PQ = 12. Find: MN.
Answer:
From the information given, we can draw the following diagram:
S ------- P
/ \ / \
/ \ QR / \
/ \ / / \
/ Q \
R-----------------P
MN
Here, SP || QR, so we have ∠QSP = ∠PQR and ∠SPQ = ∠QRP.
Since m∠QRS = 120, we have m∠QRP = 360 - m∠QRS = 360 - 120 = 240.
Since m∠QPS = 135 and ∠QSP = ∠PQR, we have m∠PQR = 360 - m∠QPS = 360 - 135 = 225.
Let x = MN. Since MN is the median of the trapezoid, we have MP = NR = x.
Now, consider the triangles SPQ and RPQ. We have:
tan(∠SPQ) = x/12 (using the tangent ratio in triangle SPQ)
tan(∠RPQ) = x/12 (using the tangent ratio in triangle RPQ)
Since ∠SPQ = ∠RPQ (as they are corresponding angles), we have:
x/12 = x/12
x = 12
Therefore, MN = x = 12.
Please Help! 50 Points!!! This composite figure is created by placing a sector of a circle on a triangle. What is the area of this composite figure? Use 3.14 for . Round to the nearest hundredth. Show your work
The number of animals in a population at the start of year t is Pt
The number of animals at the start of year 1 is 400
Given that
Pt + 1 = 1.01Pt
work out the number of animals at the start of year 3
Therefore, the number of animals at the start of year 3 is approximately 408.04.
What is equation?An equation is a mathematical statement that shows the equality of two expressions, usually with an equal sign "=" in between them. An equation can contain variables, constants, and operators. The variables are represented by letters and can take on different values, while constants are fixed values that do not change. Operators include mathematical symbols like plus, minus, multiplication, and division, as well as exponents and logarithms.
Here,
We are given that the population at the start of year 1 (P1) is 400. Using the recurrence relation Pt + 1 = 1.01Pt, we can find the population at the start of year 2 (P2) and year 3 (P3) as follows:
P2 = 1.01 * P1 = 1.01 * 400 = 404
P3 = 1.01 * P2 = 1.01 * 404 = 408.04
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help with this example, I'm from Kazakhstan, sorry, help in any way you can
The inequalities are factorized to;
1. x > 5 and x > -1
x > 2 and x > -6
2. x < 2
x < -4 and x < 2
3. x > 7
x> 1 and x> 3
How to solve the inequalitiesFrom the information given, we have the inequalities;
(x - 5)(x + 1)> 0
expand the bracket, we get;
x² + x - 5x - 5> 0
x(x + 1) - 5(x + 1) > 0
x > 5 and x > -1
2. x-2/x + 3 < 0
cross multiply the values
x - 2< 0
make 'x' the subject
x < 2
3. x -7/x + 4 > 0
cross multiply the values
x - 7> 0
make 'x' the subject
x > 7
x² + 4x - 12 > 0
x² + 6x - 2x - 12> 0
Factorize
x(x + 6) - 2(x + 6)> 0
x > 2 and x > -6
x² - 2x - 8< 0
x² -2x + 4x - 8 < 0
Factorize
x(x - 2) + 4(x -2 )< 0
x < -4 and x < 2
x² - 4x + 3 > 0
x² - 3x - x + 3> 0
factorize
x(x - 3) - 1(x - 3)> 0
x> 1 and x> 3
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3n + 2n + 7 + 3n = 7solve for n
Answer: 0
Step-by-step explanation:
1. Bring the Variables to one side and the constants to the other (3n + 2n + 3n = 7 - 7)
2. Solve ( 8n = 0, n = 0/8, n = 0)
A triangle with vertices (2, 3), (4, 7), and (3, 3) is translated a distance of 5 units to the right. What are the new coordinates?
(-3, 3), (-1, 7), (-2, 3)
(2, -2), (4, 2), (3, -2)
(2, 8), (4, 12), (3, 8)
(7, 3), (9, 7) and (8, 3)
The answer is (D) (7, 3), (9, 7) and (8, 3) when A triangle with vertices (2, 3), (4, 7), and (3, 3) is translated a distance of 5 units to the right.
To translate a two-dimensional figure such as a triangle, we move all its points a certain distance in a certain direction. In this case, we need to move the triangle 5 units to the right. This means we need to add 5 to the x-coordinate of each vertex, while leaving the y-coordinate unchanged.
In this case, we need to add 5 units to the x-coordinates of each vertex of the triangle.
The new coordinates of the triangle's vertices would be:
(2+5, 3) = (7, 3)
(4+5, 7) = (9, 7)
(3+5, 3) = (8, 3)
Therefore, the answer is (D) (7, 3), (9, 7) and (8, 3).
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The complete question is :
A triangle with vertices (2, 3), (4, 7), and (3, 3) is translated a distance of 5 units to the right. What are the new coordinates?
A.(-3, 3), (-1, 7), (-2, 3)
B.(2, -2), (4, 2), (3, -2)
C.(2, 8), (4, 12), (3, 8)
D.(7, 3), (9, 7) and (8, 3)
Translate and solve: 5 less than m is at most 70.
Write your solution in interval notation
The interval notation for the inequality statement "5 less than m is at most 70" is (-∞, 75].
What is an inequality?
In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.
The statement given is - 5 less than m is at most 70.
The given sentence can be translated into a mathematical inequality as -
m - 5 ≤ 70
To solve for m, we can add 5 to both sides of the inequality -
m - 5 + 5 ≤ 70 + 5
m ≤ 75
Therefore, m is less than or equal to 75.
In interval notation, we can represent this solution as -
(-∞, 75]
Therefore, the interval value is (-∞, 75].
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I WILL GIVE BRAINLIEST AND 5 STARS
Triangle ABC is similar to triangle DEF.
What is the length of AC?
Answer:
a
Step-by-step explanation:
16÷4 is 4 and 12÷4 is 3. I matched up the angles.
After calc, the lenght of AC segment is 3 cm. Alternative A
Similarity of trianglesThe similarity of triangles property has a way of calculating how much an unknown segment of the triangle is worth through measures of available equivalent segments of the two triangles and matching angles.
Let's see calc[tex]\begin{array}{l}\raisebox{8pt}{$\sf \dfrac{AC}{BC}=\dfrac{DF}{EF}$}\\\raisebox{8pt}{$\sf \dfrac{AC}{4}=\dfrac{\red{\diagup\!\!\!\!\!\!12}^{\div4}}{\red{\diagup\!\!\!\!\!\!16}^{\div4}}$}\\\raisebox{8pt}{$\sf \dfrac{AC}{\red{\diagup\!\!\!\!4}}=\dfrac{3}{\red{\diagup\!\!\!\!4}}$}\\\bf\therefore AC=3\end{array}[/tex]
Found the length of segment AC, which is 3cm
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Iodine 131 has a half life of 8 days. If you start with a sample of 100 grams, how much iodine 131 would be left after 10 days. (3 points)
b. Let f:x=3x² +1 and g: x=x-5
Find
i. fog(x)
ii. go f(x)
iii. (ƒ o g)−¹(x)
2.
P
When trying to determine whether triangles ABC and DEF are similar, Mae notices in the image that angles B and E are congruent. Next, she determines the ratio of BC to EF and
obtains 3/5. What else does she need to do?
A A
4
53°
B
53°
C E
3
5
Option (C) is correct i.e., To determine the ratio of AB to DE and use SAS to conclude the triangle(s) are similar
What is Triangle?A triangle is a polygon that has three(3) edges and three(3) vertices. It is one of the fundamental forms in geometry. Triangle DEF refers to a triangle with vertices D, E, and F. In Euclidean geometry, any three points that are not collinear identify a distinct triangle and a distinct plane at the same time.
Triangle ABC and Triangle DEF are similar if one of this property satisfy are , AAA, SAS, ASA and RHS.
It is Given that ∠B and ∠E are Similar and She determine ratio of BC to EF is 3/5.
Therefore, for similarity of two triangle she have to To determine the ratio of AB to DE and use SAS to conclude the triangles are similar.
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Complete Question:
When trying to determine whether triangles ABC and DEF are similar, Mae notices in the image that angles B and E are congruent. Next, she determines the ratio of BC to EF and obtains 3/5. What else does she need to do?
A. To determine the ratio of DE to AB and use SSS to conclude the triangles are similar
B. To determine the ratio of AB to DE and conclude the triangles are not similar
C. To determine the ratio of AB to DE and use SAS to conclude the triangles are similar
D. To determine the ratio of DE to AB and use AA to conclude the triangles are not similar.
Option (C) is correct i.e., To determine the ratio of AB to DE and use SAS to conclude the triangle(s) are similar
What is Triangle?A triangle is a polygon that has three(3) edges and three(3) vertices. It is one of the fundamental forms in geometry. Triangle DEF refers to a triangle with vertices D, E, and F. In Euclidean geometry, any three points that are not collinear identify a distinct triangle and a distinct plane at the same time.
Triangle ABC and Triangle DEF are similar if one of this property satisfy are , AAA, SAS, ASA and RHS.
It is Given that ∠B and ∠E are Similar and She determine ratio of BC to EF is 3/5.
Therefore, for similarity of two triangle she have to To determine the ratio of AB to DE and use SAS to conclude the triangles are similar.
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Complete Question:
When trying to determine whether triangles ABC and DEF are similar, Mae notices in the image that angles B and E are congruent. Next, she determines the ratio of BC to EF and obtains 3/5. What else does she need to do?
A. To determine the ratio of DE to AB and use SSS to conclude the triangles are similar
B. To determine the ratio of AB to DE and conclude the triangles are not similar
C. To determine the ratio of AB to DE and use SAS to conclude the triangles are similar
D. To determine the ratio of DE to AB and use AA to conclude the triangles are not similar.
Help with math problems
The value of the expressions are 1. Distributive property: -3 √3 (2 + √6) = -6 √3 - 9 √2. [2] (-2 √3 + 2)(√3 - 5) = 12 √3 - 16. [3] (-2 -3 √5)(5 - √5) = -15 √5 + 5.
What is distributive property?The distributive property in algebra says that the sum or difference of the products of the number and each term in the sum or difference is equal to the product or difference of the number and each phrase in the sum or difference. To put it another way, a(b + c) = ab + ac and a(b - c) = ab - ac are true for any value of a, b, and c. The distributive property is frequently employed in parenthetical expansion and algebraic simplification.
1. Distributive property:
-3 √3 (2 + √6)
= -6 √3 - 3 √3 √6
= -6 √3 - 3 √18
= -6 √3 - 9 √2
2. Multiplying:
(-2 √3 + 2)(√3 - 5)
= -2 √3 (√3) + 2 (√3) - 10 + 10 √3
= -2 (3) - 10 + 12 √3
= 12 √3 - 16
3. Multiplying:
(-2 -3 √5)(5 - √5)
= -2 (5) - 3 √5 (5) + 2 √5 + 15
= -10 - 15 √5 + 2 √5 + 15
= -15 √5 + 5
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Let f(x)=x+8 and g(x)=x^2-7x-9 find f(g(-1))
Answer:
First, we need to evaluate g(-1):
g(-1) = (-1)^2 - 7(-1) - 9
= 1 + 7 - 9
= -1
Now, we can plug in g(-1) into f(x):
f(g(-1)) = f(-1)
= -1 + 8
= 7
Therefore, f(g(-1)) = 7.
Please help, I can’t figure this out!
22. You take out two loans totaling $5000: one from the bank at 8% interest and the other from your uncle at only 2% interest.
a. How much did you borrow from the bank?
b. What interest do you owe to your uncle?
c. What interest do you owe the bank?
d. What is the total interest that you owe?
e. If the total interest is $300. Write a "total interest" equation
f. How much did you borrow from each? Solve the equation.
(X = how much you borrow from your uncle)
So you borrowed $3333.33 from the bank and $1666.67 from your uncle.
What is percent?Percent is a way of expressing a number as a fraction of 100. The word "percent" comes from the Latin per centum, which means "out of a hundred". Percentages are often used to express proportions, rates, or changes in quantities. For example, if you got 80 questions correct out of 100 on a test, you got 80% correct. In other words, 80 is 80% of 100. Percentages can also be used to express changes, such as if the price of a product increases from $10 to $12, that's a 20% increase ($2 increase is 20% of the original price of $10). Percentages are represented using the % symbol, for example, 50% means 50 out of 100, or 0.5 as a decimal.
Here,
a. Let's say you borrowed x dollars from the bank. Then, you borrowed (5000 - x) dollars from your uncle. Since you borrowed from the bank at 8% interest, the interest on this loan would be 0.08x dollars.
b. Since you borrowed from your uncle at 2% interest, the interest on this loan would be 0.02(5000 - x) dollars.
c. The total interest owed to the bank is 0.08x dollars, as calculated in part (a).
d. The total interest owed is the sum of the interest owed to the bank and the interest owed to your uncle:
Total interest = 0.08x + 0.02(5000 - x)
e. If the total interest is $300, we can write the equation:
0.08x + 0.02(5000 - x) = 300
f. To solve this equation for x, we can simplify and solve for x:
0.08x + 0.02(5000 - x) = 300
0.08x + 100 - 0.02x = 300
0.06x = 200
x = 3333.33 (rounded to the nearest cent)
Therefore, the answers are:
a. You borrowed $3333.33 from the bank.
b. You owe your uncle 0.02(5000 - 3333.33) = $133.33 in interest.
c. You owe the bank 0.08(3333.33) = $266.67 in interest.
d. The total interest you owe is $133.33 + $266.67 = $400.
e. The total interest equation is 0.08x + 0.02(5000 - x) = 300.
f. You borrowed $3333.33 from the bank and $1666.67 from your uncle.
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A circular hot spring has a diameter of 110 meers. Over time, the diameter of the spring decreases by 3 meters. By how many square meters does the area of he hot spring decrease?
The area οf the hοt spring decreases by apprοximately 540.24 square meters.
What is area οf a circle?The area οf a circle is fοund οut using the fοrmula. Area = πr²
Given that, r = d/2 = 110/2 = 55 meters
The οriginal area οf the hοt spring can be calculated as:
[tex]A = \pi r^2 = \pi(55)^2 = 9,525.69[/tex] square meters (rοunded tο twο decimal places)
After the diameter οf the hοt spring decreases by 3 meters, its new diameter is 110 - 3 = 107 meters. Therefοre, its new radius is:
r' = d'/2 = 107/2 = 53.5 meters
The new area οf the hοt spring can be calculated as:
[tex]A' = \pi r'² = \pi(53.5)² = 8,985.45[/tex] square meters (rοunded tο twο decimal places)
The decrease in the area οf the hοt spring is the difference between the οriginal area and the new area:
ΔA = A - A' = 9,525.69 - 8,985.45 = 540.24 square meters (rοunded tο twο decimal places)
Therefοre, the area οf the hοt spring decreases by apprοximately 540.24 square meters.
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need to Graph y≥13x−3.
The dοtted line is the line y=13x3, and the shaded area is abοve it.
what the slοpe οf the line?As is well knοwn, the equatiοn fοr a straight line is y = mx + c. Where 'c' is the y-axis intercept and 'm' is the slοpe οn the y-axis. The fοrmula fοr a hοrizοntal line is y = 3.
In οrder tο graph the inequality y≥13x−3, we can first plοt the line y=13x3
(which has a slοpe οf 13 and a y-intercept οf -3) as a dοtted line.
Since the inequality is y≥13x−3, we must shade the area abοve the line.
The dοtted line is the line y=13x3, and the shaded area is abοve it.
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Answer this question please
Either a rotation of a reflection is used to transform Shape P so that exactly one vertex is invariant.
What are transformations on a figure?The possible transformations on a figure are listed as follows:
Translation: Translation left/right or down/up, moving all vertices.Reflections: Over one of the axes or over a line.Rotations: Over a degree measure.Dilation: Coordinates of the vertices of the original figure are multiplied by the scale factor, hence all vertices are changed.An invariant vertex means that the coordinates of the vertex remain constant even after the transformation.
A translation and a dilation will always change the coordinates of every vertex, as:
The translation means that at least one of the x-coordinate of the y-coordinate is added/subtracted.The dilation means that both the x-coordinate and the y-coordinate of each vertex is multiplied by the scale factor.For reflection and rotation, they can happen over one vertex of the shape, hence that vertex would remain constant.
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A basketball player has made 12 of his last 20 free throws, which is a free throw percentage of 60%.
How many more consecutive free throws would he need to make to raise his free throw percentage to 75%?
A bag has 3 blue cubes, 8 red cubes, and 9 green cubes. If you draw a cube and replace it in the bag 100 times, which of the following options would you expect to occur? Select all that apply.
A) Pull a blue cube 15 times
B) Pull more than two times as many red cubes as blue cubes
C) Pull a red cube 40 times
D) Pull a green cube 45 times
E) Pull 50 blue cubes
Options A and B are the most likely to occur, but they still have a small probability of occurring. The other options are much less likely to occur.
what is probability?
Probability is a branch of mathematics that deals with the study of random events or outcomes.
Since there are 20 cubes in the bag, each cube has a probability of 1/20 of being drawn on any given draw, assuming that each cube is equally likely to be drawn.
A) The probability of drawing a blue cube on any given draw is 3/20. Drawing a blue cube 15 times in 100 draws would have a probability of (3/20)^15 * (17/20)^85 * 100C15, which is an extremely small probability. However, since the question asks what we would expect to occur, we might still select this option.
B) The probability of drawing a red cube on any given draw is 8/20, or 2/5. Drawing more than two times as many red cubes as blue cubes in 100 draws would mean drawing at least 10 red cubes and at most 5 blue cubes. This is equivalent to drawing at least 10 red cubes in 90 draws, which has a probability of approximately (2/5)^10 * (3/5)^90 * 100C90. This is also a small probability, but again, we might still select this option.
C) The probability of drawing a red cube on any given draw is 8/20, or 2/5. Drawing a red cube 40 times in 100 draws would have a probability of (2/5)^40 * (3/5)^60 * 100C40, which is also a small probability.
D) The probability of drawing a green cube on any given draw is 9/20. Drawing a green cube 45 times in 100 draws would have a probability of (9/20)^45 * (11/20)^55 * 100C45, which is again a small probability.
E) Drawing 50 blue cubes in 100 draws is equivalent to drawing a blue cube on every draw. This would have a probability of (3/20)^100, which is an extremely small probability.
Therefore, options A and B are the most likely to occur, but they still have a small probability of occurring. The other options are much less likely to occur.
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the question is in the picture attached
A line of best fit, also known as a regression line, is a straight line that best represents the relationship between two variables in a scatter plot. The association between the variables is strong.
How do you know strong association using line of best fit?To determine whether there is a strong association between the two variables, you need to examine the pattern of the data points in relation to the line of best fit.
If the data points are tightly clustered around the line of best fit, this indicates a strong association between the two variables. On the other hand, if the data points are more spread out and do not follow a clear pattern, this suggests a weak association.
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Four hundred fifty-nine
Answer:
459..
hope you understand
HELPP ! , Use substitution to solve each system of equations , find x and y and do checks
A spinner has three sections which are coloured black, red and yellow. Lincoln spun the spinner 100 times in total.
The frequency of spins that landed on red was
35.
The ratio of the frequency of black to the frequency of red was 2: 7.
What is the estimated probability of the spinner landing on yellow?
Give your answer as a decimal.
PLEASE HELP WILL GIVE BRAINLISET WORHT 30 POINTS !!
The mapping of the functions h(x) = |x|/2, f(x) = 2x + 8, and f(x) = x² - 5x + 2 is defined at;
1). -2, f(-2) = 1
2). 6, f(6) = 20
3). 2, f(2) = -4
What is a functionA function is a rule that defines a relationship between one variable. It is the mapping whose codomain is the set of real numbers
For x = -2;
h(-2) = |-2|/2
h(x) = 2/2 {absolute value of -2 is 2 units}
h(x) = 1
for x = 6;
f(6) = 2(6) + 8
f(6) = 12 + 8
f(6) = 20
for x = 2;
f(2) = (2)² - 5(2) + 2
f(2) = 4 - 10 + 2
f(2) = -4
Therefore, the mapping of the functions h(x) = |x|/2, f(x) = 2x + 8, and f(x) = x² - 5x + 2 is defined at;
1). -2, f(-2) = 1
2). 6, f(6) = 20
3). 2, f(2) = -4
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The Japanese automobile company Lexus has established a reputation for quality control. Recents statistics indicate that a newly purchased Lexus ES 300 will have:
0 defects with probability 0.12
1 defect with probability 0.18
2 defects with probability 0.25
3 defects with probability 0.20
4 defects with probability 0.15
5 defects with probability 0.10
If you purchase a new Lexus ES 300, find:
a) the probability that it will have 2 or fewer defects.
b) the probability that it will have 4 or more defects.
c) the probability that it will have between 1 and 3 (all inclusive) defects.
d) the expected number of defects
a) The probability that it will have 2 or fewer defects is: 0.55 = 55%.
b) The probability it will have 4 or more defects is: 0.25 = 25%.
c) The probability it will have between 1 and 3 defects is: 0.63 = 63%.
d) The expected number of defects is: 2.38.
How to calculate a probability?A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.
For this problem, we are given the frequencies, hence we must identify the outcomes and then add their frequencies.
The probability that it will have 2 or fewer defects is obtained as follows:
P(X = 0) + P(X = 1) + P(X = 2) = 0.12 + 0.18 + 0.25 = 0.55.
The probability it will have 4 or more defects is obtained as follows:
P(X = 4) + P(X = 5) = 0.15 + 0.10 = 0.25.
The probability it will have between 1 and 3 defects is obtained as follows:
P(X = 1) + P(X = 2) + P(X = 3) = 0.18 + 0.25 + 0.20 = 0.63.
The expected value is given by the sum of each value multiplied by it's respective probability, hence:
E(X) = 0 x 0.12 + 1 x 0.18 + 2 x 0.25 + 3 x 0.20 + 4 x 0.15 + 5 x 0.1
E(X) = 2.38.
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Anybody answer!!!!, I'm trying to do my IXL. Have to get it to 90!!
What kind of transformation converts the graph of f(x)=x–1 into the graph of g(x)=9x–9?
Horizontal Shrink
Horizontal Stretch
Vertical Shrink
Vertical Stretch
The transformation that converts the graph of f(x) = x - 1 into the graph of g(x) = 9x - 9 is a vertical stretch by a factor of 9, followed by a vertical shift downwards by 8 units.
What are some instances of graphs?
Some instances of graphs include line graphs, bar graphs, scatter plots, pie charts, and histograms, which are visual representations of data or mathematical functions.
To see this, we can rewrite g(x) as:
g(x) = 9(x - 1) = 9x - 9
This shows that g(x) is a vertical stretch of f(x) by a factor of 9, which means that every y-value of g(x) is 9 times the corresponding y-value of f(x). The horizontal axis remains the same.
Next, we can see that g(x) is shifted downward by 8 units compared to f(x). This is because g(x) has a y-intercept of -9, while f(x) has a y-intercept of -1. This means that every y-value of g(x) is 8 units less than the corresponding y-value of f(x).
Therefore, the transformation that converts the graph of f(x) = x - 1 into the graph of g(x) = 9x - 9 is a vertical stretch by a factor of 9, followed by a vertical shift downwards by 8 units.
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Solve for Y
3. The Mighty Drop ski slope has an angle of elevation of 75°. If the top of the slope has an elevation of 800 feet, what is the length of the run?
4. Little Judy let go of her red balloon. The balloon is now 2240 feet up in the air and 350 feet to the west. What is the angle of elevation when little Judy looks at her red balloon?
Therefore , the solution of the given problem of angle comes out to be Judy is therefore looking at her red balloon at a height of about 81.06°.
An angle meaning is what?In Euclidean space, the top and bottom of the wall divide the two circular faces that constitute the sides of a tilt. When two beams collide, they can form a junction point. Angle is another outcome of two entities interacting. They mirror dihedral forms the most. A two-dimensional curve can be created by arranging two line beams in various configurations at their ends.
Here,
We must use trigonometry to determine the run's duration. Call the run's duration "y" for now. We can use the tangent function since we are aware that the angle of height is 75°:
=> tan(75°) = opposite / adjacent
The slope's 800-foot height is on the opposing side. We want to determine the run's length, which is on the opposite side:
=> tan(75°) = 800 / y
We can multiply both sides by y and then split both sides by tan(75°) to find the answer to y:
=> y = 224.12 feet / tan(75°) = 800
The course therefore measures about 224.12 feet in length.
We must once more turn to trigonometry to determine the angle of height.
=> tan(θ) = opposite / adjacent
=> tan(θ) = 2240 / 350
We can use the inverse tangent (or arctan) of both sides to find :
=> 81.06° = arctan(2240 / 350).
Little Judy is therefore looking at her red balloon at a height of about 81.06°.
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=2x-1
7) through: (-4,-4), parallel to y=-
3
=2x-4
9) through: (-4,-4), parallel to y =
11) through: (5,-2), perp. to y =
y=²x-5
13) through: (-2, 5), perp. to y = x + 2
8) through: (1, 0), parallel to y = 2x - 1
5
10) through: (5, 5), perp. to y = --
y = ²√x + 4
12) through: (3,-2), perp. to y = -
14) through: (-4, 0), perp. to y = -
The equation of the lines parallel or perpendicular to another line and that pass through a point are listed below:
Case 7: y = (7 / 4) · x + 3
Case 8: y = 2 · x + 1
Case 9: y = (3 / 2) · x + 2
Case 10: y = (6 / 5) · x - 1
Case 11: y = - (2 / 5) · x
Case 12: y = - (5 / 2) · x - 19 / 2
Case 13: y = - x + 3
Case 14: y = - (5 / 4) · x - 5
How to derive the equation of a line
In this problem we have eight cases of line equations parallel or perpendicular to a line and that pass through a point. The equation of a line is an expression of the form:
y = m · x + b
Where:
x - Independent variable.y - Dependent variable.m - Slopeb - InterceptThere are the following relationships between two lines:
Two lines are parallel when they have the same slope.Two lines are perpendicular when the product of their slopes is equal to - 1.Now we proceed to determine the line equations:
Case 7
Slope
m = 7 / 4
m' = 7 / 4
Intercept
b = y - m · x
b = - 4 - (7 / 4) · (- 4)
b = - 4 + 7
b = 3
Equation of the line
y = (7 / 4) · x + 3
Case 8
Slope
m = 2
m' = 2
Intercept
b = 1 - 2 · 0
b = 1
Equation of the line
y = 2 · x + 1
Case 9
Slope
m = 3 / 2
m' = 3 / 2
Intercept
b = - 4 - (3 / 2) · (- 4)
b = - 4 + 6
b = 2
Equation of the line
y = (3 / 2) · x + 2
Case 10
Slope
m = - 5 / 6
m' = 6 / 5
Intercept
b = 5 - (6 / 5) · 5
b = 5 - 6
b = - 1
Equation of the line
y = (6 / 5) · x - 1
Case 11
Slope
m = 5 / 2
m' = - 2 / 5
Intercept
b = - 2 - (- 2 / 5) · 5
b = 0
Equation of the line
y = - (2 / 5) · x
Case 12
Slope
m = 2 / 5
m' = - 5 / 2
Intercept
b = - 2 + (- 5 / 2) · 3
b = - 2 - 15 / 2
b = - 4 / 2 - 15 / 2
b = - 19 / 2
Equation of the line
y = - (5 / 2) · x - 19 / 2
Case 13
Slope
m = 1
m' = - 1
Intercept
b = 5 - (- 1) · (- 2)
b = 3
Equation of the line
y = - x + 3
Case 14
Slope
m = 4 / 5
m' = - 5 / 4
Intercept
b = 0 - (- 5 / 4) · (- 4)
b = 0 - 5
b = - 5
Equation of the line
y = - (5 / 4) · x - 5
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how do scatter graphs work
The scatter graph is used to show correlation between variables
How do scatter graphs work A scatter graph, also known as a scatter plot or scatter chart, is a type of data visualization tool that uses coordinates to display the values of two variables for a set of data.
In a scatter graph, each point on the graph represents a single data point, with one variable plotted on the x-axis and the other variable plotted on the y-axis.
The position of each point is determined by the values of the two variables for that data point.
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22.
Number of Cookies
YA
72
60
48
36
24
12
Origin O
Bake Sale
2 4 6 8 10 12 X
Packages
a) k = 6
c) k = 1/6
e) none (not proportional)
Identify the constant of proportionality.
b) k = 12
d) k=1/12
Since the ratio of Y to X is not constant for different values of X, the relationship between Y and X is not proportional, and there is no constant of proportionality.
What is proportion?Proportion is a mathematical concept that expresses the relationship between two or more quantities that are related to each other in a consistent way. A proportion can be expressed as an equation in which two ratios are set equal to each other, and it states that the ratios of two quantities are always the same. In other words, if we know that two quantities are proportional, we can use this relationship to find one quantity given the other.
Here,
Based on the given table, we can see that the number of cookies (Y) is directly proportional to the number of packages (X). Thus, we can write:
Y = kX
where k is the constant of proportionality.
a) When k = 6, we can find the value of Y for different values of X as follows:
When X = 12, Y = kX = 6(12) = 72
When X = 10, Y = kX = 6(10) = 60
When X = 8, Y = kX = 6(8) = 48
When X = 6, Y = kX = 6(6) = 36
When X = 4, Y = kX = 6(4) = 24
When X = 2, Y = kX = 6(2) = 12
Thus, the constant of proportionality is k = 6.
b) When k = 12, we can find the value of Y for different values of X as follows:
When X = 12, Y = kX = 12(12) = 144
When X = 10, Y = kX = 12(10) = 120
When X = 8, Y = kX = 12(8) = 96
When X = 6, Y = kX = 12(6) = 72
When X = 4, Y = kX = 12(4) = 48
When X = 2, Y = kX = 12(2) = 24
Thus, the constant of proportionality is k = 12.
c) When k = 1/6, we can find the value of Y for different values of X as follows:
When X = 12, Y = kX = (1/6)(12) = 2
When X = 10, Y = kX = (1/6)(10) = 5/3
When X = 8, Y = kX = (1/6)(8) = 4/3
When X = 6, Y = kX = (1/6)(6) = 1
When X = 4, Y = kX = (1/6)(4) = 2/3
When X = 2, Y = kX = (1/6)(2) = 1/3
Thus, the constant of proportionality is k = 1/6.
e) Since the ratio of Y to X is not constant for different values of X, the relationship between Y and X is not proportional, and there is no constant of proportionality.
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