A 14.2 million donation was made to a charity. The overall amount of the charity's contributed funds is 34 million if which amounts to 42% of those monies.
Let x represent the total amount of donations received by the charity.
According to the problem, 14.2 million represents 42% of the charity's donated funds:
14.2 million = 0.42x
We must isolate x on one side of the equation in order to find it. We can do this by dividing both sides of the equation by 0.42:
x = 14.2 million / 0.42
x = 33.80952381 million
Rounding this answer to the nearest million gives:
x ≈ 34 million
Therefore, the total amount of the charity's donated funds is approximately 34 million.
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Do the cylinders shown have equivalent volumes? Explain.
The two cylinders have the exact same volume given by:
V = pi*r²*h
Do the cylinders shown have equivalent volumes?The volume of a cylinder of radius r and height h is given by:
V = pi*r²*h
Where pi = 3.14
While the volume for a slanted cylinder, like the one in the image, is computed in the same way, as long as the height h is measured with a line normal from the bases (like in the image).
Now, notice that the first cylinder has a diameter of 2r and height h, so its radius is 2r/2 = r, and the volume is:
v = 3.14*r²*h
The second cylinder has a radius r and a height h, so its volume is
v = 3.14*r²*h
So yea, both cylinders have the same volume.
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Find the area of the shaded region
The area of the shaded region is evaluated to be equal to the expression 24z² - 22z + 3
How to evaluate for the area of the shaded regionTo get the area of the shaded region, we subtract the area of the small square from the area of the bigger square as follows:
area of bigger square = (5z - 2)(5z - 2)
area of bigger square = 25z² - 20z + 4
area of small square = (z + 1)(z + 1)
area of bigger square = z² + 2z + 1
area of the shaded region = 25z² - 20z + 4 - (z² + 2z + 1)
area of the shaded region = 25z² - 20z + 4 - z² - 2z - 1
area of the shaded region = 24z² - 22z + 3
Therefore, the area of the shaded region is evaluated to be equal to the expression 24z² - 22z + 3
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Unsure about this question! Please help me.
The factors of [tex]16x^2yz[/tex] are [tex]4xyz, xz, and 12.[/tex]yz, xz, and 12.
What is the expression?The factors of the given expression 16x^2yz are:
[tex]4xyz:[/tex]This is a valid factor of [tex]16x^2yz[/tex] because it contains all the variables x, y, and z, and the exponent of x is 1, the exponent of y is 1, and the exponent of z is 1. Additionally, it contains a numerical factor of 4.
[tex]2xy:[/tex]This is not a factor of 16x^2yz because it does not include the variable z, which is present in the given expression.
[tex]xz:[/tex] This is not a factor of [tex]16x^2yz[/tex] because it does not include the variable y, which is present in the given expression.
[tex]12:[/tex] This is not a factor of [tex]16x^2yz[/tex] because it is a numerical factor and does not contain any of the variables x, y, or z.
[tex]xyz:[/tex] This is not a factor of [tex]16x^2yz[/tex] because it does not contain any numerical factor. It only includes the variables x, y, and z without any exponents.
Therefore, the factors of [tex]16x^2yz[/tex] are [tex]4xyz, xz, and 12.[/tex]
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A jewelry store purchases a diamond ring from a wholesaler for $375.50. The store plans to mark up the ring by 175%. What will be the
selling price of the diamond ring? Round to the nearest cent.
Answer:
$1,033.13.
Step-by-step explanation:
To find the selling price of the diamond ring, we need to add the markup amount to the original cost.
Markup amount = 175% of $375.50
= 1.75 x $375.50
= $657.63
Selling price = Original cost + Markup amount
= $375.50 + $657.63
= $1,033.13
Therefore, the selling price of the diamond ring will be $1,033.13.
Answer:
$1031.63
Step-by-step explanation:
Mark-up refers to the amount or percentage added to the cost price of a product to determine its selling price. It represents the increase in price that a business applies to cover its expenses, generate profit, and account for other factors like overhead costs, operating expenses, and desired profit margin.
To calculate the selling price of the diamond ring, we need to add the mark-up amount to the cost price and round the result to the nearest cent.
To find the mark-up amount, multiply the cost price by the mark-up percentage:
[tex]\begin{aligned}\textsf{Mark-up amount}&=\textsf{Cost price} \times \textsf{Mark-up percentage}\\\\&=\$375.50 \times 175\%\\\\&=\$375.50 \times \dfrac{175}{100}\\\\&=\$375.50 \times 1.75\\\\&=\$657.125\end{aligned}[/tex]
Therefore, the ring will be marked-up by $657.125.
To find the selling price, add the mark-up amount to the cost price:
[tex]\begin{aligned}\textsf{Selling price}&=\textsf{Cost price} + \textsf{Mark-up amount}\\\\&=\$375.50 +\$657.125\\\\&=\$375.50 +\$657.125\\\\&=\$1032.625\end{aligned}[/tex]
Finally, round the selling price to the nearest cent:
[tex]\large\boxed{\textsf{Selling Price} = \$1031.63}[/tex]
Therefore, the selling price of the diamond ring will be $1031.63, rounded to the nearest cent.
What is the missing length?
Answer:
Using the Pythagorean theorem, we know that in a right-angle triangle, we have:
a^2 + b^2 = c^2
where c is the hypotenuse.
From the given information, we have:
ab = 11 - x
a = x
b = 11
c = 11 (given)
We can use the given values to eliminate a or b from the Pythagorean theorem:
a^2 + (11 - x)^2 = 11^2
Expanding and simplifying, we get:
a^2 + 121 - 22x + x^2 = 121
a^2 + x^2 - 22x = 0
Substituting a = x into the above equation, we get:
x^2 + x^2 - 22x = 0
2x^2 - 22x = 0
2x(x - 11) = 0
So, either x = 0 or x - 11 = 0.
Since x cannot be zero (as it represents a length), we have x - 11 = 0.
Therefore, x = 11.
Hence, the value of x in its simplest form with a rational denominator is 11/1 or just 11.
Answer:
[tex]x=11\sqrt{2}[/tex]
Step-by-step explanation:
The given right triangle is an isosceles right triangle since its legs are equal in length (denoted by the tick marks).
Side x is the hypotenuse of the isosceles right triangle.
Given both legs are 11 units in length, we can use Pythagoras Theorem to calculate the length of the hypotenuse.
[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]
As a and b are the legs, and c is the hypotenuse, substitute the following values into the formula and solve for x:
a = 11b = 11c = xTherefore:
[tex]\implies 11^2+11^2=x^2[/tex]
[tex]\implies 121+121=x^2[/tex]
[tex]\implies 242=x^2[/tex]
[tex]\implies x^2=242[/tex]
[tex]\implies \sqrt{x^2}=\sqrt{242}[/tex]
[tex]\implies x=\sqrt{242}[/tex]
To simplify the radical, rewrite it as a product of prime numbers:
[tex]\implies x=\sqrt{11^2 \cdot 2}[/tex]
[tex]\textsf{Apply the radical rule:} \quad \sqrt{ab}=\sqrt{a}\sqrt{b}[/tex]
[tex]\implies x=\sqrt{11^2}{\sqrt{2}[/tex]
[tex]\textsf{Apply the radical rule:} \quad \sqrt{a^2}=a, \quad a \geq 0[/tex]
[tex]\implies x=11\sqrt{2}[/tex]
Therefore, the length of side x in simplest radical form is 11√2.
it takes brian 2/3 of hour to wash his dog sport. if he washes him once a week how many hours will brian spend washing sport over 4 weeks
Answer: 2 hours 40 minutes
Step-by-step explanation: 2/3 x 4 = 8/3 simplified to 2 2/3
2 hours and 2/3 of an hour = 2 hours 40 minutes
cot (x + y)=
cotx coty-1/cotx+coty
Actually, the correct identity is: cot(x + y) = (cot x cot y - 1) / (cot x + cot y) This identity can be derived from the sum formulas for tangent and cotangent: tan(x + y) = (tan x + tan y) / (1 - tan x tan y) cot(x + y) = (cot x cot y - 1) / (cot x + cot y) We can start by substituting cot(x + y) with its equivalent expression: cot(x + y) = cot x cot y - 1 / (cot x + cot y) Multiplying both sides by (cot x + cot y), we get: cot(x + y) (cot x + cot y) = cot x cot y - 1 Expanding the left side, we get: cot x cot y + cot x cot
A souvenir shop sells t-shirts. The shop determines the price of each shirt by adding $3.75 to the price that it pays for the item. Then, that amount is doubled.
Before tax is added to the purchase, how much will a customer pay for a t-shirt that costs the souvenir shop $16.88?
Responses
A $20.63
B $36.95
C $45.35
D $41.26
The souvenir store pays $12.99 for a t-shirt. The store uses its usual price markup and adds $1.34 for sales tax., Which choice is the total amount a customer pays for the t-shirt?
Responses
A $34.82
B $23.71
C $28.40
D $14.33
First question:
The cost of the t-shirt for the souvenir shop is $16.88. The shop adds $3.75 to the cost and then doubles the amount to get the price that the customer pays. So, the price that the customer pays before tax is:
2 x (16.88 + 3.75) = 2 x 20.63 = 41.26
Therefore, the answer is (D) 41.26.
Second question:
The cost of the t-shirt for the souvenir shop is $12.99. The shop adds $3.75 to the cost to get the price of the t-shirt. Then, the shop adds sales tax of $1.34 to the price to get the total amount that the customer pays. So, the total amount that the customer pays is:
12.99 + 3.75 + 1.34 = 18.08
Therefore, the answer is (C) 28.40.
PLEASE HELP ASAP!!!!
A cylinder with a diameter of 8 yards has a volume of 452.16 yd3. What is the height of the cylinder? Use 3.14 for π.
2 yards
8 yards
9 yards
18 yards
Answer:
The peak of the cylinder is about 9 yards.
Explanation:
From the above question,
They have given:
A cylinder with a diameter of 8 yards has a volume of 452.16 yd³.
Here we need to find what is the height of the cylinder
Here,
[tex]\pi = 3.14[/tex]. 44 yards 11 yards 8 yards 3 yards
To locate the radius of the cylinder, which is half of of the diameter.
[tex]\text{radius = 8 yards} \div \text{2}[/tex]
[tex]\text{radius} = 4 \ \text{yards}[/tex]
Then we can use the formulation for the quantity of a cylinder to locate the height:
[tex]\text{volume} = \pi \times \text{radius} \times \text{height}[/tex]
[tex]452.16 = 3.14 \times 4^2 \times\text{height}[/tex]
[tex]452.16 = 50.24 \times \text{height}[/tex]
[tex]\text{height} = 452.16 \div 50.24[/tex]
[tex]\text{height} = 9 \ \text{yards}[/tex]
Therefore, the peak of the cylinder is about 9 yards.
Answer: 9 yards
The first person to answer was correct.
Step-by-step explanation:
I took the quiz so you don't have to! <3
while calculating your quiz grades you noticed Mrs. McClellan wrote the following numbers at the top of your quizzes:. 8/10; 81%; 0.805. place scores in ascending order.
Pls help me
How long is the side CD? (Hint: opposite sides are equal. Put the expressions equal to each other and solve for x. Then, plug back into the expression for CD to find CD).
Answer:
the best answer would be 27
Step-by-step explanation:
Boom
Compare the data and use the correct measure of center to determine which bus typically has the faster travel time. Round your answer to the nearest whole number, if necessary, and explain your answer. Bus 14, with a median of 14 Bus 18, with a mean of 12 Bus 14, with a mean of 14 Bus 18, with a median of 12
Based on both the median and mean values, it appears that Bus 14 typically has a faster travel time compared to Bus 18.
What is the median and mean?
The median is the value that divides a data sample, a population, or a probability distribution's upper and lower halves in statistics and probability theory. It could be referred to as "the middle" value for a data set.
Here,
Cοmparing the medians, we can see that Bus 14 has a higher median οf 14, while Bus 18 has a lower median οf 12. This suggests that Bus 14 may have a faster travel time οn average.
Hοwever, we alsο need tο cοnsider the means. Cοmparing the means, we can see that Bus 14 has a higher mean οf 14, while Bus 18 has a lower mean οf 12. This further supports the idea that Bus 14 may have a faster travel time οn average.
Hence, based on both the median and mean values, it appears that Bus 14 typically has a faster travel time compared to Bus 18.
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Consider the two triangles.
15
C
20
Mark this and return
B
H.
12
G
To prove that the triangles are similar by the SAS
similarity theorem, it needs to be shown that
OZCZC
OZC=ZG
O
O
2/02/5
HI
BC
BC
HI
Save and Exit
Next
Submit
factor the following polynomials by finding a GCF.
1.) 28c - 12
Y varies inversely as x and y =38 when x =26 find y when x =24
In summary in axis when x = 24, y = 41.1667.
Why is it?
If y varies inversely as x, then y = k/x for some constant k. To find k, we can use the initial values given:
y = k/x
38 = k/26
k = 38×26
k = 988
Now we can use k to find y when x = 24:
y = k/x
y = 988/24
y = 41.1667 (rounded to four decimal places)
Therefore, when x = 24, y = 41.1667.
In mathematics, an axis is a straight line that defines a reference direction for a coordinate system. It is used to locate or plot points, and describe geometric shapes in a plane or in space. In a two-dimensional coordinate system, there are two axes: the x-axis, which is horizontal, and the y-axis, which is vertical.
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Which measurement is not equal to one mile
1 yard = 3 feet
6 yards = 18 feet
Truck = 20 feet
Difference = 2 feet = 24 inches
Cole, a caterer, is investing some money in equipment and employees to help grow his business. Recently he spent $100 on equipment and hired a server who makes $16 per hour. Cole is hoping to make up these expense at the next job that is scheduled, which pays a base of $50 in addition to $18 per hour that the server works. In theory, this event could pay enough to cancel out Cole's expenditures. How much would the job pay?
Claudia Pragram Technologies, Inc., has narrowed its choice of outsourcing provider to two firms located in different countries. Pragram wants to decide which one of the two countries is the better choice, based on risk-avoidance criteria. She has polled her executives and established four criteria. The resulting ratings for the two countries are presented in the table below, where 1 is a lower risk and 3 is a higher risk. SELECTION CRITERION ENGLAND CANADA Price of service from outsourcer 2 3 Nearness of facilities to client 3 1 Level of technology 1 3 History of successful outsourcing 1 2 The executives have determined four criteria weightings: Price, with a weight of 0.1; Nearness, with 0.6; Technology, with 0.2; and History, with 0.1. a) Using the factor-rating method, which country would you select? b) Double each of the weights used in part (a) (to 0.2, 1.2, 0.4, and 0.2, respectively). What effect does this have on your answer? Why?
England receives a total weighted score of [tex]1.9[/tex] , while Canada receives a score of [tex]1.6[/tex].
What is an illustration of outsourcing?Instead of hiring an own accountant, a small business may choose to contract bookkeeping responsibilities to an accounting company in to cut expenses. Some companies find it advantageous to outsource HR department duties including payroll and medical coverage.
What justifies outsourcing?Outsourcing has several benefits, including better systems and service quality, access to the world's best workers, improved scheduling, avoiding personnel concerns, and many more.
a) Using the factor-rating method, we can calculate the total weighted score for each country as follows:
For England:
Total score [tex]= (2 * 0.1) + (3 * 0.6) + (1 * 0.2) + (1 * 0.1) = 1.9[/tex]
For Canada:
Total score [tex]= (3 * 0.1) + (1 * 0.6) + (3 * 0.2) + (2 * 0.1) = 1.6[/tex]
Therefore, based on the risk-avoidance criteria and weightings established by the executives, we would select Canada as the better choice.
b) If we double each of the weights used in part (a), the total weighted score for each country changes as follows:
For England:
Total score [tex]= (2 * 0.2) + (3 * 1.2) + (1 * 0.4) + (1 * 0.2) = 4.0[/tex]
For Canada:
Total score [tex]= (3 * 0.2) + (1 * 1.2) + (3 * 0.4) + (2 * 0.2) = 2.2[/tex]
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2log12 3 + 4log12 2 simplifying
Hello, I was happy to solve the problem. If you find a bug, post in the comments or click on the report, I will see it and try to fix it as soon as possible.
The answer to this problem: 2
Our class is the best of all of the MA 110 sections, so we average an 80 on our tests. Assuming a standard deviation of 6:
What is the z-score of a student who scores 97?
Is this student in the top 1% of performers? Explain how you arrived at your answer here.
Show all work
The z-score of a student who scores 97 is 2.83. In this case, since the z-score is greater than 1.96, the student is in the top 1% of performers.
What is z-score?It is used to indicate how many standard deviations a certain value is from the mean. It is calculated by subtracting the mean from the observed value and then dividing the result by the standard deviation.
This is calculated by subtracting the mean (80) from the student’s score (97) and dividing the result by the standard deviation (6).
97- 80/6
= 2.83
The z-score of 2.83 indicates that the student is in the top 1% of performers. This is because the z-score of 2.83 is greater than the z-score of 1.96, which is the z-score that is associated with the 1% percentile.
A z-score of 1.96 means that it is higher than 1% of the scores.
In this case, since the z-score is greater than 1.96, the student is in the top 1% of performers.
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does thes formula for surface area of a cylinder work for right cylinders and oblique cylinders true or false
The height h, on the other hand, denotes the slant height for oblique cylinders, which is the distance between the two circular bases on a straight line.
how can you describe cylinder ?Two identical parallel spherical bases and a curving surface connecting the bases make up the three-dimensional geometric object known as a cylinder. It can be compared to a prism with rounded bases.
A line connecting the centres of the circle bases is the axis of a cylinder. The possibility that the axis is straight to the bases determines whether a cylinder is right or oblique.
Diagonal cylinders have bases that aren't perpendicular to the axis while right cylinders have bases that are perpendicular to the axis.
given
Both right and oblique cylinders can be calculated using the surface area of a cylinder formula, A = 2r2 + 2rh.
The height h, on the other hand, denotes the slant height for oblique cylinders, which is the distance between the two circular bases on a straight line.
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What is the diameter of a hemisphere with a volume of
62617cm[tex]What is the diameter of a hemisphere with a volume of 62617cm3, to the nearest tenth of a centimeter[/tex]
this is answer 44rrrrrrrrrrrrrrrrrrrrfrrrrrrrrrrr
For each pair of triangles below state whether the triangles are congruent not congruent or impossible to tell
Answer:
a) congruent
b) not congruent
c) congruent
Step-by-step explanation:
You want to know which sets of isosceles triangles are congruent.
Isosceles triangleAn isosceles triangle has two congruent sides and two congruent angles. If those sides and angles match corresponding sides and angles in another isosceles triangle, then the triangles will be congruent.
Unless the triangle is equilateral, the two congruent base angles will have different measures than the angle between the congruent sides. Once any of the angle measures is known, the others are determined—provided that we also know which angle it is that is known.
a) Third angle knownThe congruent side measures are the same, and the apex angle is the same. This triangle pair is congruent.
b) Angles givenThe base angle of one triangle is the same measure as the apex angle of the other. (Neither is 60°.) These triangles cannot be congruent.
c) Base angle knownThe congruent side measures are the same, and the base angles are the same. This triangle pair is congruent.
are the traiangles shown in the diagram at the right similar? Use angle relationships to explain how you know.
A<65
C< 70
B?
F<45
E<115
D< ?
The two triangles have similar angles, thus the two triangles are similar using the AA criteria.
What are congruent triangles?The dimensions of the sides and angles of two or more triangles determine whether they are congruent. A triangle's size and shape are determined by its three sides and three angles, respectively. If pairings of corresponding sides and corresponding angles are identical, two triangles are said to be congruent. These are the exact same size and form. Triangles can satisfy a wide variety of congruence requirements.
For the triangle ABC using the sum of interior angles of triangle we have:
A + B + C = 180
65 + 70 + B = 180
135 + B = 180
B = 45
Now, for triangle DEF we have:
F = 45.
E = 180 - 115 = 65
Thus, D = 70
The two triangles have similar angles, thus the two triangles are similar using the AA criteria.
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70 adults with gum disease were asked the number of
times per week they used to floss before their
diagnoses. The (incomplete) results are shown below:
# of
times
floss
per
week
0
1
2
3
4
5
6
7
4
11
15
11
6
7
(6 is blank)
6
Relative
Frequency
(4 is blank)
0.1571
0.1429
0.2143
0.1571
0.0857
0.1
0.1429
0.0857
Cumulative
Frequency
4
15
30
41
47
(5 is blank)
64
70
a. Complete the table (Use 4 decimal places when
applicable)
b. What is the cumulative relative frequency for
flossing 3 times per week?
%
The cumulative relative frequency for flossing 3 times per week is 58.57%.
How to solvea. Let's complete the table.
We can calculate the missing relative frequency for 0 times flossing per week by dividing the frequency (4) by the total number of people (70).
Relative Frequency for 0 times flossing = 4/70 = 0.0571 (rounded to 4 decimal places)
For 6 times flossing per week, the frequency is missing.
We can calculate it by summing up the frequencies for all other days and subtracting that from the total number of people.
Missing frequency for 6 times flossing = 70 - (4 + 11 + 15 + 11 + 6 + 7 + 6) = 70 - 60 = 10
Now, let's calculate the cumulative frequency for 5 times flossing per week.
Cumulative Frequency for 5 times flossing = Cumulative Frequency for 4 times flossing + Frequency for 5 times flossing = 47 + 7 = 54
The completed table looks like this:
of times floss per week | Frequency | Relative Frequency | Cumulative Frequency
0 | 4 | 0.0571 | 4
1 | 11 | 0.1571 | 15
2 | 15 | 0.2143 | 30
3 | 11 | 0.1571 | 41
4 | 6 | 0.0857 | 47
5 | 7 | 0.1 | 54
6 | 10 | 0.1429 | 64
7 | 6 | 0.0857 | 70
b. The cumulative relative frequency for flossing 3 times per week is the cumulative frequency for 3 times flossing divided by the total number of people (70).
Cumulative Relative Frequency for 3 times flossing = Cumulative Frequency for 3 times flossing / Total number of people = 41 / 70 = 0.5857 (rounded to 4 decimal places)
The cumulative relative frequency for flossing 3 times per week is 58.57%.
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The mean age of 25 people in a cinema is 24 year-old. Another person arrives at the cinema and the mean age becomes 25 year-old. Determine the age of the new person.
According to the given information, the mean age of 25 people in the cinema is 24, which means the total age of all 25 people is:
25 people * 24 years = 600 years
When the new person arrives, the total age becomes:
26 people * 25 years = 650 years
The difference between the two totals is the age of the new person:
650 years - 600 years = 50 years
Therefore, the age of the new person is 50 years old.
You're studying different function tools and begin working with this function: y = -200 - 6(x) Use the equation to answer the question. y = -200 - 6(x) What is the y-value when x equals -12? y = -200 - 6(x) Enter the correct answer. DONE 00
Answer:
Y value is -128
Step-by-step explanation:
To find the y-value when x = -12, sub -12 for x in the equation.
y = -200 - 6(x)
y = -200 - 6(-12)
y = -200 + 72
y= -128
D
Are the two triangles similar?
28°
R
110°
Yes
No
E
V
28°
T
42°
A
Answer:
Yes
Step-by-step explanation:
Angle RDE =angle TVA=28°
REV=TAV=42°
DRE=VTA=110°
THEREFORE TRIANGLE DER IS SIMILAR TO TRIANGLE ATV
HELP PLEASE!!!!
Tell me what to shade in! And also tell me what word is formed
50 point reward!***
Answer:
7/8, 1/5, 1/3, 5/9, 11/12, 2/5, 3/5, 3/20, 9/25, 3/7, 3/8, 4/9, 8/15, 4/5, 5/7, 1/8, 7/12, 21/25, 1/6, 3/14, 1/4, 8/27, 3/4, 2/3, 5/6, 2/15, 27/40, 9/10, 5/12
the secret message spells "YES"
Step-by-step explanation:
you cant divide any of the numbers above by the same number and not get a decimal.
Answer: The word is yes
Step-by-step explanation:
Shade in 7/8, 1/5, 2/5, 3/7, 8/15, 3/14, 3/4, 5/9, 3/8, 4/5, 1/4, 2/3, 1/3, 3/5, 5/7, 1/8, 5/6, 2/15, 11/12, 3/20, 9/25, 4/9, 7/12, 21/25, 1/6, 8/27, 9/10, 27/40, and 5/12.
Considering only the values of θ for which the expression is defined, which of the following is equivalent to the expression below?
cos(−θ)⋅tan(−θ)⋅cscθ
Select the correct answer below:
−sinθ
1
sinθ
−1
The cos(θ) term cancels out and we are left with -1. Therefore, the equivalent expression is -1.
We can start by using the trigonometric identities:
cos(-θ) = cos(θ)
tan(-θ) = -tan(θ)
csc(θ) = 1/sin(θ)
Substituting these identities into the original expression, we get:
cos(θ) * (-tan(θ)) * (1/sin(θ))
Simplifying this expression, we can cancel out the cos(θ) and the sin(θ) terms:
-1 * cos(θ) * (1/(cos(θ))) The cos(θ) term cancels out and we are left with -1. Therefore, the equivalent expression is -1.
In other words, the original expression simplifies to -1 for all values of θ where it is defined (i.e. θ ≠ (2n + 1)π/2, where n is an integer). This means that as θ varies, the value of the expression will always be -1 when it is defined. Trigonometric identities are mathematical equations that involve trigonometric functions and are true for every possible value of the variables involved. There are various types of trigonometric identities, including:
Pythagorean Identities:
[tex]sin^2a + cos^2a= 1\\tan^2a + 1 = sec^2a\\1 + cot^2a = csc^2a[/tex]
Angle Sum and Difference Identities:
sin(α±β) = sin α cos β ± cos α sin β
cos(α±β) = cos α cos β ∓ sin α sin β
tan(α±β) = (tan α ± tan β) / (1 ∓ tan α tan β)
Double Angle Identities:
sin 2θ = 2 sin θ cos θ
cos 2θ =[tex]cos^2[/tex]θ - [tex]sin^2[/tex]θ = 2 [tex]cos^2[/tex]θ - 1 = 1 - 2[tex]sin^2[/tex]θ
tan 2θ = (2 tan θ) / (1 - [tex]tan^2[/tex]θ)
Half Angle Identities:
sin (θ/2) = ± √[(1 - cos θ) / 2]
cos (θ/2) = ± √[(1 + cos θ) / 2]
tan (θ/2) = ± √[(1 - cos θ) / (1 + cos θ)]
Product-to-Sum Identities:
sin α sin β = (1/2) [cos (α-β) - cos (α+β)]
cos α cos β = (1/2) [cos (α-β) + cos (α+β)]
sin α cos β = (1/2) [sin (α+β) + sin (α-β)]
Sum-to-Product Identities:
sin α + sin β = 2 sin [(α+β)/2] cos [(α-β)/2]
sin α - sin β = 2 cos [(α+β)/2] sin [(α-β)/2]
cos α + cos β = 2 cos [(α+β)/2] cos [(α-β)/2]
cos α - cos β = -2 sin [(α+β)/2] sin [(α-β)/2]
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