The solutions of the equation tan(3) + 1 = sec(3) in the interval [0, 2) are approximately:
3 ≈ 0.523599 radians or 30 degrees (in the first quadrant)
3 + π ≈ 3.66444 radians or 210 degrees (in the third quadrant)
How did we get these values?The equation tan(3) + 1 = sec(3) can be rewritten as:
tan(3) + 1 = 1/cos(3)
Multiplying both sides by cos(3), we get:
sin(3) cos(3) + cos(3) = 1
Using the identity sin(2x) = 2sin(x)cos(x), we can rewrite the left-hand side as:
sin(2*3) + cos(3) = 1
Simplifying this expression, we get:
2sin(3)cos(3) + cos(3) = 1
Factorizing out cos(3), we get:
cos(3)(2sin(3) + 1) = 1
Dividing both sides by 2sin(3) + 1, we get:
cos(3) = 1/(2sin(3) + 1)
We know that sin^2(3) + cos^2(3) = 1, so we can substitute cos^2(3) = 1 - sin^2(3) into the above equation and simplify:
1/(2sin(3) + 1) = cos(3) = sqrt(1 - sin^2(3))
Squaring both sides and simplifying, we get:
(2sin(3) + 1)^2 = 1 - sin^2(3)
Expanding the left-hand side and simplifying, we get:
4sin^3(3) - 3sin^2(3) - 3sin(3) + 1 = 0
This is a cubic equation in sin(3), which can be solved using various methods, such as Cardano's formula or numerical methods. However, the solutions are quite complicated and involve complex numbers.
In the interval [0, 2), we can use a numerical method, such as Newton's method, to find an approximate solution. Starting with an initial guess of sin(3) = 0.5, we can iteratively apply the formula:
sin(3)_n+1 = sin(3)_n - f(sin(3)_n)/f'(sin(3)_n)
where f(sin(3)) = 4sin^3(3) - 3sin^2(3) - 3sin(3) + 1 and f'(sin(3)) = 12sin^2(3) - 6sin(3) - 3.
After several iterations, we find that sin(3) ≈ 0.464758. Substituting this into the equation cos(3) = 1/(2sin(3) + 1), we get cos(3) ≈ 0.885005.
Therefore, the solutions of the equation tan(3) + 1 = sec(3) in the interval [0, 2) are approximately:
3 ≈ 0.523599 radians or 30 degrees (in the first quadrant)
3 + π ≈ 3.66444 radians or 210 degrees (in the third quadrant)
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What inequality matches this graph?
A. x > -5
B. x > -5
C. x < -5
D. x < -5
Answer:
x ≥ -5
Step-by-step explanation:
Since we see the dot is filled in, we know the sign is either ≤ or ≥. Then, we can see that x is greater than -5 as the arrow is pointing in the positive direction (to the right). Therefore, it would be x ≥ -5
A local city government wants to display the ages of the city citizen. Would it be better to organize the data using a dot plot or a histogram, and why?
If local city government wants to display the ages of the city citizen, It would be better to organize the data using a histogram.
A histogram is a graph that shows the distribution of a set of continuous data. The ages of city citizens can be considered as continuous data, as there are many possible values for age within a certain range.
A dot plot, on the other hand, is a graph that shows the distribution of a set of discrete data. Discrete data are values that can only take on specific, isolated values, like shoe sizes or the number of siblings a person has.
A histogram would be more effective for displaying the distribution of the ages of city citizens because it can show the range of ages in a more continuous manner, and provide a more precise representation of the frequency of the different age groups.
A dot plot, on the other hand, would be less effective, as it could be difficult to represent every age value as a single dot, and the resulting graph may not provide a clear indication of the distribution of ages.
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(PLEASE HELP! Im using a ton of points)
A new curved projection system for a small media room states the optimal viewing angle is
74° (m/ACB = 74°) when you sit 10 feet from the center of the TV (length of CD = 10 ft). The
depth of the curved screen is 2 feet (length of DE = 2 ft). Find the length of line segment AB, in feet, that corresponds to the width of the television. (Round your answer to the nearest foot)
(Thx for any help!)
The value of the length of line segment AB, in feet, that corresponds to the width of the television is,
⇒ 12 feet
We have to given that;
A new curved projection system for a small media room states the optimal viewing angle is 74° (m/ACB = 74°) when you sit 10 feet from the center of the TV (length of CD = 10 ft). The depth of the curved screen is 2 feet (length of DE = 2 ft).
Now, We can formulate;
CE = 10 - 2 = 8 feet
∠ACD = 74/2 = 37°
Hence, We can formulate;
tan 37° = AE / CE
0.75 = AE / 8
AE = 0.75 × 8
AE = 6
Hence, The value of the length of line segment AB, in feet, that corresponds to the width of the television is,
⇒ AB = 6 x 2
⇒ AB = 12 feet
Thus, The value of the length of line segment AB, in feet, that corresponds to the width of the television is,
⇒ 12 feet
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Find the principal needed now to get the given amount; that is, find the present value. To get $4000 after 2 1/4 years at 9% compounded daily The present value of $4000 is $? (Round to the nearest cent as needed.)
Step-by-step explanation:
Compounding formula
FV = PV ( 1 + i )^n FV = Future value (4000) PV = present value
i = decimal interest per PERIOD = .09/365
n = periods = 2 1/4 yrs * 365 days/yr = 821.25 periods
4000 = PV ( 1 + .09/365)^821.25
Solve for PV = $ 3266.83
Alan and his mom are planting vegetables in their garden. Alan has finished planting 4 rows of carrots so far and is planting new rows at a rate of 2 rows per hour. His mom has finished 7 rows of tomatoes and will continue planting at 1 row per hour. Once the pair get to the point where they have finished the same number of rows, they will take a break and decide what to plant next. How many rows will they each have completed? Write a system of equations, graph them, and type the solution.
Alan and his mom will each have completed 10 rows after 3 hours.
What do you mean by term Expressions ?An expression is a combination of numbers, variables, and/or operations (such as addition, subtraction, multiplication, division, exponents, and roots) that are grouped together to represent a value.
Let x be the number of hours it takes for Alan and his mom to have planted the same number of rows.
Alan will have planted 4 + 2x rows of carrots, and his mom will have planted 7 + x rows of tomatoes.
Setting these two expressions equal to each other gives:
4 + 2x = 7 + x
Solving for x, we get:
x = 3
Thus, after 3 hours, Alan will have planted:
4 + 2(3) = 10 rows of carrots
And his mom will have planted:
7 + (3) = 10 rows of tomatoes
Therefore, they will each have completed 10 rows of vegetables after 3 hours.
To graph these equations, we can plot them on a coordinate plane, where the x-axis represents the number of hours and the y-axis represents the number of rows planted.
Alan's equation is y = 2x + 4, and his mom's equation is y = x + 7.
The graph of these two equations intersect at the point (3, 10), which represents the moment when they have each planted 10 rows.
Therefore, Alan and his mom will each have completed 10 rows after 3 hours
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A list contains the names of six anthropology students, two sociology students, and four psychology students. If one name is selected at random to assist in the professor's new study, find the probability that the chosen student is
P(anthropology)=
Someone help i really need help with this
Step-by-step explanation:
we just need to find out how many percentage of the daily goal each day, so let's assume that $50 is 100% of the daily goal.
Can someone please help me with this? I'll mark anyone who answers for helping
Graph and solve y=1/2x-6 and y=1/2x+2
Answer:
No solution.
Step-by-step explanation:
To solve this problem, all you have to do is graph the lines and see where they intersect.
This is what you would do to solve a normal problem. However, these lines are parallel this means that the solution to these lines is no soultion
A yoga studio sells monthly memberships. The function f(x) = −x^2 + 50x − 264 models the profit in dollars, where x is the number of memberships sold.
Determine the zeros, and explain what these values mean in the context of the problem.
x = 6, x = 44; The zeros represent the number of monthly memberships that produces a maximum profit.
x = 6, x = 44; The zeros represent the number of monthly memberships where no profit is made.
x = 25, x = 361; The zeros represent the number of monthly memberships where no profit is made.
x = 25, x = 361; The zeros represent the number of monthly memberships that produces a maximum profit.
The zeros of the quadratic profit function are explained as:
x = 6, x = 44; The zeros represent the number of monthly memberships where no profit is made
How to find the zeros of a quadratic function?The quadratic function is given as:
f(x) = −x² + 50x − 264
The general form of expression of a quadratic function is:
f(x) = ax² + bx + c
The quadratic formula to find the zeros is:
x = [-b ± √(b² - 4ac)]/2a
Thus, we have the zeros as:
x = [-50 ± √((-50)² - 4(-1 * -264))]/2(-1)
x = 6 and x = 44
This represents number of monthly memberships where no profit is made.
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Express the product
11 10
7 6 5 4
using factorial notation.
If we express the 9! we have 9×8×7×6×5×4×3×2×1.
What is factorial operation?A factorial operation is described as a mathematical formula represented by an exclamation mark. (!). This operation is carried out when a whole number is being multiplied by its successive smaller numbers until it gets to 1.
for Example
n! = n(n-1)(n-2)........(1)
From the question, it is are asked to evaluate 9!
9! = 9×8×7×6×5×4×3×2×1
Hence, when 9! is evaluated, we get 9×8×7×6×5×4×3×2×1.
#Probable complete question;
Use the factorial operation to evaluate 9!.
8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
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I am half as old as my sister. In 6 years’ times will be 22. How old am now?
Answer:15
Step-by-step explanation:
The decimal -2.97 is the solution to which subtraction problem?
The decimal - 2. 97 is the solution to the subtraction problem of D. - 6. 1 - ( - 3. 13 ).
How to find the solution ?When subtracting decimal numbers, the effective method is to align their decimal points and begin removing each corresponding digit from one another. You may extend any decimals with fewer digits by appending zeros on the right-hand side as required for parity.
= - 6. 1 - ( -3. 13 )
= - 6. 1 + 3. 13
= - 2. 97
The decimal 2.97 is the solution to the subtraction problem - 6. 61 - (- 3. 13).
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box of chocolates contains five milk chocolates, five dark chocolates and five white chocolates. You randomly select and eat three chocolates. What is the probability that the first piece is a milk chocolate, the second is a dark chocolate and the third is a white chocolate?
Answer: 25/546 chance
Step-by-step explanation:
total is 15 chocolates.
Five are milk, so chance of getting milk is 1/3. second is dark, so 5/14 chance. Then white, so 5/13 chance
1/3*5/14*5/13=25/546
Need the slope(m) of the function.
the y intercept of the function.
and the slope intercept form f(x)=mx+b of the function.
Answer:
We can calculate the slope by taking 2 points on the line. I took A(4,4) and B(-4,0), so we get m=1/2.
2. y-intercept is the point where a line crosses the y-axis which is (0,2)
3. We have f(x)=1/2x +b
By taking a particular point on the line,we get :
2=1/2(0)+b
==》 b=2
So the slope intercept form is
y=1/2x+2
Write an inequality that matches the number line model. Then write
a situation that the inequality could represent.
-2.75 is greater than or equal to x
Answer:
x<=-2.75
Step-by-step explanation:
Samantha is designing a new three-dimensional puzzle in the form of a square pyramid. She knows that the design and the general formula for finding the volume are as follows, where h is the height and a is the side length of the base. All of her measurements are in centimeters.
An illustration of a pyramid, with the height of h and width a.
Samantha's puzzle must have a certain volume and height, but she needs to determine the side length of the base to complete the design. Which equation can Samantha use to find the side length of the base in terms of the volume and height of her puzzle?
An equation which Samantha can use to find the side length of the base in terms of the volume and height of her puzzle is: A. a = √(3V/h)
How to calculate the volume of a square pyramid?In Mathematics and Geometry, the volume of a square pyramid can be calculated by using the following formula:
Volume of a square pyramid, V = 1/3 × a² × h
Where:
h represent the height of a square pyramid.a represent the side length of the base of a square pyramid.By making "a" the subject of formula, the side length of the base of a square pyramid can be determined as follows;
V = 1/3 × a² × h
3V = a²h
a² = 3V/h
By taking the square root of both sides of the equation, we have:
a = √(3V/h)
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PLS HURRY I AM GIVING BRAINLIEST!!!
the question is in the photo!!
A. Area of the rectangle, in standard form is: 14x² + 19x - 3
B. The degree of a polynomial is: 4. It has 4 terms (4x², 3y, -6x^4y, and 4.); C ALWAYS result in another polynomial.
What is a Polynomial?A polynomial is a mathematical expression consisting of variables and coefficients that involves only the operations of addition, subtraction, and multiplication of non-negative integer exponents.
Part A:
Area of the rectangle = (7x - 1)(2x + 3) = 7x(2x + 3) -1(2x + 3)
= 14x² + 21x - 2x - 3
Combine like terms:
Area of the rectangle = 14x² + 19x - 3 (standard form)
Part B: Given the expression, 4x² + 3y - 6x^4y + 4,
The degree of a polynomial is the highest exponent of its variable.
Therefore, the degree is 4.
It has 4 terms, which are: 4x², 3y, -6x^4y, and 4.
Part C: When two polynomials are subtracted, multiplied, or added, it will ALWAYS result in another polynomial.
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PLS HELP WILL GIVE BRAINLIEST
what will the coordinates be if Point A (-4,-5) of the trapezoid was reflected across the y-axis and shifted -6 units in the x direction and 7 units in the y direction.
Answer:
If point A (-4,-5) of the trapezoid is reflected across the y-axis, the x-coordinate will change sign while the y-coordinate remains the same. This gives us the point A' (4, -5).
Next, if we shift the point A' by -6 units in the x-direction and 7 units in the y-direction, we get the new point A":
A" = (4 - 6, -5 + 7) = (-2, 2)
Therefore, the coordinates of point A after it is reflected across the y-axis and shifted -6 units in the x direction and 7 units in the y direction are (-2, 2).
Step-by-step explanation:
Answer:
(-2, 2)
Step-by-step explanation:
I did the test
Hope this helps :)
The students in Ms. Deans chemistry class are exploring chemistry by creating milk art with food coloring, milk, and dish soap. Ms. Dean plans to give each of her 28 students 0.5 cups of milk. How many pints of milk does Ms. Dean need
Solve for x. −−9x+5<17 AND13x+25<−1
Answer: x<4/3
Step-by-step explanation: 17-5=12/9
Roger can finish his math homework in 6 hours. Trish can finish the same homework in 5 hours. What part of the homework will Roger and Trish finish if they work together for 1 hours?
Roger and Trish will have completed 11/30 of the homework if they work together for 1 hour.
To solve this problem
First, we need to find their individual rates of work, which is the reciprocal of the time it takes for each to complete the homework:
Roger's rate: 1/6 of the homework per hour
Trish's rate: 1/5 of the homework per hour
When they work together, their combined rate of work is the sum of their individual rates:
Combined rate: 1/6 + 1/5 = 11/30 of the homework per hour
After working together for 1 hour, they will have completed:
11/30 * 1 = 11/30 of the homework
Therefore, Roger and Trish will have completed 11/30 of the homework if they work together for 1 hour.
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AC has endpoints A (3,6) and C (-2, 10). What is the length of AC ?
The number of bacteria present after 13 hours
The number of bacteria present after 13 hours is 5,306.
What is the number of bacteria after 13 hours?
The number of bacteria present after 13 hours is calculated by applying half-life equation.
N = N₀ x 2^(t / d)
Where:
N₀ is the initial number of bacteria (3,000)N is the final number of bacteria we want to findt is the time elapsed d is the doubling time of the bacteria populationSince the bacteria population grew from 3,000 to 3,600 in 6 hours, the value of d is calculated as follows;
3,600 = 3,000 x 2^(6 / d)
3600/3000 = 2^(6 / d)
1.2 = 2^(6 / d)
log2 (1.2) = 6/d
0.38 = 6/d
d = 6/0.38
d = 15.8 hours
The number of bacteria after 13 hours is calculated as follows;
N = 3,000 x 2^(13 / 15.8)
N = 5,306.5 ≈ 5,306
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The complete question is below:
A culture started with 3,000 bacteria. After 6 hours, it grew to 3,600 bacteria. Predict how many bacteria will be present in 13 hours. round your answer to the nearest whole number.
Express the following probability as a simplified fraction and as a decimal.
If one person is selected from the population described in the table, find the probability that the person
graph the cirlces (x+1)+(y+2)=9
Center :
Note that the graph of the function ((x+1)²+(y+2)²=9) is attached accordingly
What is a graph?A coordinate graph is one that has two axes that are perpendicular to each other. These two axes are known as the x- and y-axes. The y-axis is represented by the vertical line, while the x-axis is represented by the horizontal line. A coordinate plane, a Cartesian plane, and a Cartesian coordinate system are other names for it.
The given circle has a radius of 3 units and centre is (-1,-2).
This circle can thus be drawn by first plotting the point (-1,-2) and then with this point as centre draw a circle of radius 3.
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Full question:
graph the circle
((x+1)²+(y+2)²=9)
The members of the Junior National Honors Society at Franklin Middle School are required to complete community service hours. The box plots show the volunteer service hours performed by the 7th and 8th graders. 7th Graders 8th Graders 3.5 4 4.5 Community Service 5 5.5 6 6.5 7 7.5 Number of Hours Which statement is best supported by the information in the box plots?
Answer:
The statement that is best supported by the information in the box plots is:
* The 8th graders completed more community service hours than the 7th graders.
The 8th grade box plot is shifted to the right of the 7th grade box plot, which indicates that the 8th graders had a higher median number of community service hours. The 8th grade box plot also has a wider spread, which indicates that there was more variation in the number of community service hours completed by the 8th graders.
However, it is important to note that the box plots do not provide information about the total number of community service hours completed by each group of students. It is possible that some 7th graders completed more community service hours than some 8th graders.
Step-by-step explanation:
Answer:The 8th graders completed more community service hours than the 7th graders
Step-by-step explanation:
Help please!!!!
Whoever answers right gets brainliest!
I don’t get 4 and 5 so can someone please help me
a) We are given that f(x) = x^3 - 10x^2 + 19x + 30 and x - 6 is a factor of f(x). Using long division or synthetic division, we can divide f(x) by x - 6 to get the other factor(s):
6 │ x^3 - 10x^2 + 19x + 30
x^3 - 6x^2
-------------
-4x^2 + 19x
-(-4x^2 + 24x)
---------------
5x + 30
5x - 30
-------
0
Therefore, we can write:
f(x) = (x - 6)(x^2 - 4x + 5)
The quadratic factor x^2 - 4x + 5 cannot be factored further using real numbers.
b) We are given that f(x) = x^3 + 6x^2 + 5x - 12 and x + 4 is a factor of f(x). Using long division or synthetic division, we can divide f(x) by x + 4 to get the other factor(s):
-4 │ x^3 + 6x^2 + 5x - 12
-x^3 - 4x^2
--------------
2x^2 + 5x
2x^2 + 8x
------------
-3x - 12
-3x - 12
--------
0
Therefore, we can write:
f(x) = (x + 4)(x^2 + 2x - 3)
The quadratic factor x^2 + 2x - 3 can be factored further using the product-sum method:
x^2 + 2x - 3 = (x + 3)(x - 1)
Therefore, we have:
f(x) = (x + 4)(x + 3)(x - 1)
The nine members of a coed intramural volleyball team are to be randomly selected from nine college men and ten college women. To be classified as coed the team must include at least one player of each gender. What is the probability the selected team includes more women than men?
The probability that selected team includes more women then men is [tex]\frac{Σ^{8} _{n=5} (^{10} _{n} )(^{9} _{9-n} )}{(^{19} _{9} ) - (^{10} _{9} ) - (^{9} _{9} )}[/tex]
The likelihood that the selected team contains more women than males matches the probability that the team contains 5, 6, 7, or 8 women (there would therefore be 4, 3, 2, or 1 man in the squad, respectively).
The likelihood of there being 5 women on the team is equal to
[tex]\frac{(^{10} _{5} )(^{9} _{4} )}{(^{19} _{9} ) - (^{10} _{9} ) - (^{9} _{9} )}[/tex]
Since, we can choose 5 women out of 10 in [tex](^{10} _{5} )[/tex] ways, 4 men out of 9 in [tex](^{9} _{4} )[/tex] ways, and 9 persons out of 19 in [tex](^{19} _{9})[/tex] - [tex](^{10} _{9})[/tex] - [tex](^{9} _{9})[/tex] ways (we must reject the possibility of having all men ( [tex](^{9} _{9})[/tex] ways) or all women ( [tex](^{10} _{9})[/tex] ways) because it is a must).
Similarly, we compute the odds of having 6, 7, or 8 women on the team, and because these are disjoint alternatives, the final result is the total of their probabilities, which is
[tex]\frac{Σ^{8} _{n=5} (^{10} _{n} )(^{9} _{9-n} )}{(^{19} _{9} ) - (^{10} _{9} ) - (^{9} _{9} )}[/tex]
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