The model of the total cost of the Family Plan will be 90 + 30P.
How to illustrate the information?It should be noted that Family Plan has $90 monthly fee, unlimited talk and text on up to 5 lines, and data charges of $30 for each device for up to 2 GB of data per device.
Therefore, the model will be:
= 90 + 30P
where p = number of devices
In conclusion, the model of the total cost of the Family Plan will be 90 + 30P.
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Let [tex]\alpha[/tex] and [tex]\beta[/tex] be real number such that [tex] - \frac{\pi}4 < \beta < 0 < \alpha < \frac{\pi}{4} .[/tex] If [tex]\sin( \alpha + \beta ) = \frac{1}{3}[/tex] and [tex]\cos( \alpha - \beta ) = \frac{2}{3}[/tex] , then the greatest integer less than or equal to [tex] \bigg( \frac{ \sin( \alpha ) }{ \cos( \alpha ) } + \frac{ \cos( \beta ) }{ \sin( \alpha ) } + \frac{ \cos( \alpha ) }{ \sin( \beta ) } + \frac{ \sin( \beta ) }{ \cos( \alpha ) } \bigg) {}^{2} \\[/tex] is
Step-by-step explanation:
We have,
[tex]\begin{gathered} \bullet \: \bold{ - \dfrac{\pi}{4} < \beta < 0 < \alpha < \dfrac{\pi}{4} } \\ \\ \implies \: - \dfrac{\pi}{4} < \alpha + \beta < \dfrac{\pi}{4} \end{gathered}[/tex]
[tex]\begin{gathered} \rm\bullet \: \: \: \sin( \alpha + \beta ) = \dfrac{1}{3} \: \: \: \: \: and \: \: \: \: \: \cos( \alpha - \beta ) = \dfrac{2}{3} \\ \end{gathered} [/tex]
Now,
[tex]\begin{gathered} y= \bigg( \dfrac{ \sin( \alpha ) }{ \cos( \beta ) } + \dfrac{ \cos( \beta ) }{ \sin( \alpha ) } + \dfrac{ \cos( \alpha ) }{ \sin( \beta ) } + \dfrac{ \sin( \beta ) }{ \cos( \alpha ) } \bigg)^{2} \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \bigg( \dfrac{ \sin( \alpha ) }{ \cos( \beta ) } + \dfrac{ \sin( \beta ) }{ \cos( \alpha ) } + \dfrac{ \cos( \beta ) }{ \sin( \alpha ) } + \dfrac{ \cos( \alpha ) }{ \sin( \beta ) } \bigg)^{2} \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \bigg( \dfrac{ \sin( \alpha ) \cos( \alpha ) + \sin( \beta \cos( \beta ) ) }{ \cos( \beta ) \cos( \alpha) } + \dfrac{ \sin( \alpha) \cos( \alpha ) + \sin( \beta ) \cos( \beta ) }{ \sin( \alpha ) \sin( \beta ) } \bigg)^{2} \\\end{gathered}[/tex]
[tex]\begin{gathered} \implies y= \bigg( \dfrac{ \sin( \alpha + \beta ) }{ \cos( \beta ) \cos( \alpha) } + \dfrac{ \sin( \alpha + \beta ) }{ \sin( \alpha ) \sin( \beta ) } \bigg)^{2} \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \sin^{2} ( \alpha + \beta ) \bigg( \dfrac{ 1 }{ \cos( \beta ) \cos( \alpha) } + \dfrac{ 1 }{ \sin( \alpha ) \sin( \beta ) } \bigg)^{2} \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \sin^{2} ( \alpha + \beta ) \bigg( \dfrac{\cos( \beta ) \cos( \alpha) + \sin( \alpha ) \sin( \beta ) }{ \cos( \beta ) \cos( \alpha) \sin( \alpha ) \sin( \beta )} \bigg)^{2} \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \sin^{2} ( \alpha + \beta ) \bigg( \dfrac{\cos( \alpha - \beta ) }{ \cos( \beta ) \cos( \alpha) \sin( \alpha ) \sin( \beta )} \bigg)^{2} \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \dfrac{4\sin^{2} ( \alpha + \beta ) \cdot\cos^{2} ( \alpha - \beta ) }{ \left \{2\cos( \alpha ) \cos( \beta ) \cdot 2\sin( \alpha ) \sin( \beta ) \right \}^{2} } \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \dfrac{4\sin^{2} ( \alpha + \beta ) \cdot\cos^{2} ( \alpha - \beta ) }{ \left \{\cos( \alpha + \beta ) + \cos( \alpha + \beta ) \right \} ^{2} \left \{ \cos( \alpha - \beta ) - \cos( \alpha + \beta ) \right \}^{2} } \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \dfrac{4\sin^{2} ( \alpha + \beta ) \cdot\cos^{2} ( \alpha - \beta ) }{ \left \{ \cos^{2} ( \alpha - \beta ) - \cos^{2} ( \alpha + \beta ) \right \}^{2} } \\\end{gathered}[/tex]
[tex]\begin{gathered} \implies y= \dfrac{4\sin^{2} ( \alpha + \beta ) \cdot\cos^{2} ( \alpha - \beta ) }{ \left \{ \cos^{2} ( \alpha - \beta ) - 1 + \sin^{2} ( \alpha + \beta ) \right \}^{2} } \\\end{gathered}[/tex]
Putting the values given above, we get,
[tex]\begin{gathered} \implies y= \dfrac{4 \cdot \dfrac{1}{9} \cdot\dfrac{4}{9} }{ \left \{ \dfrac{4}{9} - 1 + \dfrac{1}{9} \right \}^{2} } \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \dfrac{\dfrac{16}{81} }{ \left \{ \dfrac{5}{9} - 1\right \}^{2} } \\\end{gathered}[/tex]
[tex]\begin{gathered} \implies y= \dfrac{\dfrac{16}{81} }{ \left \{ \dfrac{5 - 9}{9}\right \}^{2} } \\\end{gathered} [/tex]
[tex]\begin{gathered} \implies y= \dfrac{\dfrac{16}{81} }{ \left \{ \dfrac{- 4}{9}\right \}^{2} } \\\end{gathered}[/tex]
[tex]\begin{gathered} \implies y= \dfrac{\dfrac{16}{81} }{ \dfrac{16}{81}} \\\end{gathered} [/tex]
[tex]⟹y=1[/tex]
What property is 3x=x3
Step-by-step explanation:
So, the expression “three times the variable x” can be written in a number of ways: 3x, 3(x), or 3 · x. Use the distributive property to expand the expression 9(4 + x).
9.949 round to the nearest tenth and hundredth
9.9 and 9.95 are the result of rounding off 9.949 to the nearest tenth and hundredth.
What is Rounding off?When a number is rounded off, its value is maintained but is brought closer to the next number, simplifying the number. For whole numbers as well as decimals at different places of hundreds, tens, tenths, etc., it is done. An integer with one or more "0"s at the end in a specific base is said to be round. In this way, 590 is more rounded than 592 but less rounded than 600. A round number is frequently understood to stand for a value or values close to the nominal value expressed in both formal and informal language.So, rounding off:
9.949 (nearest tenth) = 9.99.949 (nearest hundredth) = 9.95Therefore, rounding off of 9.949 to the nearest tenth and hundredth is 9.9 and 9.95.
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Line AB contains points A(4, 5) and B(9, 7). What is the slope of ?
– negative StartFraction 5 Over 2 EndFraction
– negative StartFraction 2 Over 5 EndFraction
StartFraction 2 Over 5 EndFraction
StartFraction 5 Over 2 EndFraction
The slope of line AB with points A(4, 5) and B(9, 7) is 2/5.
What is slope?The slope or gradient of a line is a number that describes both the direction and the steepness of the line.
We have,
A(4, 5) and B(9, 7)
The slope of a line with points A and B is given by:
= d - b / c - a
Where A(a, b) and B(c, d) are the coordinates of the points.
We have the points:
A(4, 5) = (a, b)
B(9, 7) = (c, d)
The slope of the line AB:
= (7 - 5) / (9 - 4) = 2 / 5
Therefore the slope of line AB with points A(4, 5) and B(9, 7) is 2/5.
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Leslie can type 56 words per minute. Each page of a report contains an average of 420 words. How many pages of the report can Leslie type in one hour?
If Leslie is about to type 56 words per minute, she would be able to type 8 pages in one hour
How many words can Leslie type in one hour?
The fact that Leslie can type 56 words per minute means that he is able, means that the number of words she is able to type in one hour is determined as 56 words multiplied 60 minutes which make an hour
number of words in one hour=56*60
number of words in one hour=3360
The number of pages typed is determined as the 3360 words typed in one hour divided by the number of words in a page
number of pages type=3360/420
number of pages typed=8 pages
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A recent student poll showed that 18% of the
high school students are in a music class.
What is the ratio of high school students
who are in a music class to students who
are not? (Example 2)
The ratio of high school students who are in a music class to students who are not is 9 : 41.
What do we mean by ratio?A ratio in mathematics indicates how many times one number contains another. For example, if a bowl of fruit contains eight oranges and six lemons, the orange-to-lemon ratio is eight to six. Similarly, the lemon to orange ratio is 6:8, and the orange to total fruit ratio is 8:14.So, 18% of high school students are in a music class which means the rest 82% are not in a music class.
Then the ratio:
18 : 82Simplest form:
18/82 = 9/41 = 9 : 41Therefore, the ratio is 9 : 41.
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Juan and his children went into a restaurant where they sell drinks for $2 each and tacos for $4 each. Juan has $40 to spend and must buy a minimum of 11 drinks and tacos altogether. If xx represents the number of drinks purchased and yy represents the number of tacos purchased, write and solve a system of inequalities graphically and determine one possible solution.
The system of inequalities for the given situation is x+y≤11 and 2x+4y≥40.
Given that, the cost of each drink = $2 and the cost of each taco = $4.
What is a system of inequalities?A system of inequalities is a set of two or more inequalities in one or more variables. Systems of inequalities are used when a problem requires a range of solutions, and there is more than one constraint on those solutions.
x represents the number of drinks purchased and y represents the number of tacos purchased.
Juan has $40 to spend and must buy a minimum of 11 drinks and tacos altogether.
So inequalities are x+y≤11 and 2x+4y≥40
Therefore, the system of inequalities for the given situation is x+y≤11 and 2x+4y≥40.
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5. The Cupcake Café makes 4 and 1/2 times as much revenue on doughnuts as muffins. If total sales were $44,000 for May, what dollar amount of each was sold?
Answer:
Step-by-step explanation:
According to the question, the cupcake cafe makes 1/2 times as much revenue on doughnuts as muffins and the total revenue is total sales were $44,000. Let us assume that the revenue on muffins is X and then the revenue on doughnuts is 412X. 4 1 2 X . The amount of money earned from muffins is 20000 dollars
2) What is the area of figure ABCD, in
square centimeters?
8 cm.
6 cm
6 cm
B
15 cm
Answer:
138cm^2
Step-by-step explanation:
Trianle area = (b*h)/2
b = 8cm + 15cm
h = 6cm
Triangle area = (23*6)/2
138/2 = 69cm^2
Upper triangle area
--------------------------------
since both triangles share the same base and height
69*2 = 138cm^2
If 8x + 7y = 6 is a true equation, what
would be the value of 5 + 8x + 7y?
Answer:
11
Step-by-step explanation:
Based on the graph, which statement could describe Janelle’s trip home from school?
Based on the graph, which statement could describe Janelle’s trip home from school is that D. Janelle rode the bus to the bus stop, talked with a friend, and then walked home.
What is a graph?It should be noted that a graph is a diagram that is used to represent a system of connections or interrelations that is among two or more things by a number of distinctive dots, bars, etc
Therefore, based on the graph, which statement could describe Janelle’s trip home from school is that Janelle rode the bus to the bus stop, talked with a friend, and then walked home.
In conclusion, the correct option is D.
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Based on the graph, which statement could describe Janelle's trip home from school?
Janelle waited for the bus, rode the bus, and then walked home.
Janelle walked the opposite direction from home to the library, rode the bus, and then walked to a friend's house.
Janelle walked home at a constant speed.
Janelle rode the bus to the bus stop, talked with a friend, and then walked home
Eula needs to buy binders that cost $4 each and notebooks that cost $2 each. She has $20. The graph of the inequality 4x + 2y ≤ 20, which represents the situation, is shown. What is the greatest number of binders Eula can buy? What is the greatest number of notebooks Eula can buy? If Eula buys 7 notebooks, what is the greatest number of binders she can buy?
The greatest number of binders Eula can buy is 5
The greatest number of notebooks Eula can buy is 10
The greatest number of binders she can buy if Eula buys 7 notebooks is 3/2
What is inequality?It is an order relationship that is greater than, greater than, or equal to, less than, or less than or equal to—between two numbers or algebraic expressions.
We have,
4x + 2y ≤ 20
x = number of binders
y = number of notebooks
The greatest number of binders Eula can buy:
Put y = 0.
4x + 2 x 0 ≤ 20
4x ≤ 20
x ≤ 20/4
x ≤ 5
The greatest number of notebooks Eula can buy:
Put x = 0.
4x + 2y ≤ 20
4 x 0 + 2y ≤ 20
2y ≤ 20
y ≤ 10
Eula buys 7 notebooks then, the greatest number of binders she can buy:
4x + 2y ≤ 20
4x + 2 x 7 ≤ 20
4x + 14 ≤ 20
4x ≤ 20 - 14
4x ≤ 6
x ≤ 6/4
x ≤ 3/2
Thus,
The greatest number of binders Eula can buy is 5
The greatest number of notebooks Eula can buy is 10
The greatest number of binders she can buy if Eula buys 7 notebooks is 3/2
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Answer:
Eula needs to buy binders that cost $4 each and notebooks that cost $2 each. She has $20. The graph of the inequality 4x + 2y ≤ 20, which represents the situation, is shown.What is the greatest number of binders Eula can buy? What is the greatest number of notebooks Eula can buy? If Eula buys 7 notebooks, what is the greatest number of binders she can buy? ⇒ 1
Step-by-step explanation:
Which answer is it? I need an answer asap!!
Answer:
[tex]2(35)+4=74[/tex]
∠SQR = 74
Step-by-step explanation:
[tex](2m+4)+(3m+1)=180\\5m+5=180\\ -5\\5m=175\\/5\\175/5=35[/tex]
Answer:
SQR = 74
Step-by-step explanation:
2m + 4 + 3m + 1 = 180
5m + 5 = 180
5m = 175
m = 35
35 * 2 + 4
70 + 4 = 74
Angle SQR = 74
Use prime factors to find the square root of 777924.
Using prime factors the root of given variety of 777924 is 882 .
The root of any variety is adequate to variety, that once square offers the first variety.The process of writing variety because the product of prime numbers is prime factorization. Prime numbers area unit the numbers that have solely 2 factors, one and therefore the variety itself.The image of prime factorization of 777924 is given below
By prime factorization of 777924 we have a tendency to follow five easy steps:
1. we have a tendency to write variety 777924 higher than a 2-column table
2. we have a tendency to divide 777924 by the tiniest attainable factor
3. we have a tendency to write down on the left facet of the table the factor and next variety to factorize on the ride facet
4. we have a tendency to still consider this fashion (we subsume odd numbers by making an attempt little prime factors)
5. we have a tendency to continue till we have a tendency to reach one on the ride facet of the table
Prime factorization of 777924 = 1×2×2×3×3×3×3×7×7×7×7
On doing square root , we get
[tex]\sqrt{777924} =\sqrt{ 1\times2\times2\times3\times3\times3\times3\times7\times7\times7\times7}[/tex]
= 882
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A yard has a perimeter of 400 feet. If eight times the length of the yard equals seventeen times the width
The length and width of the yard with perimeter of 400 feet are 136 ft and 64 ft respectively.
How to find perimeter of a rectangular yard?The yard has a perimeter of 400 feet.
Eight times the length of the yard equals seventeen times the width.
Therefore,
perimeter of a rectangular yard = 2(l + w)
where
l = lengthw = widthTherefore,
8l = 17w
Hence,
l = 17 / 8 w
perimeter of a rectangular yard = 2(17 / 8 w + w)
perimeter of a rectangular yard = 2( 25 /8 w)
perimeter of a rectangular yard = 50/ 8 w
400 = 50 / 8 w
cross multiply
3200 = 50w
divide both sides by 50
w = 3200 / 50
w = 64 ft
l = 17 / 8 × 64 = 136 ft
Therefore, the length and width of the yard with perimeter of 400 feet are 136 ft and 64 ft respectively.
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what’s the answer????
Answer:Domar range x
Step-by-step explanation:
A farmer has 150 yards of fencing to place around a rectangular garden. The fence will have an opening that is 1/3 of the garden's length. Write a function A(x) that describe the area of the garden, where x is the length of the garden. Find the dimensions if that has a maximum area, and find the maximum area
The garden has a length of 37.5 yards and width of 37.5 yards as well as an opening of 12.5 yards.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let x represent the length and y represent the width.
150 yards of fencing is available, hence:
2(x + y) = 150
x + y = 75
y = 75 - x (1)
The area (A) of the garden is given as:
A = xy
A = x(75 - x)
A = 75x - x²
The maximum area is at A' = 0
A' = 75 - 2x
75 - 2x = 0
x = 37.5 yards
y = 75 - x = 75 - 37.5 = 37.5
Opening of the garden = 1/3 * 37.5 = 12.5 yards
The garden has a length of 37.5 yards and width of 37.5 yards as well as an opening of 12.5 yards.
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what is the circumference of a circle whose diameter is 49m
Answer:
The circumference is 153.93804... or 153.94...
Step-by-step explanation:
Hope it helps! =D
Answer:
153.94 meters.
Step-by-step explanation:
The formula for finding the circumference of a circle is C = πd, where d is the diameter of the circle. With a diameter of 49m, the circumference can be calculated as C = π(49) ≈ 153.94m. Therefore, the circumference of the circle is approximately 153.94 meters.
Select the correct answer. What is the solution to |2x − 8| < 2? A. 3 < x < 5 B. -5 < x < -3 C. x > 5 or x < 3 D. x > -3 or x < -5 Reset Next
Answer:
3 < x < 5
Step-by-step explanation:
Expression is |2x − 8| < 2
The absolute rule says that if |u| < a then -a < u < a
Here u = 2x-8
So we get -2 < 2x - 8 < 2
This means
2x - 8 > -2
==> 2x > -2 + 8 (add 8 to both sides)
==> 2x > 6 (simplify)
==> x > 3 (divide by 2 both sides)
and,
2x - 8 < 2 gives
==> 2x < 2 + 8 (add 8 to both sides)
==> 2x < 10 (simplify)
==> x < 5 (divide by 2 both sides)
So the solution to |2x − 8| < 2
is 3 < x < 5
I really need help with this question can anyone please help me thank you
Answer:
between 4 and 5
Step-by-step explanation:
Solve the equation, 13|x−8|=10. Select each correct answer. Responses x=−30 x equals negative 30 x=−22 x equals negative 22 x=103 x equals 10 over 3 x = 8 x, = 8 x = 30 x, = 30 x = 38 x, = 38
The solution to the equation 1/3|x - 8| = 10 is x = 38
What are expressions?Expressions are mathematical statements that are represented by variables, coefficients and operators
How to determine the solution to the equation?The equation is given as
1/3|x - 8| = 10
Multiply through by 3
So, we have
|x - 8| = 30
Remove the absolute bracket
So, we have
x - 8 = 30
Add 8 to both sides of the equation
So, we have
x = 38
Hence, the solution to the equation is x = 38
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cynthia besch wants to buy a rug for a room that is 25ft wide and 33ft long. She wants to leave a uniform strip of floor around the rug. she can afford to buy 513 square feet of carpeting. what dimensions should the rug have
The dimensions of the rug would be 27 feet long and 19 feet wide with a 6 foot space around the carpet uniformly.
How to find the dimension of the rug?She wants to buy a rug for a room that is 25 ft wide and 33 ft long.
She wants to leave a uniform strip of floor around the rug.
she can afford to buy 513 square feet of carpeting.
Cynthia room area = lw
where
l = lengthw = widthTherefore,
Cynthia room area = 25 × 33
Cynthia room area = 825 ft²
She wants to leave a uniform strip of floor all around the rug and has affordability to buy 513 square feet of carpet.
Therefore, the dimension the rug should have is as follows:
(25 - 6) (33 - 6) = 513
19 × 27 = 513 ft²
Therefore, the dimensions of the rug would be 27 feet long and 19 feet wide with a 6 foot space around the carpet uniformly.
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What is the base 10 representation of 142^5 (142 in base 5)? I know it's 47, but I need to know why. None of the answers to this question have an actual explanation, they all just say "Because I did the assignment earlier and it's 47".
The base 10 representation of the number 142 base 5 as required to be determined in the task content is; 47.
What is the base 10 representation of the number 142 which is to base 5 as required in the task content?Since, the conversion of numbers from any base to base 10 requires imaginary exponents to represent place value in such number as follows;
142₅ can therefore be written as follows; 1²4¹2⁰₅.
On this note the evaluation is carried out by summing the product of each digits and 5 to the corresponding power as follows;
1²4¹2⁰₅ = (2 × 5⁰) + (4 × 5¹) + (1 × 5²)
= (2 × 1) + (4 × 5) + (1 × 25)
= 2 + 20 + 25.
= 45.
Ultimately, the base 10 representation of the number 142 base 5 as required to be determined in the task content is; 47.
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11)Dan’s school is selling tickets to the spring musical. On the first day of ticket sales, the school sold 8 senior citizen tickets and 12 child tickets for a total of $264. The school took in $237 on the second day by selling 11 senior citizen tickets and 6 child tickets. Find the price of each type of ticket. A)Define your variables. C)Solve the system using a method of your choice. State your final answer in a complete sentence.
If the school earns $264 for 8 senior tickets and 12 child tickets and $237 for 11 senior tickets and 6 child tickets then the price of 1 senior ticket be $15 and the price of child ticket be $12.
Given that on the first day of ticket sales, the school sold 8 senior citizen tickets and 12 child tickets for a total of $264 and the school earns $237 on IInd day by selling 11 senior citizen tickets and 6 child tickets.
We are required to define the variables and solve the system of the equations.
Suppose the price of 1 ticket of senior citizen be x.
Suppose the price of 1 ticket of child be y.
The equations will be:
8x+12y=264--------1
11x+6y=237--------2
Multiply equation 1 by 11 and multiply equation 2 by 8 and then subtract equation 2 from equation 1.
88x+132y-88x-48y=2904-1896
84y=1008
y=1008/84
y=12
Use the value of y in equation 1 to get the value of x.
8x+12y=264
8x+12*12=264
8x+144=264
8x=264-144
8x=120
x=120/8
x=15
Hence if the school earns $264 for 8 senior tickets and 12 child tickets and $237 for 11 senior tickets and 6 child tickets then the price of 1 senior ticket be $15 and the price of child ticket be $12.
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The cost of a Senior citizen ticket is $15, while a children's ticket cost $12.
On the first day:
The school sold 12 child tickets and 8 senior citizen tickets for a total amount of $264.
On the second day:
The school sold 11 senior citizen tickets and 6 child tickets for a total of $237.
Let A be the price of a senior citizen ticket and B be the price of a child ticket.
So, the equation for the first day:
8A + 12B = 264
The equation for the second day,
11A + 6B = 237
Multiplying the equation for the second day by 2 and subtracting the equation for the first day.
We get,
22A + 12B - 8A - 12B = 474 - 264
14A = 210
A = 15
Substituting the value of A in the equation 8A + 12B = 264,
8A + 12B = 264
8(15) + 12B = 264
120 + 12B = 264
12B = 144
B = 12
Therefore, the price of a senior citizen ticket is $15 and the price of a child ticket is $12.
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Please help me geometry
[tex] \large \bf \implies \angle{BAC} = 40 \degree[/tex]
Step-by-step explanation :[tex] \bf \implies \angle{ABC} + \angle{BAC} = 90\degree[/tex] [The acute angles of a right triangle are complementary]
[tex] \bf \implies 50x + 40x = 90\degree[/tex]
Substitute :
[tex]\angle{BAC} = 40x \: \: , \: \: \angle{ABC} = 50x \: into \: \angle{ABC} + \angle{BAC} = 90\degree[/tex]
[tex]\sf{x = 1}[/tex]Calculate 50x + 40x = 90° ↑
[tex]\sf{\angle{BAC} = 40}[/tex]Substitute x = 1 into [tex]\bf{\angle{BAC} = 40x}[/tex] ↑
[tex] \boxed{ \bold{\angle{BAC} = 40} }\: \mathfrak{ans.}[/tex]
What is the least possible degree of the polynomial graphed above?
Answer:
2
Step-by-step explanation:
The graph looks to be like a parabola. Due to the end behaviour of the graph (Up in both quadrants 1 and 2) it can be said that the graph has a positive leading coefficient.
Hope that helps
For [tex]\rm x \in \mathbb{R}[/tex], let the function y(x) be the solution of the differential equation
[tex] \rm \frac{dy}{dx} + 12y = \cos \bigg( \frac{\pi}{12}x \bigg ) , \: \: \: \: y(0) = 0 \\ [/tex]
Then, which of the following statements is/are TRUE?
(A) y(x) is an increasing function
(B) y(x) is a decreasing function
(C) There exists a real number β such that the line y = β intersects the curve y = y(x) at infinitely many points
(D) y(x) is a periodic function
In the differential equation
[tex]\dfrac{dy}{dx} + 12y = \cos\left(\dfrac{\pi x}{12}\right)[/tex]
multiply on both sides by the integrating factor
[tex]\mu = \exp\left(\displaystyle\int12\,dx\right) = e^{12x}[/tex]
Then the left side condenses to the derivative of a product.
[tex]e^{12x} \dfrac{dy}{dx} + 12 e^{12x} y = e^{12x} \cos\left(\dfrac{\pi x}{12}\right)[/tex]
[tex]\dfrac{d}{dx}\left[e^{12x}y\right] = e^{12x}\cos\left(\dfrac{\pi x}{12}\right)[/tex]
Integrate both sides with respect to [tex]x[/tex], and use the initial condition [tex]y(0)=0[/tex] to solve for the constant [tex]C[/tex].
[tex]\displaystyle \int \frac{d}{dx} \left[e^{12x}y\right] \, dx = \int e^{12x} \cos\left(\dfrac{\pi x}{12}\right) \, dx[/tex]
As an alternative to integration by parts, recall
[tex]e^{ix} = \cos(x) + i \sin(x)[/tex]
Now
[tex]e^{12x} \cos\left(\dfrac{\pi x}{12}\right) = e^{12x} \mathrm{Re}\left(e^{i\pi x/12}\right) = \mathrm{Re}\left(e^{(12+i\pi/12)x}\right)[/tex]
[tex]\displaystyle \int \mathrm{Re}\left(e^{(12+i\pi/12)x}\right) \, dx = \mathrm{Re}\left(\int e^{(12+i\pi/12)x} \, dx\right)[/tex]
[tex]\displaystyle. ~~~~~~~~ = \mathrm{Re}\left(\frac1{12+i\frac\pi{12}} e^{(12+i\pi/12)x}\right) + C[/tex]
[tex]\displaystyle. ~~~~~~~~ = \mathrm{Re}\left(\frac{12 - i\frac\pi{12}}{12^2 + \frac{\pi^2}{12^2}} e^{12x} \left(\cos\left(\frac{\pi x}{12}\right) + i \sin\left(\frac{\pi x}{12}\right)\right)\right) + C[/tex]
[tex]\displaystyle. ~~~~~~~~ = \frac{12}{12^2 + \frac{\pi^2}{12^2}} e^{12x} \cos\left(\frac{\pi x}{12}\right) + \frac\pi{12} e^{12x} \sin\left(\frac{\pi x}{12}\right) + C[/tex]
[tex]\displaystyle. ~~~~~~~~ = \frac1{12(12^4+\pi^2)} e^{12x} \left(12^4 \cos\left(\frac{\pi x}{12}\right) + \pi (12^4+\pi^2) \sin\left(\frac{\pi x}{12}\right)\right) + C[/tex]
Solve for [tex]y[/tex].
[tex]\displaystyle e^{12x} y = \frac1{12(12^4+\pi^2)} e^{12x} \left(12^4 \cos\left(\frac{\pi x}{12}\right) + \pi (12^4+\pi^2) \sin\left(\frac{\pi x}{12}\right)\right) + C[/tex]
[tex]\displaystyle y = \frac1{12(12^4+\pi^2)} \left(12^4 \cos\left(\frac{\pi x}{12}\right) + \pi (12^4+\pi^2) \sin\left(\frac{\pi x}{12}\right)\right) + C[/tex]
Solve for [tex]C[/tex].
[tex]y(0)=0 \implies 0 = \dfrac1{12(12^4+\pi^2)} \left(12^4 + 0\right) + C \implies C = -\dfrac{12^3}{12^4+\pi^2}[/tex]
So, the particular solution to the initial value problem is
[tex]\displaystyle y = \frac1{12(12^4+\pi^2)} \left(12^4 \cos\left(\frac{\pi x}{12}\right) + \pi (12^4+\pi^2) \sin\left(\frac{\pi x}{12}\right)\right) - \frac{12^3}{12^4+\pi^2}[/tex]
Recall that
[tex]R\cos(\alpha-\beta) = R\cos(\alpha)\cos(\beta) + R\sin(\alpha)\sin(\beta)[/tex]
Let [tex]\alpha=\frac{\pi x}{12}[/tex]. Then
[tex]\begin{cases} R\cos(\beta) = 12^4 \\ R\sin(\beta) = 12^4\pi+\pi^3 \end{cases} \\\\ \implies \begin{cases} (R\cos(\beta))^2 + (R\sin(\beta))^2 = R^2 = 12^8 + (12^4\pi + \pi^3)^2 \\ \frac{R\sin(\beta)}{R\cos(\beta)}=\tan(\beta)=\pi+\frac{\pi^3}{12^4}\end{cases}[/tex]
Whatever [tex]R[/tex] and [tex]\beta[/tex] may actually be, the point here is that we can condense [tex]y[/tex] into a single cosine expression, so choice (D) is correct, since [tex]\cos(x)[/tex] is periodic. This also means choice (C) is also correct, since [tex]\beta=\cos(x)\implies\beta=\cos(x+2n\pi)[/tex] for infinitely many integers [tex]n[/tex]. This simultaneously eliminates (A) and (B).
612,315 in scientific notation
Answer: 6.12315 × 105
Step-by-step explanation: No worry's just here for the fun of it
I need help solving please help me and give me the answer or how to solve it
Answer:
108
Step-by-step explanation:
3²(2³+4) calculate to the power of 2 and get 9.
9 (2³+4) calculate 2 to the power of 3 and get 8.
9 (8+4) add 8 and 4 to get 12.
9 · 12 = 108 multiply 9 and 12 to get 108.
what is the reference angle and cosince of [tex]\frac{7\pi }{6}[/tex]?
Answer:
The reference angle is π/6.
Cosine is -(√3)/2.
Step-by-step explanation:
To find the reference angle, find the acute angle in quadrant I and use it as a reference for the given expression.
For the cosine, the cosine is the sine of the complementary angle. The complementary angle is the given angle beside it minus a right angle, which is exactly 90 degrees. If the angle is 25 degrees, its complementary angle will be double its amount, 50 degrees. Then, for angle angle measured "theta", the cosine is equal to the sine's right-angle subtracted by theta.