Let PQRS be a quardilateral in a plane, where QR=1, ∠PQR=∠QRS=70°, ∠PQS=15° and ∠PRS=40°. If ∠RPS=θ°, PQ=α and PS=β, then the intervals that contains the value of 4αβ sinθ° are
(A) (0, √2)
(B) (1, 2)
(C) (√2, 3)
(D) (2√2, 3√2)
Answers
The intervals that contains the value of 4αβ sinθ° are
(A) (0, √2)
(B) (1, 2)
What is the Interval Angle of the quadrilateral?Given that, PQRS be a quadrilateral in a plane, where;
QR = 1
∠PQR =∠QRS = 70°
∠PQS = 15°
∠PRS=40°.
Further given that, ∠RPS = θ°, PQ = α and PS = β
Now, In triangle PQR
Using sine law, we have;
PQ/sin R = QR/sin P
α/sin 30 = 1/sin 80
Thus;
α = 1/2sin80
Now, In triangle PRS
Using sine law, we have;
PS/sin R = RS/sin P
∠RPS = ∠RSQ = 55
QR = RS = 1
β/sin 40 = 1/sinθ
βsinθ = sin40
Now, Consider 4αβ sinθ° = 4 * 1/2 sin 80 * sin 40 = 2 sin 40/(2 sin 40 cos 40)
Since sin 2x = 2 sin x cos x, then;
= 1/cos 40 = sec 40
Now, we know secx is an increasing function.
Thus;
sec 40 ∈ (2/√3, √2)
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In triangle PQR, we have by the law of sines,
[tex]\dfrac{\sin(30^\circ)}\alpha = \dfrac{\sin(80^\circ)}1 \\\\ ~~~~ \implies \alpha = \dfrac{\sin(30^\circ)}{\sin(80^\circ)}[/tex]
and by the law of cosines,
[tex]PR^2 = 1 + \alpha^2 - 2\alpha \cos(70^\circ)[/tex]
In triangle PQS, the law of sines gives
[tex]\dfrac{\sin(15^\circ)}\beta = \dfrac{\sin(85^\circ-\theta)}\alpha \\\\ ~~~~ \implies \beta = \dfrac{\sin(15^\circ)\sin(30^\circ)}{\sin(85^\circ-\theta)\sin(80^\circ)}[/tex]
In triangle PRS, again with the law of sines,
[tex]\dfrac{\sin(40^\circ)}\beta = \dfrac{\sin(140^\circ - \theta)}{\sqrt{1 + \alpha^2 - 2\alpha\cos(70^\circ)}} \\\\ ~~~~ \implies \beta = \sqrt{1+\alpha^2 - 2\alpha\cos(70^\circ)} \dfrac{\sin(40^\circ)}{\sin(140^\circ-\theta)}[/tex]
Together, these give
[tex]\dfrac{\sin(15^\circ) \sin(30^\circ)}{\sin(85^\circ-\theta) \sin(80^\circ)} = \sqrt{1 + \alpha^2 - 2\alpha \cos(70^\circ)} \dfrac{\sin(40^\circ)}{\sin(140^\circ - \theta)}[/tex]
Carefully, we can rearrange terms and invoke identities (half-angle, product-to-sum) to reduce this equation to
[tex]\dfrac{\sin(140^\circ - \theta)}{\sin(85^\circ - \theta)} = \dfrac{\sqrt{2 - 3\sin(10^\circ) - \sin^2(10^\circ) + 2\sin^3(10^\circ)}}{\sin(15^\circ)}[/tex]
On the left, substitute [tex]\phi=85^\circ-\theta[/tex], and on the right, [tex]\sigma=\csc(15^\circ)\sqrt{2-3\sin(10^\circ)-\sin^2(10^\circ)+2\sin^3(10^\circ)}[/tex]. Expand the left side and rearrange terms.
[tex]\dfrac{\sin(55^\circ-\phi)}{\sin(\phi)} = \sigma \\\\ ~~~~ \implies (\sigma + \sin(35^\circ))\sin(\phi) - \cos(35^\circ) \cos(\phi) = 0[/tex]
Condense this using the identity
[tex]R\sin(\phi-\xi) = R\cos(\xi)\sin(\phi) - R\sin(\xi)\cos(\phi)[/tex]
Solve for [tex]R,\xi[/tex].
[tex]\begin{cases}R\cos(\xi) = \sigma+\sin(35^\circ) \\ R\sin(\xi) = \cos(35^\circ)\end{cases} \implies \begin{cases}R^2 = (\sigma+\sin(35^\circ))^2 + \cos^2(35^\circ) \\\\ \tan(\xi) = \dfrac{\cos(35^\circ)}{\sigma+\sin(35^\circ)} \end{cases}[/tex]
So our equation becomes
[tex]\sqrt{(\sigma+\sin(35^\circ))^2 + \cos^2(35^\circ)} \, \sin\left(\phi - \tan^{-1}\left(\dfrac{\cos(35^\circ)}{\sigma+\sin(35^\circ)}\right)\right) = 0[/tex]
Solve for [tex]\phi[/tex], bearing in mind that [tex]40^\circ<\theta<100^\circ[/tex] so [tex]-15^\circ<\phi<45^\circ[/tex], then evaluate the target expression. (Forgive the sudden switch to approximations; I do think there is a nice analytical solution.)
[tex]\phi = \tan^{-1}\left(\dfrac{\cos(35^\circ)}{\sigma+\sin(35^\circ)}\right) \approx 8.88311^\circ \\\\ ~~~~ \implies \theta \approx 76.1169^\circ \\\\ ~~~~ \implies \beta \approx 0.68309 \\\\ ~~~~ \implies 4\alpha\beta\sin(\theta) \approx 0.91334 < 1 < \sqrt2 < 2\sqrt2[/tex]
which makes (A) the correct choice.
Related Questions
Solve for m.
m – 6.82 = 9.24
Answers
Explanation: m-6.82=9.24
+6.82= +6.82
————————
M= 16.06
Just add the inverse operation of 6.82 to both sides and you get 6.82.
Helllp 20 points plus brainliest
Answers
Answer: Maxwell jogs faster.
Step-by-step explanation:
Haley jogs 3/4 miles in 10 minutes.
Maxwell jogs 2/3 miles in 8 minutes.
3/4 = .75
2/3 = .6 repeating
The table below shows the cost of gasoline based on the number of gallons purchased. Gallons Purchased Cost of Gasoline 5 $16.25 8 $26.00 11 $35.75 14 $45.50 Which equation represents the cost, y, of x gallons of gasoline? Responses
Answers
The equation that represents the cost, y, of x gallons of gasoline, is expressed as: y = 3.25x.
How to Create a Linear Equation for a Table?To create an equation in slope-intercept form, y = mx + b, for a table of values, use two pairs of values from the table to find the slope (m) and determine the y-intercept (b).
Using (5, 16.25) and (8, 26.00) find the slope (m):
Slope (m) = change in y / change in x = (26.00 - 16.25) / (8 - 5)
Slope (m) = (9.75) / (3)
Slope (m) = 3.25
Find the y-intercept (b) by substituting m = 3.25 and (8, 26.00) into y = mx + b:
26.00 = 3.25(8) + b
26.00 = 26.00 + b
26.00 - 26.00 = b
0 = b
b = 0
Substitute m = 3.25 and b = 0 into y = mx + b:
y = 3.25x + 0
y = 3.25x
Therefore, the equation that represents the cost, y, of x gallons of gasoline, is expressed as: y = 3.25x.
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What is the answer to 0.772 times 0.06
Answers
Answer: 0.04632
Step-by-step explanation:
Use the drawing tool(s) to form the correct answer on the provided graph.
Plot the axis of symmetry and the point where the maximum value occurs for this function:
hxx) = -(x + 2)² +8.
Drawing Tools
Select
Mark Feature
Line
+
Click on a tool to begin drawing.
-10
-8
-6
-4
10-
00
Delate Undo
2
4
6
8
Reset
10
Answers
Maximum value occurs at the vertex, given by (h,k)=(-2,8)
In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.
The given function is: h(x) = -(x + 2)² +8.
This function is in the form: a[tex](x-h)^{2}[/tex]+k
Axis of symmetry is a line that divides an object into two equal halves, thereby creating a mirror-like reflection of either side of the object. The word symmetry implies balance.
The axis of symmetry is given by x = h
By comparison, we have -h=2.
This implies h=-2
Therefore the axis of symmetry is x = -2
The maximum value of a function is the place where a function reaches its highest point, or vertex, on a graph. If your quadratic equation has a negative term, it will also have a maximum value.
The maximum value occurs at the vertex, given by (h,k)=(-2,8)
See attachment for the axis of symmetry and maximum point.
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HEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEELPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PLSSSSSS Will records the wins and losses for his high school basketball team each year. He notices that the ratio of wins to losses has been consistent each year. Wins Losses 18 2 45 5 36 4 27 x Using the information in the table, how many losses should he predict for next year if the number of games won is 27? Group of answer choices 9 1 3 6
Answers
Convert 29 days to weeks and days.
Answers
Answer:
4 weeks and 1 day
Step-by-step explanation:
7 x 4 = 28
28 + 1 = 29
Answer:
4weeks and 1 day
7X4=28
28+1=29
Step-by-step explanation:
hope this helps (❁´◡`❁)
Which number set is the most reasonable to describe an adult's weight?
O Integers
O Irrational numbers
O Rational numbers
O Whole numbers
Answers
Answer:
rational numbers
Step-by-step explanation:
An adult's weight can be a non-integer, but is likely of the form of a decimal.
This matches best with the set of rational numbers (of which we take the positive ones).
30-12 (mod 6) pls I need solution class ss1
Answers
Answer:
30
Step-by-step explanation:
Divide
[tex]\frac{12}{6}\\\\\bold{Long\:division}\\6 \times 2 = 12\\= 2[/tex]
[tex]\mathrm{2\:remainder\:0}[/tex]
Simplify the rest of the expression
[tex]= 30 - 0\\= 30[/tex]
Hope this helps!
How much interest would you pay on a loan of 8000 for 6 months at 14 percent interest
Answers
[tex]6000 + \frac{14 \\ }{100} \times 6000 = 6840 \times 6 = 41040[/tex]
Add. Pllssssssss help
Answers
[tex] \bf \implies \: ( - \frac{9}{4}) \: or \: (- 2 \frac{1}{4} )\: or \: ( - 2.25 )\\ [/tex]
Step-by-step explanation:Step-by-step explanation is in attachment see and go through it
Rewrite the following quadratic function in vertex form and graph it F(x)=2x^2+8x+4
Answers
In vertex form, the quadratic function F(x) = 2x² + 8x + 4 is given by:
y = 2(x + 2)² - 4
The graph of the function is given at the end of the answer.
What is the equation of a parabola given it’s vertex?The equation of a quadratic function, of vertex (h,k), is given by the following rule:
y = a(x - h)² + k
In which:
h is the x-coordinate of the vertex.k is the y-coordinate of the vertex.a is the leading coefficient.What is the vertex of a quadratic equation?A quadratic equation is modeled by:
y = ax^2 + bx + c
The vertex is given by:
[tex](x_v, y_v)[/tex]
In which:
[tex]x_v = -\frac{b}{2a}[/tex][tex]y_v = -\frac{b^2 - 4ac}{4a}[/tex]For the function F(x) = 2x² + 8x + 4, the coefficients are given as follows:
a = 2, b = 8, c = 4.
Hence the coordinates of the vertex are:
h = -8/4 = -2.k = -(8² - 4(2)(4))/8 = -4.Hence:
y = 2(x + 2)² - 4
As the leading coefficient is of a = 2.
The graph of the function is given at the end of the answer.
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C=0.4m+68C, equals, 0, point, 4, m, plus, 68 Lúcia hires a truck driver to deliver the food for her catering company. For a delivery that is mmm miles away, the total cost, CCC, in dollars, is given by the above equation. What is the meaning of the 0.4m0.4m0, point, 4, m in this equation?
Answers
The meaning of the 0.4m in this equation is the additional cost of hiring the delivery truck for m number of miles.
EquationGiven equation:
C = 0.4m + 68
Where
m = number of miles
C = total cost of hiring the delivery truck
68 = Fixed cost of hiring the delivery truck
0.48 = Additional cost per mile of hiring the delivery truck
For instance,
If the truck is hired for 10 miles
C = 0.4m + 68
= 0.4(10) + 68
= 4 + 68
C = $72
Therefore, the meaning of the 0.4m in this equation is the additional cost of hiring the delivery truck for m number of miles.
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Can someone help with this question?✨
Answers
Answer:
13
Step-by-step explanation:
According to the Pythagorean principle
[tex]c^{2} = a^{2} +b^{2}[/tex]
[tex]c^{2} = (5^{2} ) + (12^{2} ) = 25 +144= 169\\c= \sqrt{169 } = 13[/tex]
Elena gives the following transformation to show that the two figures are similar by transforming ABCD into EFGD
Answers
It should be noted that the information regarding the transformation illustrates that Elena is wrong.
How to explain the information?A point, line, or geometric figure can be transformed in a way that affects the shape and/or location of the object. Pre-Image refers to the object's initial shape, and Image, after transformation, refers to the object's ultimate shape and location.
It should be noted that the first step makes the figures be the same by size but the second step, reflection over line AE doesn't map them together.
In this case, the correct line of reflection would be a vertical line through point D.
Therefore, it should be noted that the information regarding the transformation illustrates that Elena is wrong.
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Elena gives the following sequence of transformations to show that the two figures are similar by
transforming ABCD into EFGD.
1. Dilate using center D and scale factor 2.
2. Reflect using the line AE.
The dance team lines up in 4 rows of 6 dancers each. How many dancers are on the dance team?
? dancers
6 6 6 6
4 rows →. ^
6 dancers in each row
Answers
2. Let f(x) = x4 + 2x3 - x² - 2x + 10. Given that one root of f(x) is 1 + i find all other roots.
Answers
All the roots of equation x⁴ +2x³ -x² -2x +10 are: (1 + i), (1 - i), -2i, 2i
In the given question, we have to find out all the roots or solutions for the equation x⁴ +2x³ -x² -2x +10. One of the roots given for this equation is 1+i
According to the 'Complex Conjugate' pairs, if an equation has a complex number as a root then its conjugate will also be a root of the equation.
So, we get z = 1 + i as a root then z* = 1 - i which is the conjugate of z, and z* will also be a root of the equation.
Also, the product of two roots will also be a root of the equation. So,
we know that, x = 1 + i, x = 1 - i, we get their product:
(x -(1 + i))*(x - (1 - i)) => by solving we get => x² -2x +2 = 0
=> x² -2x +1 = 1 => x² -x -x +1 = 1 (by middle term split)
=> x(x-1) -1(x-1) = 1 => we get (x - 1) as a factor
So, (x - 1) = 1 => x = 2
=> x = -2i, we get x = -2i
Again, by Complex conjugate pairs if x = -2i is a factor then it's conjugate x = 2i is also a factor.
Hence, the roots of the equation are: (1 + i), (1 - i), -2i, 2i
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HELP PLEASEEEEEEE!!!!!!!!
Answers
It should be noted that two perfect squares so that the square root will fall between consecutive numbers of 0.4 and 0.5 include 0.2025 and 0.2209.
How to illustrate the information?It should be noted that based on the information, we want to find two perfect squares so that the square root will fall between consecutive numbers of 0.4 and 0.5.
In this case, it should be noted that 0.45 × 0.45 = 0.2025. It should be noted also that 0.47 × 0.47 = 0.2209.
As a result, 0.2025 and 0.2209 are two perfect squares such that the square root will lie between consecutive numbers of 0.4 and 0.5.
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Peeling 8 potatoes in 5 minutes
Answers
Answer:
1.6 potatoes per minute
Step-by-step explanation:
8 divided by 5 is 1.6
1. find three consecutive integers such that twice the least integer is 12 more than the greater integer.
2. The perimeter of a rectangle is 150 cm the length is 15cm greater than the width. Find all of the dimensions, please.
Answers
Answer: offf
Step-by-step explanation:off
8 more than negative 10
Answers
8 is greater than -10, because 8 is farther to the right than -10 in a number line.
1. A rectangle has a length that is three feet more than twice its width.
(a) If the width of the rectangle is represented
by the variable w, give an expression for
the length of the rectangle in terms of w.
(b) What would the perimeter of this
rectangle be if the width was 10 feet?
(c) The perimeter of the rectangle is 78 feet. Set up an equation using w that models this
situation. Solve it and state the values for both the width and the length of the rectangle.
Answers
Part a: Expression for length; l = 2w + 3.
Part b: Perimeter equation; P = 2(l + 10).
Part c: Length of rectangle = 29 feet.
What is defined as the rectangle?A rectangle is a the kind of quadrilateral, a two-dimensional shape with four sides as well as four vertices. The rectangle's four angles are all right angles as well as equal to 90 degrees.The given conditions are-
Let 'l' be the length of the rectangle.
Let 'w' be the width of the rectangle.
Part a: expression for the length of a rectangle in the terms of w.
Length is three feet more than twice its width.
l = 2w + 3
Thus, the expression of l in terms of w is: l = 2w + 3.
Part b: Perimeter equation.
The width is given as 10 feet.
The formulae of perimeter if rectangle is;
P = 2(l + w)
Put the value;
P = 2(l + w)
P = 2(l + 10)
Thus, the equation for perimeter is P = 2(l + 10)
Part c: Length of rectangle.
perimeter of rectangle = 78 units.
P = 2(l + w)
w = 10 feet.
78 = 2(l + 10)
l + 10 = 39
l = 29
Thus, the length of the rectangle is found as 29 feet.
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Find the y-intercept of the line on the graph
Answers
Since no equation of the line, or graph is provided in the question statement, considering the line to be [3x +4y = 12], the y-intercept of the line is 3.
As per question statement, we are supposed to determine the y-intercept of the line on the graph, but no equation of the line, or graph is provided. Hence, we considered the equation of a random line, say,
[3x +4y = 12] and then, determine it's y-intercept.
To solve this question, we can use two methods:
One method is by converting the given equation into the format of the standard equation of a line, and then comparing it to the standard equation to determine the y-intercept, i.e.,
The standard equation of line is (y = mx + c), where, "m" is the slope of the line, and "c" is the y-intercept. Converting [3x +4y = 12] to the form of the standard equation, we get,
[tex]3x+4y=12\\or,4y=-3x+12\\or,y=\frac{-3x+12}{4}\\ or,[y=(\frac{-3}{4})x+3]...(i)[/tex]
Now comparing (i) to the standard equation of line, we can identify that
(c = 3), i.e., y-intercept of the line [3x +4y = 12] is 3.
Another way we can determine the y-intercept of a line is by, substituting (x = 0) and then solving the equation for "y".
Y-Intercept: The y-intercept is the point on a cartesian plane, where a graph crosses the y-axis, i.e., the value of y when (x = 0).To learn more about Y-Intercepts, click on the link below.
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GIVING BRAINIEST IF CORRECT!!!!!!!
Express in simplest form
Answers
Explanation:
-3 √ 40 may be simplified into -6 √10. 40 may be divided into 4 and 10, and 4 is a perfect square.
-3 √40
-3 ( √4 x √10)
-3 ( 2 x √10)
-6 √10
Then √160 can be simplified into 4 √10
√160
√16 x √10
4 √10
Finally, we put the two together
-6 √10 - 4 √10
-10 √10
Answer:
Step-by-step explanation:
-3√(4)(10) - √(16)(10)
-3(2)√(10) - 4√(10)
-6√(10) - 4√(10)
-10√(10)
7 is divided by the sum of a number and 8
Answers
Answer:
n + 8 ÷ 7
Step-by-step explanation:
I think that's the answer, but I could be wrong- lol.
A farmer sells an average of 15g bushels of corn
each day. What integer represents the change in
bushels of corn in his inventory after 6 days?
Answers
According to the solving his unused inventory will have = x - [tex]\frac{468}{5}[/tex]
What is mixed fraction?A mixed fraction is defined by its quotient as well as remainder. 2/3, for example, is a mixed fraction in which 2 is the quotient and 1 is the remainder. A mixed fraction is thus the product of a whole number as well as a proper fraction.
What is an example of a simple fraction?If the only common factor of a fraction's numerator and the denominator is 1, the fraction is considered to be in its simplest form. For example,89, because 1 is the sole common factor in this fraction of 8 and 9. We simplify fractions simply because it is usually easier to work with or compute fractions when they are in their simplest form.
According to the given data:Every day, a farmer sells on average of 15 3/5 bushels of maize.
Converting mixed fraction to proper fraction;
[tex]15\frac{3}{5}[/tex] this can be written as [tex]\frac{78}{5}[/tex]
Assume he has a total of x bushels.
As a result, he will sell in 6 days.
= 6 x [tex]\frac{78}{5}[/tex]
= [tex]\frac{468}{5}[/tex]
So,
According to the solving his unused inventory will have = x - [tex]\frac{468}{5}[/tex]
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I understand that the question you are looking for is:
A farmer sells an average of 15 3/5 bushels of corn each day. What integer represents the change in bushels of corn in his inventory after 6 days.
help!!!!!!!!!!! plsssssssssss
Answers
Answer:
1st image: -14, 4
2nd image: -0.7, 1.7
3rd image: -5/8, 7/8
Step-by-step explanation:
What I came up with...
A standard die is rolled. Find the probability that the number rolled is greater than 2.
Answers
Answer:
2/3
Step-by-step explanation:
To find this, we need to find out how many numbers are greater than 2 but no more than 6. This set includes 3, 4, 5, 6; there are 4 numbers that qualify for this. This means that out of all 6 numbers, 4 numbers are in the probability, so our answer would be 4/6 = 2/3
Answer:
2/3
Step-by-step explanation:
The probability of a die getting 3,4,5,6 over a 1 or 2 is 4 chances out of the 6 chances in a die, or 2/3.
Need help with #9! Intermediate Algebra.
Answers
Answer:12.35 = t
Step-by-step explanation:
95 + t + 3.5 = 110.85
where t = tax
12.35 = t
2198290.45 divided by 1.08