The ratio of the volume of a pyramid to the volume of a prism with the same base and height is 1 to 3; the formula for the volume of a pyramid is V = 1/3Bh.
Volume of pyramidVolume of prism, V = BhVolume of pyramid, V = 1/3Bhwhere,
B = the area of the base and h = the heightVolume of prism : Volume of pyramid
Bh : 1/3Bh
Therefore, the ratio of the volume of a pyramid to the volume of a prism with the same base and height is 1 to 3.
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In the expansion of (2a + 3b)8, which of the following are possible variable terms? Explain your reasoning.
a2b3; a5b3; ab8; b8; a4b4; a8; ab7; a6b5
The possible variable terms are a^5b^3, b^8, a^4b^4, a^8, ab^7
How to determine the possible variable term?The expression is given as:
(2a + 3b)^8
The power of the above expression is 8
When expanded, the sum of powers on each variable must be 8.
Using the above explanation, we have the following possible variable terms:
a^5b^3, b^8, a^4b^4, a^8, ab^7
This is because the sum of powers equals 8
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Point R divides PQin the ratio 1:3. If the x-coordinate of Ris -1 and the x-coordinate of Pis -3, what is the x-coordinate of Q?
Since Point R divides PQ in the ratio 1:3. the x-coordinate of Q is 3
The question has to do with line division
What is line division?This is the division of a line into a given ratio.
How to find the coordinate of Q?
Since Point R divides PQ in the ratio 1:3. If the x-coordinate of R is -1 and the x-coordinate of P is -3.
Let
x₁ = coordinate of P = -3, x₂ = coordinate of R = -3 and x₃= coordinate of Q.Since R divides PQ in the ratio, 1:3, we have that
PR/RQ = 1/3
(x₂ - x₁)/(x₃ - x₂) = 1/3
Substituting the values of the variables into the equation, we have
(x₂ - x₁)/(x₃ - x₂) = 1/3
(-1 - (-3))/(x₃ - (-3)) = 1/3
(-1 + 3)/(x₃ + 3) = 1/3
2/(x₃ + 3) = 1/3
x₃ + 3 = 2 × 3
x₃ + 3 = 6
x₃ = 6 - 3
x₃ = 3
So, the x-coordinate of Q is 3
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On a number line, the directed line segment from q to s has endpoints q at –14 and s at 2. point r partitions the directed line segment from q to s in a 3:5 ratio. which expression correctly uses the formula (startfraction m over m n endfraction) (x 2 minus x 1) x 1 to find the location of point r?
The coordinates of point r(-8,0).
What is midpoint formula in coordinate geometry?The coordinates of the point r(x,y) which divides the line segment joining the points p([tex]x_{1}[/tex],[tex]y_{1}[/tex]) and q([tex]x_{2}[/tex],[tex]y_{2}[/tex]) internally in the ratio :[tex]m_{1}[/tex][tex]m_{2}[/tex] are
[tex]\left(\frac{m_{1}x_{2}+ m_{2}x_{1} }{m_{1}+m_{2} } ,\frac{m_{1}y_{2}+ m_{2}y_{1} }{m_{1}+m_{2} }\Rifgt)[/tex]
Given that,
Two end points on the line q(-14,0) and s(2,0). point r(x,y) is the partition of the line segment from q to s in a ratio 3:5
[tex]m_{1}[/tex] = 3 and [tex]m_{2}[/tex] = 5
By using the midpoint formula
[tex]\left(\frac{m_{1}x_{2}+ m_{2}x_{1} }{m_{1}+m_{2} } ,\frac{m_{1}y_{2}+ m_{2}y_{1} }{m_{1}+m_{2} }\Rifgt)[/tex]
[tex]\left(\frac{3(2)+ 5(-14) }{3+5 } ,\frac{3(0)+ 5(0) }{3+5 }\Rifgt)[/tex]
[tex]\left(\frac{-64}{8 } ,\frac{0 }{8 }\Rifgt)[/tex]
[tex]\left(-8 ,0\Rifgt)[/tex]
Hence, The coordinates of point r(-8,0).
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In 4 weeks, Mark earns an average of $400 per
week. If, after working another 2 weeks at a
constalat salary, his average carnings per week
are $500, how much did he earn in the last 3
weeks?
By which number -7÷25 should be divided to get -1÷15
Answer:
Step-by-step explanation:
Set it up like an equation first:
[tex]\frac{\frac{-7}{25} }{x}=-\frac{1}{15}[/tex] and then bring up the lower fraction, flip it, and multiply:
[tex]-\frac{7}{25}*\frac{1}{x}=-\frac{1}{15}[/tex]
Simplify to
[tex]-\frac{7}{25x}=-\frac{1}{15}[/tex] and cross multiply to get
-25x = -105 and divide both sides by -25 to get that
[tex]x=\frac{105}{25}=\frac{21}{5}[/tex]
The following are the ages of 13 history teachers in a school district. 24, 27, 29, 29, 35, 39, 43, 45, 46, 49, 51, 51, 56 Notice that the ages are ordered from least to greatest. Give the five-number summary and the interquartile range for the data set. Five-number summary
Minimum:
Lower quartile:
Median:
Upper quartile:
Maximum:
Interquartile range:
Use the method of undetermined coefficients to find the general solution to the de y′′−3y′ 2y=ex e2x e−x
I'll assume the ODE is
[tex]y'' - 3y' + 2y = e^x + e^{2x} + e^{-x}[/tex]
Solve the homogeneous ODE,
[tex]y'' - 3y' + 2y = 0[/tex]
The characteristic equation
[tex]r^2 - 3r + 2 = (r - 1) (r - 2) = 0[/tex]
has roots at [tex]r=1[/tex] and [tex]r=2[/tex]. Then the characteristic solution is
[tex]y = C_1 e^x + C_2 e^{2x}[/tex]
For nonhomogeneous ODE (1),
[tex]y'' - 3y' + 2y = e^x[/tex]
consider the ansatz particular solution
[tex]y = axe^x \implies y' = a(x+1) e^x \implies y'' = a(x+2) e^x[/tex]
Substituting this into (1) gives
[tex]a(x+2) e^x - 3 a (x+1) e^x + 2ax e^x = e^x \implies a = -1[/tex]
For the nonhomogeneous ODE (2),
[tex]y'' - 3y' + 2y = e^{2x}[/tex]
take the ansatz
[tex]y = bxe^{2x} \implies y' = b(2x+1) e^{2x} \implies y'' = b(4x+4) e^{2x}[/tex]
Substitute (2) into the ODE to get
[tex]b(4x+4) e^{2x} - 3b(2x+1)e^{2x} + 2bxe^{2x} = e^{2x} \implies b=1[/tex]
Lastly, for the nonhomogeneous ODE (3)
[tex]y'' - 3y' + 2y = e^{-x}[/tex]
take the ansatz
[tex]y = ce^{-x} \implies y' = -ce^{-x} \implies y'' = ce^{-x}[/tex]
and solve for [tex]c[/tex].
[tex]ce^{-x} + 3ce^{-x} + 2ce^{-x} = e^{-x} \implies c = \dfrac16[/tex]
Then the general solution to the ODE is
[tex]\boxed{y = C_1 e^x + C_2 e^{2x} - xe^x + xe^{2x} + \dfrac16 e^{-x}}[/tex]
Jaxon is training for a climbing competition.
He breaks his training up into strength and
endurance training in the ratio 5:3.
If he spends 4 hours on endurance training,
how long should he spend on strength training?
======================================================
Work Shown:
(x hours strength)/(4 hours endurance) = 5/3
x/4 = 5/3
3x = 4*5
3x = 20
x = 20/3
x = (18+2)/3
x = 18/3 + 2/3
x = 6 + 2/3
x = 6 & 2/3
He needs to spend 6 full hours, plus an additional 2/3 of an hour, on strength training.
1 hr = 60 min
2/3 hr = (2/3)*60 min
2/3 hr = 40 min
6 hr + 2/3 hr = 6 hr + 40 min
Therefore, he needs to spend 6 hours, 40 minutes on strength training.
This is equivalent to 6*60+40 = 360+40 = 400 minutes.
HELP WITH MATH PLS!!!!!
The difference of 32 · x² / (x² + 8 · x + 15) - 14 · x² / (x² - 9) is equal to [18 · x³ - 166 · x²] / [(x + 3) · (x + 5) · (x - 3)]. (Correct choice: B)
What is the result of the subtraction between two algebraic rational functions?
In this question we have a subtraction between two rational functions, which have to be simplified by algebra properties. The complete procedure is presented below:
32 · x² / (x² + 8 · x + 15) - 14 · x² / (x² - 9) Given
32 · x² / [(x + 3) · (x + 5)] - 14 · x² / [(x + 3) · (x - 3)] Factorization
[x² / (x + 3)] · [32 / (x + 5) - 14 / (x - 3)] Distributive and associative properties
[x² / (x + 3)] · [32 · (x - 3) - 14 · (x + 5)] / [(x + 5) · (x - 3)] Subtraction of rational numbers with distinct denominators
[x² / (x + 3)] · [32 · x - 96 - 14 · x - 70] / [(x + 5) · (x - 3)] Distributive property / (- 1) · a = - a
[x² / (x + 3)] · (18 · x - 166) / [(x + 5) · (x - 3)] Distributive property / Definitions of addition and subtraction
[18 · x³ - 166 · x²] / [(x + 3) · (x + 5) · (x - 3)] Mutiplication between rational numbers / Multiplication between powers / Distributive property
The difference of 32 · x² / (x² + 8 · x + 15) - 14 · x² / (x² - 9) is equal to [18 · x³ - 166 · x²] / [(x + 3) · (x + 5) · (x - 3)]. (Correct choice: B)
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Aisha wants to paint the four walls of her living room.
Each wall is 2.4 m high and 5.2 m long.
One wall has a door of 2 m by 0.8 m.
Tins of paint cost £12 per 2 litre tin.
Each litre of paint can cover 12 m2 of wall.
There is an offer of: Buy 2 tins get the 3rd at half price.
How much will Aisha pay to paint her living room?
The total amount Aisha would pay to paint her room is $54.
What is the cost of painting the rooms?The first step is to determine the area of the walls. The walls have the shape of a rectangle. Thus, the formula for the area of a rectangle would be used to determine the area of the wall.
Area of the three walls without a door : 3 x (length x width)
3 x (2.4 x 5.2) = 37.44 m²
Area of the fourth wall with a door : area of the wall - area of the door
(2.4 x 5.2) - (2 x 0.8) = 10.88m²
Total area of the walls : area of the three walls + area of the wall with the door
10.88m² + 37.44 m² = 48.32 m²
The next step is to determine the number of tins that would be needed:
48.32 /12 = 4.03
5 tins would needed
Cost of the 5 tins: (4 x 12) + (0.5 x 12) = $54
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Solve for x.
°x = 196
Enter your answers in the boxes. Enter the smallest answer first.
x= ____ and x= ____
4 andfirst find x by finding its smallest factor
The solution to the given system of equation is -7 and 7
What is linear equation?A linear equation is an algebraic equation of the form y=mx+b. involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept. Occasionally, the above is called a "linear equation of two variables," where y and x are the variables.
Given:
4x² = 196
4x²/4= 196/4
x² = 49
x=√49
x= ±7
Hence, the solution to the given system of equation is -7 and 7
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In the year 2000, there were 200,000 cell phone subscribers in a city in new york. the number of subscribers increased by 60 percent per year after 2000. which equation can be used to model the number of subscribers, y, in the city t years after 2000?
The equation can be used to model the number of subscribers, y, in the city t years after 2000 is y = 200,000(1+60)t.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign. For example, 3x – 5 = 16 is an equation. Solving this equation, we get the value of the variable x as x = 7.
The first step is to calculate the percentage increase for each year.
If the subscribers increase by 60%, it means that the subscribers will be (100+60)% of the previous year.
(100+60)% = (100/100 + 60/100)
= (1 + 0.6)
After 2,000, the subscribers will be 200,000×(1+0.6).
For the following years the 200,000×(1+0.6)t
So, the value of y will be given by,
Y=200,000(1+0.6)t
Hence,The equation Y=200,000(1+0.6)t .
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Tanya is training a turtle for a turtle race. For every 1/3 of an hour that the turtle is crawling, he can travel 2/23 of a mile. At what unit rate is the turtle crawling?
The unit rate with the turtle crawling is 6/23 mile per hour.
According to the statement
we have given that the every 1/3 of an hour that the turtle is crawling, he can travel 2/23 of a mile. and
we have to find the unit rate with the turtle crawling
So, For this purpose,
we know that the
The unit rate of speed in miles per hour the number of miles in one hour.
So, to find the unit rate:
Divide distance in miles with the time which is in hours.
So,
unit rate of speed = distance / time
unit rate of speed = (2/23) / (1/3)
unit rate of speed = 6/23 mile per hour.
So, The unit rate with the turtle crawling is 6/23 mile per hour.
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Find the maximum and minimum values attained by f(x, y, z) = 2xyz on the unit ball x2 y2 z2 ≤ 1
The maximum and minimum values of f(x,y,z) = 2xyz are [tex]\frac{2}{\sqrt{3} } and \frac{-2}{\sqrt{3} }[/tex]
The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function. Differentiate the given function.
f(x,y,z)=2xyz
Computation:
Differentiating the given equation up to second order
[tex]\begin{gathered}f_x=2yz\Rightarrow 2yz=0 either y=0 or z=0\\f_y=0\Rightarrow 2xz=0 either x=0 or z=0\\f_z=0\Rightarrow 2xy=0 either x=0 or y=0\end{gathered}[/tex]
So, the critical point is (0,0,0)
Now, using the Lagrange's on the boundary,
[tex]\begin{gathered}g(x,y,z)=x^2+y^2+z^2-1=0\\g_x=2x\\g_y=2y\\g_z=2z\end{gathered}[/tex]
[tex]So, \bigtriangledown f=\lambda \bigtriangledown g\left < 5yz,5xz,5xy \right > =\lambda \left < 2x,2y,2z )[/tex]
By solving we get,
[tex]x^2=y^2=z^2[/tex] then,
[tex]\begin{gathered}x^2+y^2+z^2=1\\3z^2=1\\z=\pm \frac{1}{\sqrt{3} } \\y=\pm \frac{1}{\sqrt{3} } \\\\x=\pm \frac{1}{\sqrt{3} } \\\end{gathered} x 2+y 2+z 2=13z 2=1z=± 31[/tex]
[tex]So, the critical points are (0,0,0),(\frac{1}{\sqrt{3} } ,\frac{1}{\sqrt{3} } ,\frac{1}{\sqrt{3} } )and (\frac{-1}{\sqrt{3} }, \frac{-1}{\sqrt{3} },\frac{-1}{\sqrt{3} })[/tex]
So, by substituting the critical points we get,
[tex]\begin{gathered}f(0,0,0)=0\\f(\frac{1}{\sqrt{3} }, \frac{1}{\sqrt{3} },\frac{1}{\sqrt{3} })=2(\frac{1}{\sqrt{3} })(\frac{1}{\sqrt{3} })(\frac{1}{\sqrt{3} })\\=\frac{2}{3\sqrt{3} } \\f(-\frac{1}{\sqrt{3} },-\frac{1}{\sqrt{3} },-\frac{1}{\sqrt{3} })=2(-\frac{1}{\sqrt{3} })(-\frac{1}{\sqrt{3} })(-\frac{1}{\sqrt{3} })\\=-\frac{2}{3\sqrt{3} }\end{gathered}[/tex]
Hence the maximum and minimum values of f(x,y,z) are [tex]\frac{2}{\sqrt{3} } and \frac{-2}{\sqrt{3} }[/tex]
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1. Use the figure to determine the value of x and y.
Answer:
Step-by-step explanation:
x=100 bc like 180-80=100
y=80 bc vertical angles
Answer:
y = 80*
x = 100*
Step-by-step explanation:
y is the same to the 80* angle and therefore x is 100*
180* - 80 = 100*
It is October and that means it is State Fair time again! Your uncle took you and 2 of your friends to the
fair. Admission for everyone to get into the fair was $20.00. You then went to the Midway, where they
have all of the rides, and he bout all-day ride passes for you and your friends. If your uncle spent a total
of $65.00 at the fair, how much did each of the all-day ride passes cost?
Answer: All-day ride passes cost $15 each
Step-by-step explanation:
It was $20 in total for everyone to get in, so subtract that from $65 to just focus on the all day passes
65-20=45
So the all day passes all together cost $45
Since your uncle bought it for you and 2 friends that's 3 people in total so divide 45 by 3
45 divided by 3 = 15
So each of the all day passes cost $15
Hope this helped :)
Does the graph represent a function? Why or why not?
A. No, because it fails the horizontal line test
B. Yes, because it passes the horizontal line test
C. No, because it fails the vertical line test.
•
D. Yes, because it passes the vertical line test.
Answer:
D.
Step-by-step explanation:
Yes, because it passes the vertical line test.
Kaitlin, Joe, and Miguel have a total 96 of in their wallets. Kaitlin has 6 less than Joe. Miguel has 4 times what Joe has. How much does each have?
Answer:
15, 11, 60
Step-by-step explanation:
Let Joe's amount be x
x + x - 6 + 4x = 96
6x - 6 = 96
6x = 90
x = 15
Joe has 15
Kaitlin has 11
Miguel has 60
PLEASE PLEASE PLEASE HELP ME ITS A MATH PROBLEM!!! EXPLANATIONS WELCOMED
Answer:
8
Step-by-step explanation:
The given ratios can be combined to a composite ratio using least common multiples.
Given ratiosWe can identify each of the entities with a 2-letter designation:
TT - Tumtum tree; TW - tulgey wood; BS - frumious Bandersnatch;
ST - slithy tove; BG - borogoves; MR - mome raths; JW - Jabberwocky;
JB - Jubjub birds.
Then the given ratios are ...
20 TT : 1 TW1 TW : 1 BS5 ST : 2 BG2 MR : 1 JW2 JB : 200 TT . . . . (reduces to 1 : 100)200 MR : 1 BG5 JB : 1 STComposite ratiosThe path from BS to JW is unique, so we can work our way there one ratio at a time. Having done this once, we can go back and do it again with the least possible multiplier.
Here, that turns out to be 125 Bandersnatches. We can combine these to the composite ratio ...
125 BS : 125 TW : 2500 TT : 25 JB : 5 ST : 2 BG : 400 MR : 200 JW
Then the ratio of Bandersnatches to Jabberwocks is 125 : 200 = 5 : 8.
If there are 5 frumious Bandersnatches, there should be 8 Jabberwocks.
__
Additional comment
We know 1 JW : 2 MR and 200 MR : 1 BG. In order to get a common value of 200 MR in these ratios, we need to multiply the first by 100:
100 JW : 200 MR : 1 BG
Next, we notice the other ratio involving BG is 2 BG : 5 ST, which means we need to multiply by 2 again:
200 JW : 400 MR : 2 BG : 5 ST
The number of JB is 5 times the number of ST, so this extends to ...
... 5 ST : 25 JB
The number of TT is 100 times the number of JB, so we have ...
... 5 ST : 25 JB : 2500 TT
Continuing in like fashion, we get the composite ratio shown above. We originally worked this starting from BS. Reducing the JB/TT ratio from 2/200 to 1/100 helped eliminate an extra factor of 2.
Step-by-step explanation:
it's not really a math problem per se, it is more like a regular riddle with numbers.
true, everything involving numbers is essentially math, but then what will you do in life with things like earning a salary, going shopping, checking a calendar and schedules for bus, train and flights or business meetings, sharing a pizza with friends, playing computer games, ...
life is for a big part numbers and therefore math.
so, don't be afraid, just use your brain and common sense !
let's think this through (just be strict, concentrated, and follow each line literally) :
5 fr. Band. means 5 tulgey woods.
which means 20×5 = 100 tumtum trees.
as there are 2 jubjub birds in 200 tumtum trees, that means that there is 1 jubjub bird in the given 100 tumtum trees.
as there are 5 jubjub birds in 1 slithy toves, but we have only 1 bird, that means we have only 1/5 slithy toves.
there are 5 slithy toves in 2 borogoves. that means 1 slithy toves in 2/5 borogoves. and 1/5 slithy toves in 2/5 / 5 = 2/25 borogoves.
there are 200 mome raths in each borogove. and we have 2/25 borogoves. that means we have
200 × 2/25 = 400/25 = 16 mome raths.
and there are 2 mome raths per jabberwocky. we have 16 raths. how many pairs of raths does that make ?
16/2 = 8 pairs of raths = jabberwocks.
so, there should be 8 jabberwocks.
you see ? that's all that was needed. this was like a labyrinth riddle, where you need to find the way from an inside starting point to the exit without crossing a line (or wall).
Distinguish between a correlation and an autocorrelation. how are these measures similar? how are they different?
A correlation is a causative association between two or more variables, whereas an autocorrelation is a correlation of variables in successive ranges.
What is correlation?A correlation is defined as a causative association between two or more variables (different variables), which can be used to make predictions about a given outcome.
The correlation coefficient enables the estimation of this association between different variables.
Moreover, an autocorrelation is a special type of correlation between variables that can be found in successive ranges (e.g. range time intervals).
In conclusion, a correlation is a causative association between two or more variables, whereas an autocorrelation is a correlation of variables in successive ranges.
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Rounding off 6 400mm
The rounding off a given number implies approximating the number to the required value. Thus the answer to the given question is 6 000 mm.
Rounding off a given number implies approximating the number to the required value. This can be done in two major ways: rounding up or rounding down.
Rounding up implies considering the required digit if it is up to or greater than 5, which turns to 1. Rounding down requires no consideration of digits less than 5 which turns to 0.
Thus considering the given question, Since the second digit is 4 which is lesser than 5, then it turns to a zero. So that;
rounding off 6 400 mm would give 6 000 mm.
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Which line of best fit accurately represents the given scatter plot?
Mrs. Greenaway has decided to open a new preschool in her town. To decide whether she will have enough students to be successful, she hires a researcher to conduct a survey of the town’s population. Which of the following groups would make up the best sample for this survey?
A. parents of preschool children
B. parents of elementary school children
C. registered voters in the town
D. members of the town council
The answer choice which represents the group which makes up the best sample for the researcher's survey as described in the task content is; Choice A; Parents of preschool children.
Which answer choice best represents a sample for the intended survey by Mrs Greenaway's researcher?It follows from the task content that; Mrs. Greenaway has decided to open a new preschool in her town.
The need for a researcher therefore arises from the decision to either open the preschool or not.
Consequently, in a bid to choose a sample, Mrs. Greenaway's researcher must make sure the sample chosen is from the population space.
Therefore, answer choice A represents the correct sample.
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HELPPPP WITH ALL OF THE 3 PLSSS
Answer:
See picture.
C is the point that does not move because it is on the line of reflection
Yes, a reflection is a rigid motion. It does not change size. A dilation changes the size.
Step-by-step explanation:
In the picture, you will see that I traced the triangle on tracing paper and then folded (flipped) the drawing over the line of reflection.
Someone please help me with Graph!!
(My answers keep coming wrong)
Topic= Solving systems of equations by Graphing
See below for the solution to the systems of equations
How to solve the systems of equations?System of equations 1
Here, we have
y = -5x/3 + 3
y = x/3 -3
We start by plotting the graph of both equations
Next, we write out the point of intersection of the equations
See attachment for the equation of the system
From the attached graph, the point of intersection of the equations is (3, -2)
Hence, the solution to the system is (3, -2)
System of equations 2
Here, we have
y = 4x + 3
y = -x -2
We start by plotting the graph of both equations
Next, we write out the point of intersection of the equations
See attachment for the equation of the system
From the attached graph, the point of intersection of the equations is (-1, -1)
Hence, the solution to the system is (-1, -1)
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pls help solve math problem!!! lots of points
If csc(x)=7, for 90 deg
sin(x/2)=
cos(x/2)=
tan(x/2)=
The values of trigonometric functions sin(x/2) is [tex]\sqrt{\frac{7+4\sqrt{3}}{14}}[/tex], cos(x/2) is [tex]\frac{1}{\sqrt{14(7+4\sqrt{3})}}[/tex] and tan(x/2) is 7+4√3.
Given that the value of trigonometric function csc(x)=7 for 90°<x<180°.
We are given:
csc(x)=7 Where: 90° < x < 180°
This interval indicates that the angle x in the second quadrant and we know that at that quadrant sin(x) and CSC(x) functions are positive and all other trigonometric functions sign are negative.
Now:
sin(x)=1/csc(x)
sin(x)=1/7
Using the trigonometric identity sin²x+cos²x=1, we get
cosx=±√(1-sin²x)
cosx=±√(1-(1/7)²)
cosx=±√((49-1)/49)
cosx=±(4√3)/7
As x is in second quadrant.
Therefore, cosx=-(4√3)/7
consider the inequality, 90°<x<180°
Divide by 2, we get
45°<x/2<90°
This is the angle x/2 in the first quadrant and in that quadrant all functions sign are positive.
Substitute the value of cosx in half angle formula for sine function,
[tex]\begin{aligned}\cos x&=1-2\sin^2\left(\frac{x}{2}\right)\\ -\frac{4\sqrt{3}}{7}&=1-2\sin^2\left(\frac{x}{2}\right)\\ 2\sin^2 \frac{x}{2}&=1+\frac{4\sqrt{3}}{7}\\\sin \frac{x}{2}&=\pm\sqrt{\frac{7+4\sqrt{3}}{14}}\end[/tex]
As x/2 lies in first quadrant then [tex]\sin \frac{x}{2}=\sqrt{\frac{7+4\sqrt{3}}{14}}[/tex]
Again, Substitute the value of cosx in half angle formula for sine function,[tex]\begin{aligned}\sin x&=2\sin\left(\frac{x}{2}\right)\cos\frac{x}{2}\\ \frac{1}{7}&=2\sqrt{\frac{7+4\sqrt{3}}{14}}\cos\left(\frac{x}{2}\right)\\ \cos \frac{x}{2}&=\frac{1}{14\times\frac{\sqrt{7+4\sqrt{3}}}{\sqrt{14}}}\\\cos \frac{x}{2}&=\frac{1}{\sqrt{14(7+4\sqrt{3})}}\end[/tex]
Now find tan(x/2) as shown below, we get
[tex]\begin{aligned}\tan\frac{x}{2}&=\frac{\sin\frac{x}{2}}{\cos \frac{x}{2}}\\&=\frac{\sqrt{7+4\sqrt{3}}}{\sqrt{14}}\times \frac{\sqrt{14}\sqrt{7+4\sqrt{3}}}{1}\\&=7+4\sqrt{3}\end[/tex]
Hence, when trigonometric function csc(x)=7 for 90°<x<180° then sin(x/2) is [tex]\sqrt{\frac{7+4\sqrt{3}}{14}}[/tex], cos(x/2) is [tex]\frac{1}{\sqrt{14(7+4\sqrt{3})}}[/tex] and tan(x/2) is 7+4√3.
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A bird (B) is spotted flying 6,000 feet from a tower (). An observer (0) spots the top of tower (T) at a distance of 9,000 feet. What is the angle of depression from the bird (B) to the
observer (0)?
Using relations in a right triangle, it is found that the angle of depression is of θ = 56.31º.
What are the relations in a right triangle?The relations in a right triangle are given as follows:
The sine of an angle is given by the length of the opposite side to the angle divided by the length of the hypotenuse.The cosine of an angle is given by the length of the adjacent side to the angle divided by the length of the hypotenuse.The tangent of an angle is given by the length of the opposite side to the angle divided by the length of the adjacent side to the angle.For this problem, we have that:
The opposite side to the angle of depression is the top of tower, at a height of 9000 feet.The adjacent side to the angle is the distance to the bird, of 6000 feet.Hence, considering θ as the depression angle, we have that:
tan(θ) = 9000/6000
tan(θ) = 1.5
θ = arctan(1.5)
θ = 56.31º.
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Solve the equation in the image (yr 8 math, 40pt)
Answer:
a = 3
Step-by-step explanation:
4(2a - 3) = -2(a - 9)
8a - 12 = -2a + 18
10a = 30
a = 3
Let's check if we got it correct by substituting 3 for a in the original equation.
4(2*3 - 3) = -2(3 - 9)
4(6 - 3) = -2(-6)
4(3) = 12
12 = 12
Write an equation for the nth term of the arithmetic sequence.
Step-by-step explanation:
Given [tex] \sf \: first \: term \: (a) = \dfrac{1}{4} \\ [/tex][tex] \small{\sf \: Common \: difference \: (d) = \dfrac{1}{2} - \dfrac{1}{4} = \bold{\dfrac{1}{4}}}[/tex]To findnth termSolution[tex] \sf \: T_n = a+(n-1)d[/tex]
[tex] \sf \: T_n = \frac{1}{4} +(n-1) \frac{1}{4} [/tex]
[tex]\sf \: T_n = \frac{1}{4} + \frac{n}{4} - \frac{1}{4} [/tex]
[tex]\sf \: T_n = \frac{n}{4}[/tex]
an equation for the nth term of the arithmetic sequence. is n/4
Step-by-step explanation:
TN=n/4 is the nth term of arithmetic sequence.