Answers
[tex]\bf \pm\sqrt{3^2-2^2}=b\implies \pm\sqrt{9-4}=b\implies\boxed{ \pm\sqrt{5}=b}\\\\ -------------------------------\\\\ cos^{-1}\left( \frac{2}{3} \right)=\theta \implies sin\left[ cos^{-1}\left( \frac{2}{3} \right) \right]\implies sin(\theta ) \\\\\\ sin(\theta )=\cfrac{\pm\sqrt{5}}{3}\cfrac{\leftarrow opposite}{\leftarrow hypotenuse}[/tex]
it doesn't say the angle is in a certain quadrant, thus the +/- versions of it are both valid.
Related Questions
The range of a projectile fired at an angle θ with the horizontal and with an initial velocity of v 0 feet per second is r = 1 v 20 sin 2θ where r is measured in feet. a golfer strikes a golf ball at 120 feet per second. 32 ignoring the effects of air resistance, at what angle must the golfer hit the ball so that it travels 140 feet? (round answer to nearest angle.)
Answers
There are already derived equations for this so it is already convenient. However, your given equation for range is incomplete. The correct equation is:
R = v₀²sin2θ/g
where
R = 140 ft
v₀ = 120 ft/s
g = 32.2 m/s²
Substituting the values,
sin 2θ = Rg/v₀²
sin 2θ = (140)(32.2)/(120²)
sin 2θ = 0.313
2θ = sin ⁻¹ (0.313)
2θ = 18.24°
θ =18.24°/2
θ = 9.12°
The area of a rectangular wall of a barn is 75 75 square feet. its length is 10 10 feet longer than the width. find the length and width of the wall of the barn.
Answers
A = 75
L = W + 10
75 = W(W + 10)
75 = W^2 + 10W
W^2 + 10W - 75 = 0
(W + 15)(W - 5) = 0
W + 15 = 0
W = -15...not this one because it is negative
W - 5 = 0
W = 5 <=== width is 5 ft
L = W + 10
L = 5 + 10
L = 15 <=== length is 15 ft
First, represent the unknown width as 'x'. Then, because the length is 10 feet more than the width, represent the length as 'x + 10'. Since the area is 75 square feet, set up and solve the equation x*(x + 10) = 75 for 'x'. This will give the width and the length will be this width plus 10 feet.
Explanation:This problem can be solved using algebra. First, let's denote the width of the wall as x in feet. If the length of the wall is 10 feet longer than the width, then we can express the length as x + 10. We're told that the area of the rectangle formed by the wall is 75 square feet. The area of a rectangle is calculated by multiplying its length by its width. Therefore:
x*(x + 10) = 75
You can solve this quadratic equation for x. The positive solution will be the width of the wall and the length will be that value plus 10 feet.
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Shawn wants to buy a cd player that costs $48.00. If he has already saved $30.00, what percent of the price of the cd player has he saved?
Answers
the answer is 62.5%
100%/x%=48/30
Multiply both sides by x.
(100/x)x=(48/30)x
100=1.6x
Divide both sides by 1.6.
100/1.6=x
62.5=x
30 is 62.5% of 48
if a metal tool chest has the shape of a prism with a pentagonal base of 6194in squared in area, calculate the volume of the metal chest and pick where length c = 6 inches and length m = 19 inches
Answers
The volume of the metal tool chest is 37,164 cubic inches. To calculate the volume of the metal tool chest, we use the formula for the volume of a prism: Volume = Base area × Height
Given that the pentagonal base has an area of 6194 square inches, and the length c (which represents the height) is 6 inches, we can calculate the volume as follows:
Volume = 6194 in² × 6 in
Volume = 37,164 cubic inches
So, the volume of the metal tool chest is 37,164 cubic inches.
As for the lengths provided:
c = 6 inches (height)
m = 19 inches (unknown side length of the pentagon)
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The sum of two real numbers is 18 and the product is 70. what are the two numbers?
Answers
x+y=18......i
xy=70.........ii
from i
y=18-x.....iii
from ii
y=70/x......iv
substituting iii in iv we get:
70/x=18-x
multiplying through by x;
70=18x-x^2
x^2-18x+70=0
x=12.3 or 5.7
thus:
x=12.3 and y=5.7
a bag contains 5 red marbles, 3 yellow marbles, and 2 blue marbles. Once a marble is drawn, it is not replaced.what is the probability of drawing a blue on the first draw and a second blue marble on the next draw ?
Answers
P(first one being blue) = 2/10 which reduces to 1/5
not replaced
P(second one is blue) = 1/9...because u already pick a blue, meaning there is only 1 left, and there is one less marble, so there is a total of 9 marbles.
P(both events happening) = 1/5 * 1/9 = 1/45 <==
What is the square root of pi?
Answers
Hope this helps! :)
In which section of the number line is √55?
Answers
________________________________________________________
Explanation:
________________________________________________________
The question asks:
________________________________________________________
"In which section of the number line is √55?"
________________________________________________________
We know that: 7² = 49 ; 8² = 64 .
So, 7 < √55 < 8 ; So, √55 will be in between 7 and 8.
So, √49 < √55 < √64 .
55 - 49 = 6 .
64 - 55 = 9 .
So, there is greater difference between the radicands in
√64 and √55 ; compared to the difference in radicands between
√49 and √55 .
________________________________________________
This tells us that the correct answer is:
________________________________________________
[A] { in the figure shown: "7.0 - 7.5" } ;
________________________________________________
rather than: [B]: { "7.5 - 8.0" } .
________________________________________________
Answer:
A. Section A
Step-by-step explanation:
imagine said it was correct
What is the factored form of the expression
Answers
In other words, D) (3g²-2)(5g+6)
What polynomial identity should be used to prove that 20 = 36 − 16
Answers
Diffrence of squares is the correct answer just score pract q
The sum of six consecutive odd numbers is 204. What is the fourth number in the sequence
Answers
6x + 30 = 204
6x = 174
x = 29
six consecutive odd numbers are 29,31,33,35,37,39
answer
the fourth number in the sequence is 35
2n+1 , 2n+3, 2n+5, 2n+7, 2n+9,2n+ 11 and these add up to 204
12n + 36 = 204
12n = 168
n = 14
so fourth number in the sequence is 2(28) + 7 = 63
The point (-2, 6) is on the line given by which of the equations below?
A.
y = 4x
B.
y = 2x
C.
y = 3x
D.
y = -3x
Answers
for (-2, 6) = (x, y) which also means that x = -2 and y = 6
A/ y = 4x => 6 = 4.(-2) <=> 6 = (-8) Absurd
B/ y = 2.x => 6 = 2.(-2) <=> 6 = (-4) Absurd
C/ y = 3x => 6 = 3.(-2) <=> 6 = (-6) Absurd
D/ y = -3x => 6 = (-3).(-2) <=> 6 = 6 Right Answer
find the value of x if f(x) equals 7 f(x) equals 4x minus 1
Answers
Don is 6 feet tall. at a given time of day, he measures his shadow to be 13 feet long. at the same time, he measures the shadow length of a nearby tree to be 51 feet. how tall is the tree
Answers
the equation for this problem would be...
6/13=x/51
you then multpli the denominators on both sides and get...
6*51=13x
you then solve...
306=13x
and then you divide 13 on both sides...
x=23.54ft
the tree is 23.54ft tall
(leave a thanks and a rating if i helped) c:
The height of the tree maintaining the given ratio is 23.54 feet.
What are the ratio and proportion?The ratio is the division of the two numbers.
For example, a/b, where a will be the numerator and b will be the denominator.
Proportion is the relation of a variable with another. It could be direct or inverse.
Given that,
Don height = 6 feet
His shadow = 13 feet
The ratio of height to shadow = 6/13
Tree height = x (let's say)
Tree shadow = 51 feet
Ratio = x/51
Both ratios must be the same, therefore,
x/51=6/13
x=23.54 feet
Hence "The height of the tree maintaining the given ratio is 23.54 feet".
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The trigonometric form of a complex number is unique - True or false?
Answers
Answer with explanation:
let, Z= a + i b,be a complex number.
Where, a = r cos A
b= r sin A
[tex]a^2 + b^2=(r cos A)^2 +(r sin A)^2\\\\ a^2 + b^2=r^2(cos^2 A+sin^2 A)\\\\ a^2 + b^2=r^2\\\\r=\sqrt{a^2 + b^2}[/tex]
[tex]Z=r cos A + i r sin A\\\\Z=r(cos A + i sin A)\\\\ Z=re^{iA},\text{Where}, e^{i\alpha }=cos \alpha +i sin\alpha[/tex]
Now, if we replace A, by, 2 kπ + A,in the above equation,where k is any positive integer,beginning from ,0.k=0,1,2,3,4,....
[tex]Z=r cos (2k\pi +A) + i r sin(2k\pi + A)\\\\Z=r[cos (2k\pi +A) + i sin(2k\pi + A)]\\\\ Z=re^{i(2k\pi +A)}[/tex]
So, for different value of , k ,there will be Different Complex number.
→So,the Statement: The trigonometric form of a complex number is unique = False
The trigonometric form of a complex number is unique is a false statement. Check the reason why it is false below.
Why is the trigonometric form of a complex number not special?The trigonometric form of a complex number is commonly seen as the polar form of that number.
This is due to the fact that there are lots of infinite choices are often made for θ and as such the trigonometric form of a complex number is said to be not unique in any way.
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Solve m=10x−x for x
help
Answers
m = 10x - x, solve for x.
First, we need to combine like-terms.
10x - x = 9x.
We now have:
m = 9x
Divide both sides by 9 to solve for x.
x = [tex] \frac{m}{9} [/tex]
I hope this helps!
Solving for x in terms of m will give [tex]\frac{m}{9}[/tex]
The given expression:
m = 10x - x
To find:
x in terms of mTo find x in terms of m, we use the following simple approach;
[tex]m = 10x - x \ \ \ (you \ can \ subtract \ 'x" \ from \ "10x" \ since \ they \ are \ similar)\\\\m = 9x\\\\divide \ both \ side s\ by \ 9\\\\\frac{m}{9} = \frac{9x}{9} \\\\\frac{m}{9} = x[/tex]
Thus, solving for x in terms of m will give [tex]\frac{m}{9}[/tex]
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How you know when there is only one solution triangle possible from a side-side-angle?
Answers
Answer:
There will be only on solution if either of these conditions is met:
the given angle is opposite the longest sidethe triangle is a right triangle (the ratio of the side opposite the angle to the other given side is equal to the sine of the angle)Step-by-step explanation:
Consider the SSA geometry with the angle placed in standard position at the origin and its adjacent given side (the second side of SSA) extending along the +x axis. Draw a circle having a radius equal to the length of the first side of SSA, centered on the vertex between the two segments on the +x axis.
There are three possibilities for the way this circle intersects the other ray of the angle (the ray that is not the x-axis):
the first side is as long or longer than the second side, so there is one point of intersection (one solution to the triangle, purple in the attachment)the first side is just long enough to be tangent to the other ray, so there is one point of intersection (one solution to the triangle, green in the attachment)the first side is between these lengths, so intersects the other ray in two places, giving two solutions to the triangle, red in the attachment.if M and N are midpoints in a diagram below, determine values for x and y
Answers
AС = 2*MN
3x + 10 = 2(x+5)
3x + 10 = 2x + 10
3x - 2x = 10 - 10
x = 0
M is a midpoint of AB ⇒
AM = BM
3y - 8 = 2(y+1)
3y - 8 = 2y + 2
3y - 2y = 2 + 8
y = 10
A point on an ellipse is 11 units from one focus and 7 units from the other. What is the length of the major axis?
Answers
VW is the major axis
ST is the minor axis
P is a point on the ellipse
A and B are the foci (or focus)
The length of major axis is given by either VA+AW or AP+BP
One useful fact to know that AP+BP is constant wherever the point P is on the ellipse
We know that the distance of the point 'P' from one focus is 11 units and from the other focus is 7 units, so we know the length of AP and BP.
Hence, the length of major axis is 11+7=18 units
(06.06) Which of the following demonstrates the Commutative Property of Multiplication? 3(4a − 2) = 12a − 6 3(4a − 2) = (4a − 2) ⋅ 3 12a − 6 = (4a − 2) ⋅ 3 (3 ⋅ 4a) − 2 = 3(4a − 2)
Answers
3(4a - 2) = (4a - 2) * 3 <==
so its the second option:-
3(4a - 2) = (4a - 2).3
Use the given graph to determine the limit, if it exists.
limf(x)
x-->2
Answers
Answer:
Step-by-step explanation:
4
The limit as x tends to 2 is -1
How to find the limit
The limit is sought by investigating the graph. More attention is directed to when the domain or the x-value is at 2.
It can be observed that at this point we have 2 open dots and one closed dot.
The closed dot implies there is a value for x = 2, hence we trace the y coordinate at this point.
Tracing the y coordinate shows that the value is -1
Which of the following ordered pairs is a solution to the equation 2x+7y=8
Select one:
a. (3, -2)
b. (2, -3)
c. (-3, 2)
d. (0, 4)
Answers
The length of a rectangle is eight times it's width. If the length was decreased by 10 meters and the width decreased by 2 meters the area would be decreased by 162 m. Find the original dimensions.
Answers
(l-10)*(w-20)=a-162
Substituting l for 8w and a for l*w and then 8w*w, we have
(8w-10)(w-20)=8w^2-162=8w^2-160w-10w+200=8w^2-170w+200
8w^2-170w+200=8w^2-162 , so we subtract 8w^2 from both sides as well as subtract 200 to both to get -170w=-362, so w=362/170 and
l= 8*362/170=2896/170
A rectangular flower garden has area 32 square feet. if the width of the garden is 4 feet less than the length, what is the perimeter, in feet, of the garden?
Answers
Area = Length x Width
where length = x ft
width = x - 4 ft
area = 32 ft^2
32 = x (x -4)
32 = x^2 - 4x
Solving for the positive value of x,
x = 8 ft = length
width = x -4 = 8 - 4 = 4 ft
Perimeter = 2(length) + 2(width) = 2(8) + 2(4) = 24 ft
The perimeter of the garden is 24 feet.
What is a Rectangle?A Rectangle is a four sided-polygon, having all the internal angles equal to 90 degrees.
How to solve it?We have, an area is 32 and width is 4,
so, the length will be 32/4 = 8 feet.
The perimeter will be 2(8+4) = 2(12)= 24 feet.
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Where do the following graphs intersect? Show how you can find the intersection point algebraically.
1) y-2x=3
2) y-3= x^2
Answers
y-2x = 3
y-2x+2x = 3+2x
y = 2x+3
Plug that into equation (2)
y - 3 = x^2
2x+3 - 3 = x^2
2x = x^2
Get everything to one side
2x = x^2
2x-2x = x^2-2x
0 = x^2-2x
x^2-2x = 0
Now factor and use the zero product property to find the solutions for x
x^2-2x = 0
x(x-2) = 0
x = 0 or x-2 = 0
x = 0 or x = 2
If x = 0, then y is...
y = 2x+3
y = 2(0)+3
y = 3
So (x,y) = (0,3) is one solution of the system. This is one point where the graphs intersect.
If x = 2, then y is...
y = 2x+3
y = 2(2)+3
y = 4+3
y = 7
So (x,y) = (2,7) is another solution to the system. This is another point where the graphs intersect.
See the attached image for a visual. The two curves cross at point A and point B
A = (0,3)
B = (2,7)
A parabola has a focus of F(2, -0.5) and a directrix of y=-1.5 P(x,y) represents any point on the parabola, while D(x, -1.5) represents any point on the directrix.
What are the steps required to find the equation of the parabola?
Answers
We have the focus [tex](a,b) = (2, -0.5)[/tex]
The point [tex]P(x,y)[/tex]
The directrix, c at [tex]y=-1.5[/tex]
The steps to find the equation of the parabola are as follows
Step 1
Find the distance between the focus and the point P using Pythagoras. We have two coordinates; [tex](2, -0.5)[/tex] and [tex](x,y)[/tex].
We need the vertical and horizontal distances to find the hypotenuse (the diagram is shown in the second diagram).
The distance between the focus and point P is given by
[tex] \sqrt{ (x-a)^{2}+ (y-b)^{2} } [/tex]
Step 2
Find the distance between the point P to the directrix [tex]c[/tex]. It is a vertical distance between y and c, expressed as [tex]y-c[/tex]
Step 3
The equation of parabola is then given as
[tex] \sqrt{ (x-a)^{2}+ (y-b)^{2} } [/tex]=[tex]y-c[/tex]
[tex] (x-a)^{2}+ (y-b)^{2}= (y-c)^{2} [/tex] ⇒ substituting a, b and c
[tex] (x-2)^{2}+ (y--0.5)^{2} = (y--1.5)^{2} [/tex]
[tex] (x-2)^{2}+ (y+0.5)^{2}= (y+1.5)^{2} [/tex]⇒Rearranging and making [tex]y[/tex] the subject gives
[tex]y= \frac{ x^{2} }{2} -2x+1[/tex]
1. In how many different ways can 5 people be seated at a round table?
2. Each block on the grid is 1 unit by 1 unit. We wish to walk from A to B via 7 unit path but we have to stay on the grid. How many different paths can we take?
3. How many different "words" can be made from the given word by re-arranging the letters?
a) VECTOR
B) TRUST
Answers
Assume the people A, B, C, D and E are sitting in a row.
since there are 5 positions, they can sit in 5*4*3*2*1=120 ways.
consider 1 certain sitting position, for example ABCDE, that is B has A to his right, C to his left. D has C to his right and E to his left. And so on.
the positions
ABCDE
BCDEA
CDEAB
DEABC
EABCD
while in a row are different positions, in a round table they are the same thing.
(read the letters starting from A, and when the letters finish, continue reading from the beginning of the row. They all read "ABCDE")
This means that any arrangement in a round table, is converted to 5 arrangements in a row.
So there are in total [tex] \frac{5!}{5}=4*3*2*1=24 [/tex] arrangements of 5 people around a table.
2. check picture.
Consider the case where A is at (2, 2) and B is at (5, 6)
A can move to B through 3 horizontal units to the right, we call them E (for East) and 4 vertical units up, which we call N (for north).
the total path is 7 units.
it can be done in several ways, for example:
ENNNENE (the red path shown in the figure)
NNEEENN (the black path shown in the figure)
In total there are
[tex] \frac{7!}{3!*4!}= \frac{7*6*5*4!}{3!*4!}= \frac{7*6*5}{3!}= 35 [/tex] paths.
Remark: 7! is the number of arrangements if the letters were different. We divide by 3! because of the 3 E's and 4! because of the 4 N's.
Another possibility could be A'(3, -5), B'(8,-3).
From A' to B' we go by a total of 5 E and 2 N, so there are
[tex] \frac{7!}{2!*5!}= \frac{7*6*5!}{2!*5!}= \frac{7*6}{2}= 42 [/tex] paths in total.
there is one more possibility
[tex] \frac{7!}{1!(6!)}=7 [/tex]
C.
a) consider the letters {V,E,C,T,O,R}
we can form 6*5*4*3*2*1=720 words, as the first digit can be any of the 6 letters, combined with 5 for the second letter and so on.
b) the difference with a is that we have 5 letters and we have 2 T.
we have in total 5*4*3*2*1=120 arrangements of these letters, if the 2 T's were considered as separate.
consider the arrangements:
[tex]T_1RUST_2[/tex] and [tex]T_2RUST_1[/tex].
We read bth as just TRUST, but the permutation formula 5*4*3*2*1 considers these as 2 different.
this is why we need to divide 120 by 2, to get the actual number of words that can be formed. So the number is 60.
Answers:
1) 24
2) 7, 35 or 42. it depends on the positions of A and B, check the solution
3) a-720, b-160
The distance from earth to the moon is about 22^4 miles. the distance from earth to neptune is22^7 miles. which distance is the greatest and by how much
Answers
22^7=22x22x22x22x22x22x22
This means that 22^7 is greater and by the amount of 22^3
Find the equation of the line perpendicular to -9x+4y=8 that contains the point (9,7)
Answers
Answer:
[tex]\[y=-\frac{4}{9}x+11\][/tex]
Step-by-step explanation:
Equation of the given line is [tex]\[-9x+4y=8\][/tex]
Slope of the line = [tex]\[\frac{9}{4}\][/tex]
Slope of the perpendicular line = [tex]\[-\frac{4}{9}\][/tex]
Equation of the line perpendicular to the given line is [tex]\[y=mx+c\][/tex]
[tex]\[y=-\frac{4}{9}x+c\][/tex]
But this line passes through (9,7)
Substituting the values in the equation:
[tex]\[7=-4+c\][/tex]
=> [tex]\[c=7+4=11\][/tex]
So the overall equation of the parallel line is given by [tex]\[y=-\frac{4}{9}x+11\][/tex]
Find the circumference of a circle with radius of 18 in. Leave your answer in terms of π
Answers
C=2(18)π
C=36π in.
C = 2*pi*18
C = 36pi