In the diagram, ABC is a triangle in which AB = 4 cm, BC = 6 cm and angle ABC = 150°. The line CX is perpendicular to the line ABX.
(i) Find the exact length of BX and show that angle CAB = tan^-1 (3/4+3√3)
(ii)
Show that the exact length of AC is √52 + 24√3 cm.
Answers
Part (i)
Angle ABC is 150 degrees. The angle CBX is supplementary to this, so,
(angleABC)+(angleCBX) = 180
angle CBX = 180-(angleABC)
angle CBX = 180-150
angle CBX = 30 degrees
Because of the right angle marker at point X, we can find that triangle CBX is a 30-60-90 triangle. This special type of right triangle has its hypotenuse twice as long as the short leg, which we'll call y. The long leg is equal to y*sqrt(3) units.
The diagram shows BC = 6 is the hypotenuse, so the short leg is y = 6/2 = 3. The longer leg is y*sqrt(3) = 3*sqrt(3) which is the distance from B to X.
In summary so far,
CX = 3
BX = 3*sqrt(3)
We have enough information to find the tangent of angle CAB, and then find the actual angle itself.
tan(angle CAB) = opposite/adjacent
tan(angle CAB) = (CX)/(AX)
tan(angle CAB) = 3/(4+3*sqrt(3))
angle CAB = arctan( 3/(4+3*sqrt(3)) )
I'm using arctan in place of tan^(-1) which is the same basic function, just different notation.
========================================================
Part (ii)
Focus on triangle AXC. The legs we found were
AX = 4+3*sqrt(3)
CX = 3
Use the Pythagorean theorem to find the hypotenuse AC
a^2 + b^2 = c^2
(AX)^2 + (CX)^2 = (AC)^2
(AC)^2 = (AX)^2 + (CX)^2
(AC)^2 = (4+3*sqrt(3))^2 + (3)^2
(AC)^2 = (4+3*sqrt(3))(4+3*sqrt(3)) + 9
(AC)^2 = 4(4+3*sqrt(3))+3*sqrt(3)(4+3*sqrt(3)) + 9
(AC)^2 = 16+12*sqrt(3)+12*sqrt(3)+27 + 9
(AC)^2 = 52+24*sqrt(3)
AC = sqrt( 52+24*sqrt(3) )
Note that AC is a length so AC is positive. We don't have to worry about the plus minus.
This is known as a nested radical since one square root is buried in another.
We can write that as [tex]AC = \sqrt{52 + 24\sqrt{3}}[/tex] if you're curious as to how that looks on paper.
Using your calculator, you should find that,
[tex]\sqrt{52+24\sqrt{3}} \approx 9.673118[/tex]
Related Questions
Help solve this issue with mathematics
Answers
Answer:
last one
Step-by-step explanation:
i would say the last one, but im not sure, this one is really hard
The cost $y$ (in dollars) to rent a kayak is proportional to the number $x$ of hours that you rent the kayak. It costs $27 to rent the kayak for 5 hours
Answers
Note: Consider we need to find the constant of proportionality.
Given:
The cost y (in dollars) to rent a kayak is proportional to the number x of hours that you rent the kayak.
It costs $27 to rent the kayak for 5 hours.
To find:
The constant of proportionality.
Solution:
The cost y (in dollars) to rent a kayak is proportional to the number x of hours that you rent the kayak.
[tex]y\propto x[/tex]
[tex]y=kx[/tex]
where, k is constant of proportionality.
It costs $27 to rent the kayak for 5 hours. So, put x=5 and y=27 in the above equation.
[tex]27=k(5)[/tex]
Divide both sides by 5.
[tex]\dfrac{27}{5}=k[/tex]
Therefore, the constant of proportionality is [tex]\dfrac{27}{5}[/tex].
8x-4y= -56 put in
slope-intercept form
Answers
Answer: x=2x +14
Step-by-step explanation:
1) subtract 8x from both sides: -4x= -8x -56
2) divide both sides by -4: x=2x +14
Determine the solution to the equation. 8+4x=2x+8+2x
Answers
Answer: x= all real numbers.
Step-by-step explanation: Simplify the equation first. 8+4x=2x+8+2x. Add the variables on the right side. 4x+8=4x+8. Because the two sides are exactly equal, x can be all real numbers.
Answer:
8=8
Step-by-step explanation:
8+4x=2x+8+2x
combine like terms
8+4x=4x+8
subtract 4 from both sides
8=8
I HOPE THIS HELPED!!!!
There are 500 students in the 8th grade. Three percent were absent Friday. How many 8th graders were NOT absent Friday?
Answers
Answer:
485
Step-by-step explanation:
500/100=5
5*3=15
500-15=485
Hope this helps;)
Answer:
485
Step-by-step explanation:
slope of - 1/2 and y-intercept of 3
Answers
Answer: 5/2
Step-by-step explanation:
Twelve large bottles of water cost $9.How many bottles can you buy for $3?
Answers
Answer: 4 large bottles of water for $3.
Step-by-step explanation: To find how many large water bottles you can buy with $3, you’ll have to divide 12 by 3 which equals to 4. You can check your answer by multiplying 4 x 3 which equals to 12, which is the original amount of large water bottles you can buy for $9.
Solve for x
3x−8≤23 AND −4x+26≥6
Answers
Answer:
Step-by-step explanation:
Ok so the first one is X<31/3
And the second one is X>-5
bye cus im so bad at math smh.
on top it says:
“The graph shows the amount two cleaning companies are charged based on the number of hours it takes them to clean.”
SOMEONE HELP PLS
Answers
The answer is the third one. Trust me. I don't feel like explaining.
Step-by-step explanation:
Evaluate 9/12 × 6/9 Give your answer in its simplest form.
Answers
Answer:
1/2
Step-by-step explanation:
reduce 12 with 6 and 9 with 9 and your equation is now 1/2 x 1/1
you now multiply ur franction, which gives you 1/2
hope this helped :)
Can someone solve all these answers for this polynomials worksheet?
Answers
Answer:
2. 4
3. 5
4. 8
5. 9
6. 1
7. 6
8. 5
9. 7
10. 2
Step-by-step explanation:
Count the number of terms for each. Alternately, count the number of "+" and "-", and add 1.
Fifteen more than the square of a number is eight times the number.
Answers
Answer: n = 3 or 5
n^2+15=8n
Or,
n2 - 8n + 15 = 0
(n - 3)(n - 5) = 0
So n is 3 or 5.
What is the speed of a cheetah that travels 116 m in 3.0seconds
Answers
This is because speed= distance/time
Answer:
38.6666666667
Step-by-step explanation:
speed = distance/time.
Time: 3.0
Distance: 116 miles.
116/3.0 = 38.6666666667.
What is the solution to the equation?
3 + 3x – 5 = r
A.
-2 and -7
B.
2 and 7
O c.
-2
D. 7
Answers
Which is the definition of a ray
Answers
all of the lines in which light and heat may stream through from the sun or any luminous body, or pass through small openings.
Step-by-step explanation:
Answer:
Ray: Any of a set of straight lines passing through one point. Each of the lines in which light (and heat) may seem to stream from the sun or any luminous body, or pass through a small opening.
(I'm so sorry if this is wrong)
Hope this Helps!
Austin makes a
Commission of 20% on
his sales each week.
Last week, his sales
were $520. How much
was Austin's
Commission?
Answers
Answer:
104$
Step-by-step explanation:
520/10*2
Answer:
$104
Step-by-step explanation:
cool person said it so
PLEASE ANSWER ASAP
Part I: Given the function ()-3./242)
A. What restriction is there on the value under the square root symbol? In other words, what can't you
do with a square root expression? (1 point)
Answers
Answer:
x - 2 ≥ 0
Step-by-step explanation:
The expression under the square root cannot be negative, thus
x - 2 ≥ 0 , that is
x ≥ 2
1 is 25% of what number
Answers
Answer:
4
Step-by-step explanation:
because 25% times 4 makes 100 so it is 4
A certain shade of paint is made by mixing 8 parts blue paint with 3 parts
white
paint. To get the correct shade, how many quarts of white paint should be mixed
with 64 quarts of blue paint?
Answers
Answer:
15 quarts
Step-by-step explanation:
An element's identity is determined by the number of
Answers
Answer:
number of protons
Step-by-step explanation:
The identity of an element is determined by the number of protons. One cannot alter the number of protons without altering the identity of the element. By adding a proton, the atomic number increases by one and the element identity changes. Number of neutrons can be altered to create isotopes.
if petrol cause $1.20 per gallon how much would 5 3/4 gallons cost
Answers
Answer:
$6.90
Step-by-step explanation:
1.20 x (5 3/4) = 6.9
The three-digit number "ab5" is divisible by 3. How many different three-digit numbers can "ab5" represent?
Answers
Answer:
30
Step-by-step explanation:
I can't prove this, but I can tell you the answer.
30.
a = 100 s digit
b = 10 s digit
5 is always the units digit.
I'm pretty sure that 30 is the correct answer.
What an interesting question. If I find out how it is done, I'll post it in the comments.
I can say this much.
a + b + 5 must be divisible by 3.
a = 1 b = 9 + 5
a = 1 b = 6 + 5
a = 1 b = 3 + 5
a = 1 b = 0 + 5
===============
Three more are obtained by interchanging the 100s and 10s digit except for b = 0
a = 9 b=1 + 5
a = 6 b =1 + 5
a = 3 b= 1 + 5
==================
a = 2 b = 2 + 5
a = 2 b = 5 + 5
a = 2 b = 8 + 5
only 2 more are obtained by interchanging the 10s and 100s digit
=============
a = 3 b = 4 +5
a = 3 b = 7 +5
You can't go any further. You get 2 more by interchanging the b and a digits.
=============
I hope you see what I've done here. I don't think this is an exact proof, but it is an exhaustive search.
$300 + $0.07s = $426
Answers
Answer:
0.0.0.0
Step-by-step explanation:
i just dont know it thpo jrnij4tg ihtbgotno5ybih6
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I NEED THE ANSWER NOW
Answers
Answer:
The last one
Step-by-step explanation:
68/5 - 22/5 = 46/5 = 9 1/5
Answer:
Option d
Explanation:
[tex]\frac{68}{5} - \frac{22}{5} = \frac{46}{5}[/tex]
46/5 in mixed fraction is [tex]9\frac{1}{5}[/tex]
Which linear equations have an infinite number of solutions? Check all that apply.
(x – 3\7) = 2\3 (3\2x – 9\14 )
8(x + 2) = 5x – 14
12.3x – 18 = 3(–6 + 4.1x)
1\2(6x + 10) = 7(3\7x – 2)
4.2x – 3.5 = 2.1 (5x + 8)
Answers
Answer:
a) (x – 3\7) = 2\3 (3\2x – 9\14 )
c) 12.3x – 18 = 3(–6 + 4.1x)
Step-by-step explanation:
A linear equation can be said to have an infinite number of solutions when the number of the right hand side is the same at the number on the left hand side of the equation.
For example:
2 = 2 or x = x or 0 = 0
Solving the question above:
Which linear equations have an infinite number of solutions? Check all that apply.
a) (x – 3\7) = 2\3 (3\2x – 9\14 )
x - 3/7 = 6x/6 - 18/42
x - 3/7 = x - 3/7
Collect like terms
x - x = -3/7 + 3/7
0 = 0
The linear equation in option a has an infinite number of solutions
b) 8(x + 2) = 5x – 14
8x + 16 = 5x - 14
Collect like terms
8x - 5x = -14 - 16
3x = -30
x = -30/3
x = -10.
The linear equation in Option b has a true solution.
c) 12.3x – 18 = 3(–6 + 4.1x)
12.3x - 18 = -18 + 12.3x
Collect like terms
12.3x - 12.3x = -18 + 18
0 = 0
The linear equation in Option c has infinite number of solutions
d) 1\2(6x + 10) = 7(3\7x – 2)
6x/2 + 5 = 21/7x - 14
3x + 5 = 3x - 14
Collect like terms
3x - 3x = -14 - 5
0 = -19
The linear equation Option d has no solution
e) 4.2x – 3.5 = 2.1 (5x + 8)
4.2x - 3.5 = 10.5x + 16.8
Collect like terms
4.2x - 10.5x = 16.8 + 3.5
-6.3x = 20.3
x = -20.3/6.3
x = -3.22
Hence, the linear equation in Option e has a true solution
Therefore, the linear equations have an infinite number of solutions are:
a) (x – 3\7) = 2\3 (3\2x – 9\14 )
c) 12.3x – 18 = 3(–6 + 4.1x)
Answer(x – 3\7) = 2\3 (3\2x – 9\14 ) and (12.3x – 18 = 3(–6 + 4.1x)
Step-by-step explanation:
Emily is preparing cookies for her friends she is at the market where she sees a 14 ounce bag of chocolate chips for 3.50 the market also sells a 24 ounce bag for 5.28 which size bag is better to buy
Answers
Answer:
Bag 2
Step-by-step explanation:
Given
Bag 1:
[tex]14\ ounce = 3.50[/tex]
Bag 2:
[tex]24\ ounce = 5.28[/tex]
Required
Which is a better buy
To get which buy is better, we first get the unit price of each bag.
[tex]Unit\ Price = Amount/Size[/tex]
For bag 1:
[tex]Unit\ Price = 3.50/14[/tex]
[tex]Unit\ Price = 0.25[/tex]
For bag 2:
[tex]Unit\ Price = 5.28/24[/tex]
[tex]Unit\ Price = 0.22[/tex]
The best buy is the one with smaller unit price
Hence, bag 2 is better to buy
Without multiplying, is 4 × 0.36 less
than or greater than 4? Explain its for my little sis oof
Answers
Answer:
Less
Step-by-step explanation:
Think of it like this:
Four times of 0.36
0.36 + 0.36 + 0.36 + 0.36 = 1.44
So 4 × 0.36 is less than 4
Hope I helped your little sista :)
In ARST, mZR = (5x + 15)", mZS = (6x + 8)°, and mZT = (4x + 7)º.
What is the value of x?
Answers
Answer:
10
Step-by-step explanation:
(5x + 15)° + (6x + 8)° + (4x + 7)° = 180°
15x + 30 = 180 ⇒ x = 10
What is the equation of a line passing through the point (-4, 8) that is parallel to the line y = 2x + 6
Answers
Answer:
y = 2x + 16
Step-by-step explanation:
slope is parallel and equal to y = 2x + 6
y intercept = 16
Answer:
b - y=2x+16
Step-by-step explanation:
y=y1+m(x+x1)
(-4,8)
y=8+2(x-4)
y=8+2x-8
add 8 on both sides
y=2x+16
uhh yeahh don't know lol ...
Answers
Answer:
for first blank its 1/3 and second blank is 5
Step-by-step explanation:
Answer:
y=-1/3x+5
Step-by-step explanation:
BRAINLIEST PLS