Let the lengths of each side of △ABC having area equal to 1 be as follows: AB = 2, BC = a and CA = b. Let CD be a perpendicular line from point C to AB. Answer the following questions.
(1) Given AD = x, write a²+(2√3-1)b² in the form of x.
(2) Find the value of x at which a²+(2√3 - 1)b² is the lowest and the magnitude of ∠BAC.
Need help! Please show your work too. Thanks!
Answers
Answer:
Part 1)
[tex]\displaystyle \left((2-x)^2 + 1)\right) + (2\sqrt{3} - 1 ) \left(x^2 + 1\right)[/tex]
Or simplified:
[tex]\displaystyle = 2\sqrt{3}x^2 - 4x + 4 + 2\sqrt{3}[/tex]
Part 2)
The value of x for which the given expression will be the lowest is:
[tex]\displaystyle x = \frac{\sqrt{3}}{3}\approx 0.5774[/tex]
And the magnitude of ∠BAC is 60°.
Step-by-step explanation:
We are given a ΔABC with an area of one. We are also given that AB = 2, BC = a, and CA = b. CD is a perpendicular line from C to AB.
Please refer to the diagram below.
Part 1)
Since we know that the area of the triangle is one:
[tex]\displaystyle \frac{1}{2} (2)(CD) = 1[/tex]
Simplify:
[tex]\displaystyle CD = 1[/tex]
From the Pythagorean Theorem:
[tex]\displaystyle x^2 + CD^2 = b^2[/tex]
Substitute:
[tex]x^2 + 1 = b^2[/tex]
BD will simply be (2 - x). From the Pythagorean Theorem:
[tex]\displaystyle (2-x)^2 + CD^2 = a^2[/tex]
Substitute:
[tex]\displaystyle (2-x)^2+ 1 = a^2[/tex]
We have the expression:
[tex]\displaystyle a^2 + (2\sqrt{3} - 1) b^2[/tex]
Substitute:
[tex]\displaystyle = \boxed{\left((2-x)^2 + 1)\right) + (2\sqrt{3} - 1 ) \left(x^2 + 1\right)}[/tex]
Part 2)
We can simplify the expression. Expand and distribute:
[tex]\displaystyle (4 - 4x + x^2 + 1)+ (2\sqrt{3} -1)x^2 + 2\sqrt{3} - 1[/tex]
Simplify:
[tex]\displaystyle = ((2\sqrt{3} -1 )x^2 + x^2) + (-4x) + (4+1-1+2\sqrt{3})[/tex]
Simplify:
[tex]\displaystyle = 2\sqrt{3}x^2 - 4x + 4 + 2\sqrt{3}[/tex]
Since this is a quadratic with a positive leading coefficient, it will have a minimum value. Recall that the minimum value of a quadratic always occur at its vertex. The vertex is given by the formulas:
[tex]\displaystyle \text{Vertex} = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)[/tex]
In this case, a = 2√3, b = -4, and c = (4 + 2√3).
Therefore, the x-coordinate of the vertex is:
[tex]\displaystyle x = -\frac{(-4)}{2(2\sqrt{3})} = \frac{1}{\sqrt{3}} =\boxed{ \frac{\sqrt{3}}{3}}[/tex]
Hence, the value of x at which our expression will be the lowest is at √3/3.
To find ∠BAC, we can use the tangent ratio. Recall that:
[tex]\displaystyle \tan \theta = \frac{\text{opposite}}{\text{adjacent}}[/tex]
Substitute:
[tex]\displaystyle \tan \angle BAC = \frac{CD}{x}[/tex]
Substitute:
[tex]\displaystyle \tan \angle BAC = \frac{1}{\dfrac{\sqrt{3}}{3}} = \sqrt{3}[/tex]
Therefore:
[tex]\displaystyle\boxed{ m\angle BAC = \arctan\sqrt{3} = 60^\circ}[/tex]
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An internet company reported that its earnings will be less than the 24 cents per share that was predicted
Answers
Answer:
p < 0.24
Step-by-step explanation:
If earning is 24% less Than predicted, this means that the we can represent the predicted earning with a variable, p
If the predicted earning = p
Then, the actual earning can be represented by the inequality ; p < 24%
p < 0.24
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Step-by-step explanation:
angle marked on diagram = 180 - 75 = 105° (supplementary angles)
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Which of the following should be the y-coordinate of left parenthesis (5,__) so that the ordered pair is a solution of 4x - y= -10
Answers
Answer:
y = 30
Step-by-step explanation:
Substitute x = 5 into the equation and solve for y
4(5) - y = - 10
20 - y = - 10 ( subtract 20 from both sides )
- y = - 30 ( multiply both sides by - 1 )
y = 30
Then (5, 30 ) is a solution of the equation
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х=10
I'm so sorry, i don't know how it's called (because English isn't my native language) but there are angles that have the same value
maybe u'll tell me..
Name the postulate or theorem you can use to prove the triangles congruent
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ASA
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Answer:
ASA
Step-by-step explanation:
if you understood the other, similar questions now, this one is really, really easy.
we got the confirmation for 2 angles and the side between these angles.
so, angle side angle or ASA
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Step-by-step explanation:
a.)
[tex]14*2^x[/tex]
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1. when the expression x^3+ 2kx+2 is divided by x+2, the remainder is less than the remainder when the expression is divided by x+1 find the value of k. help plz
Answers
Answer:
let the polynomial be
f(x)=x³+2kx+2
is divided by
g(x)=x+2
hence
(x+2) is a factor of f(x)
so comparing with x-a we get a=-2
so
f(a)=0
f(-2)=(-2)³+2*-2*k+2
0=-8-4K+2
4k=-6
K=-6/4
k=-3/2
again
when divided by x+1
g(x)=x+1
hence
x+1 is a factor
so a=-1
f(-1)=-(1)³+2*-1*k+2
0=-1-2k+2
0=1-2k
k=1/2
The value of k is 1/2 when divided by x+1
and
K =-3/2 when divided by x+2
Which of the following polynomial equations has 1, 4, -5 as three of its roots?
x 2+4x-5=0
x 4+2x 3-5x+10=0
x 3-21x+20=0*
Answers
Answer: x³-21x+20=0
Step-by-step explanation:
To find which polynomial has 1, 4, -5 as the roots, all we have to do is equal each root to 0 and multiply all factors together.
1 is (x-1)=0
4 is (x-4)=0
-5 is (x+5)=0
Now, we just multiply them together.
(x-1)(x-4)(x+5)=0 [FOIL]
(x²-4x-x+4)(x+5)=0 [combine like terms]
(x²-5x+4)(x+5)=0 [FOIL]
x³+5x²-5x²-25x+4x+20=0 [combine like terms]
x³-21x+20=0
Now we know x³-21x+20=0 is the polynomial with those roots.
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Answer:
Step-by-step explanation:
Is the answer I’ve chosen correct? This is really confusing for my stressed brain :(.
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Answer:
X is greater than 5.
Step-by-step explanation:
You choose the correct answer because X > 5.
Meaning that x would be any number over 5.
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Answer:
D and B
Step-by-step explanation:
2) (59.6)/180 * pi=1.04
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Answer:
smart dot company would be c=.50t+12
communication plus would be c=2.50t
Step-by-step explanation:
smart dot starts with $12 a month and adds $0.50 for each hour of internet use. communication plus has no starting fee so it just adds $2.50 for each hour of internet use.
So then for the table you would put the number given for time in for the letter t and find what c would be. This would then find your cost for that amount of hours. To get you started smart dot company starts off with 0 hours so put that in for t being c=0.50(0)+12 and c=12. Communication plus also starts with 0 so it would be c=2.50(0) and c=0
Then you graph once you have both values you can graph having x=time and y=cost
Let me know if this helps!
a student takes a rectangular piece of paper 30 cm long and 21 cm wide find the area of the biggest circle that can be cut out from the paper also find the area of the paper left after cutting out the circle
Answers
Answer:
Step-by-step explanation:
diameter of the circle=smallest of the length and width=21 cm
r=21/2
area of the circle=πr²=22/7 ×(21/2)²=(21×441)/(7×4)=1323/4=330.75 cm²
there are 160 girls and 240 boys in a school
i. find the percent of girls in the school
ii. find the percent of boys in the school.
Answers
Answer:
girls = 40%
boys = 60%
Step-by-step explanation:
Girls = 160
boys = 240
total students = 160+240
= 400
so,
percent of girls = (160/400)×100%
= (2/5)×100%
= 2×20%
= 40%
so,
percent of boys = 100% - 40%
= 60%
What is the solution to the system that is created by the equation y = - x + 6
Answers
Answer:
If you draw the graph of y = -x + 6 on the above graph the solution to system will be the coordinates of the point of intersection.
the equation of the line drawn above is y = 0.5x
so you can also find the solution by solving the system
y = 0.5x
y = -x + 6
0.5x = -x + 6
1.5x = 6
x = 6/1.5 = 4
and y = 0.5*4 = 2
The answer is (4, 2) - the third choice.
Step-by-step explanation:
If ori has 10 mangos and gives 20 mangos to lucy,how many mangos does ori need to give
Answers
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Answers
Answer:
$239
Step-by-step explanation:
20+25+22+30+15+18+20+17+30+27+15=239
Explanation: 20+25=45
22+30=52
15+18=33
20+17=37
30+27=57
45+52=97
52+33=85 > 276+15=291
37+57=94
Is the following number rational or irrational?
13
12
Choose 1 answer:
А
Rational
B
Irrational
Answers
Answer: Rational
This is because the number is in the form P/Q where P and Q are integers
We have P = -13 and Q = 12.
You can think of "rational" as in "ratio" which is very closely tied to fractions. Any rational number is a fraction of two integers. The denominator can never be zero.
Answer:
Rational
Step-by-step explanation:
A rational number is any integer, fraction, terminating decimal, or repeating decimal.
Decimal form: − 1.08 3
3 repeats itself, so it's rational.
In the diagram below ABE = 65° and DĈF = 32º. E- B 650 D 32°C A F Calculate the size of EBC
Answers
Step-by-step explanation:
lol wea is the diagram??
Which value of x makes this equation true?-9x+15=3(2-x)
Answers
Step-by-step explanation:
-9x+15=3(2-x)
expand the bracket by the right hand side6-6x
2. collect like terms
-9x+15= 6-6x
15-6 = 6x+9x
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3. divide both sides by the coefficient of X which is 15
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TWO TEST
23. Evaluate 4b2 for b
= -1/2
Answers
4b² when b = -1/2
[tex] = {4( \frac{ - 1}{2})}^{2} \\ = 4( \frac{1}{4}) \\ = \frac{4}{4} \\ = 1[/tex]
Answer:
-4
Step-by-step explanation:
4b x 2 when b = -1/2
1) put -1/2 where b is.
4 x -1/2 x 2
2) solve.
-2 x 2
-4
pls help its a optional math question
Find the length of a pendulum which describes an arc of 33cm when it turns through and angle 27º.
Answers
Answer:
70cm
Step-by-step explanation:
33=27/360*2*22/7*r
33*2520/1188=r
83160/1188=r
r=70cm
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Solve the equation √2 – 1 = 3.
Question 5 options:
A)
x = 4
B)
x = 16
C)
x = 8
D)
x = 2
Answers
Answer:
C x=8
Step-by-step explanation:
Chúc bạn ngày mới tốt lành
G.C.D of 55 and 121 is are
Answers
Common factor of 55 and 121 is 11
Answer:
= 11
Find the Greates Common Deviser :
55 = 5, 11, 55.121 = 11, 121.The 12th term of the arithmetic sequence is 10.5. The 18th term of this sequence is 13.5. Find the common difference and the first term.
Answers
let the n'th term be called x, and the value of the term y,
then there is a function y=a*x + b
that will give us all term we want.
this formula is also used for straight lines.
we just need a and b. we already got two data points. we can just plug the known x/y pairs into the formula
10.5 = a*12 + b
13.5 = a*18 + b
now lets manipulate these lines.
multiply the first line by 3 and the second line by 2
31.5 = a*36 + 3b
27 = a*36 + 2b
subtract the second line from the first line
4.5 = b
yey, we now know b, let's plus b into either line from above, I'll go with the first one, looks easier.
10.5 = a*12 + 4.5
6 = a*12
0.5 = a
now y=a*x + b can be filled with a and b
y = 0.5 * x + 4.5
for x=1 (the first term) it's f(1)=5
each step is 0.5, hence the common difference.
2and way:(13.5-10.5)/(18-12)
= 3 / 6
= 0.5
we got the comment difference by looking at the full difference over 6 steps and divined by these 6 steps.
from step 12 to step one it's 11 steps down
10.5 - 0.5*11
= 10.5 - 5.5
= 5
okay I admit... 2and way to do it might be faster and more intuitive... :D4(1-2x)>32
solve for x
Answers
Answer:
x < -7/2
Step-by-step explanation:
First, perform the indicated multiplication:
4(1-2x)>32 becomes 4 - 8x > 32.
Next, isolate the -8x term by subtracting 4 from both sides of this inequality:
-8x > 28
Finally, solve for x by dividing both sides by -8. This requires that we reverse the direction of the inequality sign:
x < -28/8, or, after reduction, x < -7/2
Which angles are supplementary to each other?
Angle 5 and Angle 2
Angle 3 and Angle 8
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Answers
angle 8 and 5 are supplementary angle
Find −4^3(5^2). Express using exponents.
Answers
Answer:
-1600
Step-by-step explanation:
−4^3(5^2)
5 × 5 = 25
-4 × -4 × -4
-16 × -4
-64
64 × 25 = -1600
The answer is -1600
Hope this helped.
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(3, -1)
(-1, -5)
(-4, 9)
Answers
Answer:
Y = 2x - 4
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gradient= 2
y-intersept = -4
Cho a + b + c = 0 và a^2 + b^2 + c^2 = 1. Tính giá trị của M = a^4 + b^4 + c^4
Answers
Step-by-step explanation:
hello you then good mujhe samajh mein nahin a rahi hai sorry
