[tex]\vec{a}[/tex] and [tex]\vec{b}[/tex] are two unit vectors. If [tex]\vec{5a} + \vec{3b}[/tex] and [tex]\vec {a} - \vec {b}[/tex] are perpendicular to each other, then the angle between a and b is ( 2n + 1)π/2.
As given in the question,
Given two unit vectors are : [tex]\vec{a}[/tex] and [tex]\vec{b}[/tex]
|a| = 1 and |b| = 1
Vectors which are perpendicular to each other are :
[tex]\vec{5a} + \vec{3b}[/tex] and [tex]\vec {a} - \vec {b}[/tex]
( [tex]\vec{5a} + \vec{3b}[/tex] ). ( [tex]\vec {a} - \vec {b}[/tex] ) = 0
⇒ 5|a|² -2a.b - 3|b|² = 0
⇒ 5(1)² - 2a.b - 3(1)² = 0
⇒ 5 - 2a.b - 3 = 0
⇒ 2[tex]\vec{a}[/tex].[tex]\vec{b}[/tex] = 2
⇒ [tex]\vec{a}.\vec{b}[/tex] = 1
Let θ be the angle between two vectors
|a||b|cosθ = 1
⇒ (1)(1)cosθ = 1
⇒cosθ = 1
⇒θ = (2n+1)π/2
Therefore, [tex]\vec{a}[/tex] and [tex]\vec{b}[/tex] are two unit vectors. If [tex]\vec{5a} + \vec{3b}[/tex] and [tex]\vec {a} - \vec {b}[/tex] are perpendicular to each other, then the angle between a and b is
( 2n + 1)π/2.
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[tex]x^{2 + 5x+6[/tex]
The resulting factors on factorizing the expression is (x+2)(x+3).
Factorizing quadratic expressionQuadratic equations are equation that has a leading degree of 2.
Writing a number or other mathematical object as the sum of several factors typically smaller or simpler things of the same kind—is known as factorization or factoring.
Given the quadratic equation below:
x^2 + 5x + 6
Factorize to have:
x^2 + 2x + 3x + 6
Group
(x^2+2x)+(3x+6)
Factor out the common variable or value
x(x + 2) + 3(x + 2)
Find the factors
(x+3)(x+2)
Hence the factored form of the expression in its simplest form is (x+2)(x+3)
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Complete question
Factorize x^2 + 5x + 6
which of these expressions is equivalent to 15C3? a) 15C12 b)12C15 c) 15C18 d) 18C15
Answer: Choice A
15C12
=============================================
Reason:
Imagine there are 15 students and we wish to form a committee of three people, where order doesn't matter. The answer to this problem is 15C3 using the nCr formula. Here n = 15 and r = 3.
The 15C3 represents the number of ways to pick the three people on the committee, while 15C12 is the number of ways to not pick a person for the committee. We can think of it like two separate committees if you want.
So 15C3 = 15C12
The numbers after the "C" add to the value of n
The identity is [tex]_nC_r = _nC_{n-r}[/tex]
The r and n-r add to n.
Write an equation of a parabola that passes through the points (-3,-5) , (-1,3) , and (0,4)
Hence the equation of the parabola is y = -x² + 4
In mathematics, a parabola is a roughly U-shaped, mirror-symmetrical planar curve. The same curves can be defined by a number of seemingly unrelated mathematical definitions, which all correspond to it.
A line and a point (the focus) are two components of one definition of a parabola (the directrix). There is less emphasis on the directrix. The parabola is the location of points in that plane that are equally spaced apart from the directrix and the focus. Another definition of a parabola is a right circular conical surface intersected by a plane parallel to another plane tangential to the conical surface, resulting in a conic section.The general equation of a parabola is given by y = ax² + bx+ c
The parabola passes through (-3,-5) , (-1,3) , and (0,4).
Now substituting the values in the parabola we get :
-5 = a(-3)² + b(-3)+ c
or , -5 = 9a -3b + c
again:
3 = a(-1)² + b(-1)+ c
or, 3 = a -b + c
again
4 = a(0)² + b(0)+ c
or, c =4
Solving the equation we get :
9a - 3b = -9 and a-b = -1
a= -1 and b = 0
Hence the equation of the parabola is
y = -x² + 4
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The graph of the derivative of a function, f, is shown. Use the graph to determine the interval on which f is increasing.
A. (-∞, 2)
B. (2, ∞)
C. (-∞, ∞)
D. (-1, ∞)
E. (-∞, 1)
Answer:
B) [tex](2,\infty)[/tex]
Step-by-step explanation:
Since we are given the graph of [tex]f'(x)[/tex], this gives us the value of the derivative, or the rate of change, of [tex]f(x)[/tex] given an x-value. For [tex]f(x)[/tex] to be increasing, then [tex]f'(x) > 0[/tex], which is a positive rate of change. As you may have noticed, when [tex]x > 2[/tex], the value of [tex]f'(x)[/tex] is positive. This means that f increases on the interval [tex](2,\infty)[/tex], or answer B.
Mrs. Byers rents a water trampoline for her son's birthday party. The delivery fee for the trampoline is $75, plus a rental fee of $8 per hour. She wants to spend less than $125 on the trampoline. Find h, the number of hours Mrs. Byers could rent the trampoline?
Answer:
h= 6
Step-by-step explanation:
To find out the value of h, start by forming an equation for the cost of renting a trampoline:
Cost of renting= $(75 +8t), where t is the rental duration.
When the cost is $125,
75 +8t= 125
Subtract 75 from both sides:
8t= 125 -75
8t= 50
Divide both sides by 8:
t= 50 ÷8
t= 6.25
Since she wants to spend less than $125, let's round the value down to the nearest hour.
∴ h= 6
Thus, she could rent it for 6 hours.
Let's check!
Total cost of renting for 6 hours
= 6(8) +75
= $123
Total cost of renting for 7 hours
= 7(8) +75
= $131
Hence, 6 hours is indeed the maximum number of hours she can rent the trampoline for, such that the total cost is less than $125.
Given ΔABC with m∠B = 62°, a = 14, and c = 16, what is the measure of A? Use the law of cosinus.
Given data,
[tex]\begin{gathered} \angle B=62 \\ a=14\text{ } \\ c=16 \end{gathered}[/tex]Use the law of cosines.
[tex]\begin{gathered} b^2=a^2+c^2-2acCos(B) \\ b^2=14^2+16^2-(2\times14\times16\times Cos(62)) \\ b^2=196+256-210.32 \\ b^2=241.68 \end{gathered}[/tex]Thus, the value of b is
[tex]b=15.54[/tex]The cosines formula for angle A isThus, angle A will be
[tex]a^2=b^2+c^2-2bcCos(A)[/tex][tex]\begin{gathered} A=cos^{-1}\lbrack\frac{b^2+c^2-a^2}{2bc}\rbrack \\ A=\cos ^{-1}\lbrack\frac{15.54^2+16^2-14^2}{2\times15.54\times16}\rbrack \\ A=52.67 \end{gathered}[/tex]Therefore, angle A is 52.67 degrees.
If 539 is invested at 17%, how much money will be in the account after 21 months
Step-by-step explanation:
[tex]i = prt \div 100[/tex]
p=539
r=17%
t=21
I=?
[tex]i = 539 \times 17 \times 21 \div 100[/tex]
Need help answering these columns, if more info is needed then i will edit the post to fit it so there's enough information
Applying the definition of the midpoint of a line and other properties, the proof that shows that AC ≅ CD is shown in the image attached below.
What is the Midpoint of a Line?A line is divided into two equal parts by its midpoint. Thus, the midpoint of that line is that point where the line segment is bisected into two parts that are equal.
Given the image showing two lines that intersect each other at point C, the proof below shows that AC is congruent to CD:
1. C is the midpoint of line AE
C is the midpoint of line BD
AE is congruent to BD
(Reason: Given)
2. AC = CE; BC = CD
(Reason: Definition of midpoints)
3. AE = BD
(Reason: Definition of congruent segments)
4. AE = AC + CE;
BD = BC + CD
(Reason: Segment addition property)
5. AC + CE = BC + CD
(Reason: substitution property)
6. AC + AC = BC + BC
(Reason: substitution property)
7. 2AC = 2BC
(Reason: Simplify)
8. AC = CD
(Reason: division property)
9. AC ≅ CD
(Reason: definition of congruent segments).
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A sub shop ordered 40 pounds of tomatoes. If the tomatoes are evenly distributed into 12 tubs, how many pounds will go into each tub?
Using unitary method we conclude that, 3.33 pounds of tomatoes will go into each tub.
In this question, we have been given a sub shop ordered 40 pounds of tomatoes.
The tomatoes are evenly distributed into 12 tubs.
We need to find the number of pounds of tomatoes will go into each tub.
Let x be the number of pounds of tomatoes will go into each tub.
By unitary method,
x = 40 / 12
x = 3.33 pounds
Therefore by using unitary method we conclude that, 3.33 pounds of tomatoes will go into each tub.
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If 3^5 = x, which of the following expressions is equal to 3^11?
A) 243x
B) 3x^2
C) 9x^4
D) x^4
Answer:
b
Step-by-step explanation:
because it is tht simpl
Solve 12-ln(3-X) = 0
Answer:
[tex]x = 3 - {e}^{12} [/tex]
Step-by-step explanation:
first let's isolate the logarythm:
[tex]ln(3 - x) = 12[/tex]
at this point we can apply the definition of logarythm to obtain
[tex]3 - x = {e}^{12} [/tex]
which, with some simple algebra returns
[tex]x = 3 - {e}^{12} [/tex]
For each equation, determine whether it shows a direct variation (that is, shows directly proportional variables).If it does, find the constant of variation and write it in simplest form.3r - 81 = -7O Direct variation ( K =__)O Not direct variation10y = -5xO Direct variation ( K =__)O Not direct variation-Picture included-
Given:
The equations are given as,
[tex]\begin{gathered} 3x-8y=-7 \\ 10y\text{ = -5x} \end{gathered}[/tex]Required:
Direct and inverse variation verification.
Explanation:
(a) The equations are given as follows:
[tex]3x-8y\text{ = -7}[/tex]On simplifying further,
[tex]3x\text{ = 8y - 7}[/tex]The given equation is not a direct variation.
(b) The equation is given as,
[tex]\begin{gathered} 10y\text{ = -5x} \\ -5x\text{ = 10y} \\ x\text{ = -2y } \end{gathered}[/tex]The given equation represents inverse variation.
Answer:
Thus the required result is,
a) Not a direct variation
b) A direct variation( k = -2)
Out of 250 racers who started the marathon, 217 completed the race, 24 gave up, and 9 were disqualified. What percentage did not complete the marathon? Round your answer to the nearest tenth of a percent.
Answer:
209 out of 230 completed that is 209/230 x 100% = 90.869 % the rest did NOT complete 100% - 90.8% = 9.13 %
HURRYYY WILL GIVE BRAINLIESTTTT!!!
1. A player throws a basketball toward a hoop, and it
follows a parabolic path through the points (2,10)
(4,12) and (10,12). Find a quadratic model, and if
the hoop is at (12,10) will the ball pass through the
hoop?
The answer is given below
credits brainly
…Ajavsbyhsbwdujdnwiowbeuhsbwb
Answer:
alphabet
Step-by-step explanation:
if x=3 and y=5 what is 2x squared+3y squared
Answer:
the answer is 93
Step-by-step explanation:
I'm very sure....
Answer: I really wish I could help you but I don't know the answer either
Step-by-step explanation:
JK
Which of the following sets of rational numbers
is ordered from least to greatest?
|-35| 18 |0.58|
The most appropriate choice for modulus of a number and ascending order will be given by-
|0.58|, 18, |-35| is in ascending order.
What is Modulus of a number and ascending order?
Modulus of a number returns the absolute value of the number.
Modulus of positive number is positive and modulus of negative number is also positive.
If the numbers in a series are arranged in increasing order of their value, then the series is arranged in ascending order
Here,
|-35| = 35
18 = 18
|0.58| = 0.58
As, 0.58, 18, 35 is in ascending order,
So, |0.58|, 18, |-35| is in ascending order
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A traffic control engineer reports that 75% of the vehicles passing through a checkpoint are from within the state. What is the probability that fewer than 2 of the next 5 vehicles are from out of state? (Round to 4 decimal places )
Using combinations, we concluded that the probability that fewer than 2 of the next 5 vehicles are from out of state is 0.907.
A checkpoint sees 75% of its traffic coming from within the state.
Therefore, there is a 0.75 percent chance that the car comes from the state.
1-0.75=0.25 represents the likelihood that the car is from outside the state.
Let x now act as the random variable. Consequently, n=9 and X=4
It is necessary to determine the likelihood that less than 2 of the following 5 automobiles are from another state.
We are to calculate the probability of x<2
p=0.25
q=0.75.
Using the formula,
P(x≤2) = ∑ ⁿCₓ pˣ q⁽ⁿ⁻ˣ⁾
P(x≤2) = ∑ ⁵C₂ p² q³ + ⁵C₁ p¹ q⁴ + ⁵C₀ p⁰ q⁵
P(x≤2) = ∑ [tex]\frac{5!}{2! 3!}[/tex] (0.25)² (0.75)³ + [tex]\frac{5!}{1! 4!}[/tex] (0.25)¹ (0.75)⁴ + [tex]\frac{5!}{0! 5!}[/tex] (0.75)⁵
P(x≤2) = 10 × 0.065 × 0.421875 + 5 × 0.25 × 0.3164 + 0.2373
P(x≤2) = 0.2742 + 0.3955 + 0.2373
P(x≤2) = 0.907
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You want to buy a 28,000 car. You can make a 10% down payment, and will finance the balance with a 2% interest for 48 months. What will your monthly payment be
The monthly payment when you want to buy a 28,000 car is $567.
How to calculate the value?It should be noted that when the 10% down payment is made, the amount left will be:
= $28000 - (10% × $28000)
= $28000 - $2800
= $25200
The interest will be:
= PRT / 100
P = $25200
R = 2%
T = 48 months = 4 years
Interest = (25200 × 2 × 4)/100
= $2016
Total amount to be paid:
= $25200 + $2016
= $27216
The monthly payment will be:
= Amount / 48 months
= $27216 / 48
= $567
The monthly payment is $567.
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Answer:
$546.72
Step-by-step explanation:
The length of a rectangular picture frame is 28 inches and its perimeter is 84 inches. What is the width of the picture frame? *
The perimeter of a rectangle can be calculated using the following formula:
[tex]P=2w+2l[/tex]Where
"w" represents the width of the rectangle
"l" represents the length of the rectangle
If you know the perimeter and length of a rectangle, you can use the formula to determine the width. The first step is to write the formula for the width:
-Pass 2l to the left side of the expression by applying the opposite operation to both sides of it:
[tex]P-2l=2w[/tex]-Divide both sides by 2
[tex]\begin{gathered} \frac{P-2l}{2}=\frac{2w}{2} \\ \frac{P-2l}{2}=w \end{gathered}[/tex]Now that we have determined a formula for w, replace the value of the length and perimeter and calculate the width:
P=84in
l=28in
[tex]\begin{gathered} w=\frac{84-2\cdot28}{2} \\ w=\frac{84-56}{2} \\ w=\frac{28}{2} \\ w=14 \end{gathered}[/tex]The width of the rectangular frame is 14 inches long
Solve the systems of equations
-8x + 2y = -4
x^2 + 2y = 5
What’s the correct answer answer asap for brainlist
The option that is unique to only skeletal muscle is Option(B.) They can fatigue with too much movement or exercise.
Why do skeletal muscles get fatigued after heavy exercise?When muscles become fatigued, they provide less force over time.The skeletal muscles that move the bones in the body can become exhausted from overuse.Oxygen levels in the muscles are depleted by overexertion, such as intense exercise.In order to produce energy, the muscles briefly switch to anaerobic metabolism. Lactic acid and energy are produced in order to accomplish this. Muscle weariness results from lactic acid buildup in the muscles.Intracellular acidosis, which is mostly caused by lactic acid buildup, has been identified as the most important contributing factor to skeletal muscle weariness.To learn more about skeletal muscles visit:
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255 divided by 13 and reaminre
Please hurry due today I will give you brainlest
Which numbers are solutions of 4p+6<3p? Choose exactly two answers that are correct
Answer: -12 and -9
Step-by-step explanation:
4p + 6 <3p
4p - 3p < -6
p < -6
So, -12 < -6 is CORRECT
-9 < -6 is CORRECT
-6 is not less than -6 SO THIS IS INCORRECT
6 is not less than -6 so this is INCORRECT
Roberto's Club has a secret knock for its Clubhouse door that only its members know. The members remember to knock on the door exactly n times in a row, were five more than three times n and is equal to the difference between 21 and n . How many times in a row do members knock on the clubhouse door?
Answer:
[tex]\text{ They knock 4 times on the clubhouse door.}[/tex]Step-by-step explanation:
Let n be the times that they knock on the door.
If five more than three times n and is equal to the difference between 21 and n;
[tex]3n+5=21-n[/tex]Solve for n, using inverse operations to solve equations:
[tex]\begin{gathered} 3n+n=21-5 \\ 4n=16 \\ n=\frac{16}{4} \\ n=4\text{ } \\ \text{ They knock 4 times on the clubhouse door.} \end{gathered}[/tex]Use the Distributive Property to simplify.
4(3+2)
help meeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Step-by-step explanation:
it is just a matter of definition that can be easily looked up.
the domain is the interval or set of all valid x values.
while the range is the interval or set of all valid y (function result) values.
so, the domain here is every value of x used in this function.
e.g. does x = -6 have a functional (y) value in the curve ?
no, it does not.
x = - 4 ? no.
x = -2 ? no.
x = -1 ? ah, yes, this is just the left-most point of the curve.
and then all x values further to the right of -1 are covered. as the arrows indicate : into all eternity ("to infinity and beyond !")
so, the domain is [-1, +infinity).
and you need to use B.
please note, the round bracket indicates that that ends of the interval is not included or undefined (after all, infinity is not a number we can work with, it is basically undefined and only represents a principle).
for the range this is tricky, as the graph does not really show us the limit. it could be y = -1 or 0 down to y = -9 or -10.
Kevin earns $6 for 4 polished rocks. The equation y = 1.5x can be used to represent this situation. What is the constant of proportionality?
The constant of proportionality in the equation y=1.5x is 1.5 .
The Constant Of Proportionality(k) in the equation can be defined as the constant value of the ratio between the two proportional quantities .
In Direct proportionality y=kx , where k is the constant of proportionality
means that if x increases then y also increases at the same rate ,
and if x decreases then y also decreases at the same rate .
For Example : constant of proportionality in y=6x , will be 6 .
In the question ,
it is given that
Kevin earns $6 for 4 polished rocks ,
the equation to represent the situation is y=1.5x
So 1.5 is the constant of proportionality .
Therefore , the constant of proportionality in the equation y=1.5x is 1.5 .
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Which expression is equivalent to 2 log4 x + 3 log4 y – 5 log4 z?
log base 4 of the quantity x squared y cubed all divided by z to the fifth end log
log base 4 of the quantity x squared z to the fifth all divided by y cubed end log
log base 4 of the quantity x times y to the fifth all divided by z to the fifth end log
log base 4 of the quantity xy divided by z end log
The expression equivalent to [tex]2 log_4 x + 3 log_4 y - 5 log_4 z[/tex] is the first option that is [tex]log_4(\frac{x^2y^3}{z^5} )[/tex].
Logarithms
A logarithm of a number with a base is equal to another number. A logarithm is just the opposite function of exponentiation.
For example, if [tex]10^2 = 100[/tex] then [tex]log_{10} 100 = 2[/tex].
In a log function of the form [tex]log_ab=x[/tex] ,
a is the base of the log
b is the argument of the log and,
x is the real number
It is the most convenient way to express large numbers.
The properties of logarithms are,
1) [tex]xlog_ab =log_ab^x[/tex]
2) [tex]log_ab-log_ac=log_a(\frac{b}{c} )[/tex]
Hence,
According to 1) , we can write,
[tex]2 log_4 x =log_4x^2\\3 log_4 y = log_4y^3\\ 5 log_4 z = log_4z^5[/tex]
According to 1) and 2), we can write,
[tex]2 log_4 x + 3 log_4 y - 5 log_4 z= log_4(\frac{x^2y^3}{z^5} )[/tex]
Hence, the answer is the first option.
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Help,,,,,,,,,,,,,,,,,,,,,