Here we go ~
[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
[tex]\qquad \sf \dashrightarrow \: (1 - \tan {}^{4} (a) ) \cos {}^{4} (a) [/tex]
[tex]\qquad \sf \dashrightarrow \: (1 + \tan {}^{2} (a) )(1 - { \tan}^{2} (a)) \cos {}^{4}(a ) [/tex]
[ a² - b² = (a + b)(a - b) ]
[tex]\qquad \sf \dashrightarrow \: ( \sec {}^{2} (a) )(1 - ( \sec{}^{2} (a) - 1) )\cos {}^{4} (a) [/tex]
[ sec² a = 1 + tan² a, so : tan² a = sec²a - 1 ]
[tex]\qquad \sf \dashrightarrow \: \bigg( \dfrac{1}{{}cos^{2} (a)} \bigg)(2 - \sec{}^{2} (a) ) \cos {}^{4} (a) [/tex]
[tex]\qquad \sf \dashrightarrow \: \bigg(2 - \dfrac{1}{ \cos {}^{2} (a) } \bigg) \cos {}^{2} (a) [/tex]
[tex]\qquad \sf \dashrightarrow \: \dfrac{2 \cos {}^{2} (a) - 1}{ \cos {}^{2} (a) } \times \cos {}^{2} (a) [/tex]
[tex]\qquad \sf \dashrightarrow \: 2 \cos {}^{2} (a) - 1[/tex]
[tex]\qquad \sf \dashrightarrow \: 2(1 - \sin {}^{2} (a) ) - 1[/tex]
[ sin²a + cos² a = 1, hence sin²a = 1 - cos²a ]
[tex]\qquad \sf \dashrightarrow \: 2 - 2 \sin {}^{2} (a) - 1[/tex]
[tex]\qquad \sf \dashrightarrow \: 1 - 2 \sin {}^{2} (a) [/tex]
Answer:
See proof below
Step-by-step explanation:
Prove [tex]\left(\:1\:-\:tan^4A\right)cos^4A\:=\:1\:-\:2\:sin^2A[/tex]
[tex]\left(1-\tan ^4\left(A\right)\right)\cos ^4\left(A\right)[/tex] can be expressed in sin, cos terms
Use the trigonometric identity [tex]\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}[/tex]
[tex]\left(1-\tan ^4\left(A\right)\right)\cos ^4\left(A\right) = \left(1-\left(\frac{\sin \left(A\right)}{\cos \left(A\right)}\right)^4\right)\cos ^4\left(A\right)[/tex]
[tex]\mathrm{Simplify}\:\left(1-\left(\frac{\sin \left(A\right)}{\cos \left(A\right)}\right)^4\right)\cos ^4\left(A\right)[/tex]
[tex]\mathrm{Apply\:exponent\:rule}:\quad \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}[/tex]
= [tex]\left(1-\frac{\sin ^4\left(A\right)}{\cos ^4\left(A\right)}\right)\cos ^4\left(A\right)[/tex]
Multiplying the expression in parentheses by [tex]\cos ^4\left(A\right)[/tex] we get
[tex]\frac{\cos ^4\left(A\right)-\sin ^4\left(A\right)}{\cos ^4\left(A\right)}\cos ^4\left(A\right)[/tex]
Cancel the common factor [tex]\cos ^4\left(A\right)[/tex]
This gives us
[tex]\cos ^4\left(A\right)-\sin ^4\left(A\right)[/tex]
Now,
[tex]\sin ^4\left(A\right)=\left(\sin ^2\left(A\right)\right)^2[/tex]
[tex]\cos ^4\left(A\right)=\left(\cos ^2\left(A\right)\right)^2[/tex]
[tex]\:\cos ^4\left(A\right)-\sin ^4\left(A\right)[/tex] = [tex]\left(\cos ^2\left(A\right)\right)^2-\left(\sin ^2\left(A\right)\right)^2[/tex]
[tex]\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}x^2-y^2=\left(x+y\right)\left(x-y\right)[/tex]
[tex]\left(\cos ^2\left(A\right)\right)^2-\left(\sin ^2\left(A\right)\right)^2=\left(\cos ^2\left(A\right)+\sin ^2\left(A\right)\right)\left(\cos ^2\left(A\right)-\sin ^2\left(A\right)\right)[/tex]
= [tex]\cos ^2\left(A\right)-\sin ^2\left(A\right)[/tex] [tex]\textrm{ since }\cos ^2\left(A\right)+\sin ^2\left(A\right)[/tex] = 1
Using the fact that [tex]\cos ^2\left(A\right)=1-\sin ^2\left(A\right)[/tex]
we get
[tex]\cos ^2\left(A\right)-\sin ^2\left(A\right) = 1-\sin ^2\left(A\right)-\sin ^2\left(A\right)\\\\= 1-2\sin ^2\left(A\right)[/tex]
Proved
Devon Davies earns an annual salary of
$43,112.00.
What is his biweekly salary?
$829.08
Step-by-step explanation:
43,112 ÷ 52 = 829.08
Find the intervals where f(x) is increasing or decreasing: √x -2x. Ans: increasing (0, 1/4) Decreasing (1/4, ∞)
Answer:
[tex]\textsf{Increasing function}: \quad \left[0, \dfrac{1}{16}\right)[/tex]
[tex]\textsf{Decreasing function}: \quad \left(\dfrac{1}{16}, \infty\right)[/tex]
Step-by-step explanation:
Given function:
[tex]f(x)=\sqrt{x}-2x[/tex]
As a negative number cannot be square rooted, the domain of the function is restricted to [0, ∞).
[tex]\textsf{A function is \textbf{increasing} when the \underline{gradient is positive}}\implies f'(x) > 0[/tex]
[tex]\textsf{A function is \textbf{decreasing} when the \underline{gradient is negative}} \implies f'(x) < 0[/tex]
Differentiating produces an algebraic expression for the gradient as a function of x. Therefore, differentiate the given function:
[tex]\begin{aligned}f(x) & = \sqrt{x}-2x\\& = x^{\frac{1}{2}}-2x\\\implies f'(x) & = \dfrac{1}{2}x^{\left(\frac{1}{2}-1\right)}-2x^{(1-1)}\\ & = \dfrac{1}{2}x^{-\frac{1}{2}}-2x^0\\ & = \dfrac{1}{2\sqrt{x}}-2\end{aligned}[/tex]
Increasing function
To find the interval where f(x) is increasing, set the differentiated function to more than zero and solve for x:
[tex]\begin{aligned}f'(x) & > 0\\\implies \dfrac{1}{2\sqrt{x}}-2 & > 0\\\dfrac{1}{2\sqrt{x}} & > 2\\\ \dfrac{1}{2} & > 2\sqrt{x}\\\dfrac{1}{2 \cdot 2} & > \sqrt{x}\\ \dfrac{1}{4} & > \sqrt{x}\\ \sqrt{x} & < \dfrac{1}{4}\\\left(\sqrt{x}\right)^2 & < \left(\dfrac{1}{4}\right)^2\\ x & < \dfrac{1}{16}\end{aligned}[/tex]
As the domain is restricted, the function is increasing when:
[tex]\textsf{Solution}: \quad 0 \leq x < \dfrac{1}{16}[/tex]
[tex]\textsf{Interval notation}: \quad \left[0, \dfrac{1}{16}\right)[/tex]
Decreasing function
To find the interval where f(x) is decreasing, set the differentiated function to less than zero and solve for x:
[tex]\begin{aligned}f'(x) & < 0\\\implies \dfrac{1}{2\sqrt{x}}-2 & < 0\\\dfrac{1}{2\sqrt{x}} & < 2\\\ \dfrac{1}{2} & < 2\sqrt{x}\\\dfrac{1}{2 \cdot 2} & < \sqrt{x}\\ \dfrac{1}{4} & < \sqrt{x}\\ \sqrt{x} & > \dfrac{1}{4}\\\left(\sqrt{x}\right)^2 & > \left(\dfrac{1}{4}\right)^2\\ x & > \dfrac{1}{16}\end{aligned}[/tex]
Therefore, the function is decreasing when:
[tex]\textsf{Solution}: \quad x > \dfrac{1}{16}[/tex]
[tex]\textsf{Interval notation}: \quad \left(\dfrac{1}{16}, \infty\right)[/tex]
Note: The answer quoted in the original question is incorrect for the quoted function (please refer to the attached graph for proof).
Differentiation Rules
[tex]\boxed{\begin{minipage}{4.8 cm}\underline{Differentiating $ax^n$}\\\\If $y=ax^n$, then $\dfrac{\text{d}y}{\text{d}x}=nax^{n-1}$\\\end{minipage}}[/tex]
Can someone answer these??
+
e. The ratio of walnuts to peanuts in a can of mixed nuts is 3 to 5. There are
walnuts, peanuts, and cashews in the mix. The mix has 100 nuts in total. If
there are 18 walnuts, use a proportion and additional math to determine how
many cashews there are.
$2
3
The number of cashews in the can are 52.
Here, we are given that the ratio of walnuts to peanuts in a can of mixed nuts is 3 to 5.
⇒ walnuts : peanuts = 3 : 5
The number of walnuts in the packet are = 18
⇒ 18 : peanuts = 3 : 5
18/ peanuts = 3/5
peanuts = (5/3) × 18
peanuts = 30
Now, we know that the total number of nuts = 100
⇒ cashews + walnuts + peanuts = 100
cashews + 18 + 30 = 100
cashews + 48 = 100
cashews = 100 - 48
cashews = 52
Thus, the number of cashews in the can are 52.
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find the measure of angle C of triangle ABC if:
Answer:
∠C = 180 - 3α
Step-by-step explanation:
The interior angles of a triangle always add up to 180°
So A + B + C = 180
C = 180 - (A + B)
= 180 - (α + 2α)
= 180 - 3α
Answer this question please
The inequality for 2/3 I 4v + 6 I - 2 ≤ 10 is -6≤v≤3.
Given the inequality is 2/3 I 4v + 6 I - 2 ≤ 10
Isolate the variable by dividing each side by factors that don't contain the variable.
2/3 I 4v + 6 I ≤ 10 +2
arrange the constants on the right side.
2/3 I4v + 6I ≤ 12
multiply the terms.
8v + 12/3 ≤ 12 or -8v - 12/3 ≤ 12
8v + 12 ≤ 36 or -8v -12 ≤ 36
8v ≤ 36 -12 or -8v ≤ 36 + 12
8v ≤ 24 or -8v ≤ 48
v ≤ 24/8 or v ≤ 48/8
v ≤ 3 or v≥-6
therefore, -6 ≤ v ≤ 3.
Hence the inequality is -6 ≤ v ≤ 3.
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C) g(x) = (4-5x)²
If it’s even or odd
Answer:
odd
Step-by-step explanation:
the answer is odd, my dead grandma just finished this question since she failed 7th grade.
Graduation gift of $5,000 average 2.5% interest. 3. what will the total value be after 10 years simple interest?
compare -3/4 and -5/16
A. -3/4 < -5/16
B. -3/4 > -5/16
C. -3/4 = -5/16
LCM can help us to compare the two of the given fractions. The correct option is A, -(3/4) < -(5/16).
What is LCM?The least common multiple that is divisible by both a and b is the smallest positive integer, lowest common multiple, or smallest common multiple of two numbers a and b, generally indicated by LCM.
Bring both the terms to the same denominator for easy comparison. Now, since the LCM of 4 and 16 is 16. Therefore, we can write the two of the fractions as,
-(3/4) = -(3×4)/(4×4) = -12/16
-(5/16) = -5/16
Further, we can write the comparison of the two fractions as,
-12/16 < -5/16
-(3/4) < -(5/16)
Hence, the correct option is A, -(3/4) < -(5/16).
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Ellen wanted to buy the following items: A DVD player for $49.95 A DVD holder for $19.95 Personal stereo for $21.95. Does Ellen have enough money to buy all three items if she has $90?
Answer: no she dosen't
Step-by-step explanation:
If you were to add up all three numbers
49.95+19.95+21.95=91.85
Which means she is $1.85 short to buy all of the items
A storage locker measures 8 feet wide,
12 feet deep, and 9 feet high. The
monthly rental price for the locker
is $3.60 per cubic yard. How much
does it cost to rent the locker each
month? Explain.
The sun of 2,5, and a number amounts to 14. Find the number
Answer:
x=is the number searched
we have to solve this equation:
(2/5)x=14
x=14*5/2=35
35
Step-by-step explanation:
9x-9y-0
-3x+3y=0
elimination process
Answer:
(-10,-10)Step-by-step
explanation:
Yes 9x-9y=03x-4y=10In
elimination, we want both equations to have the same form and like terms to be lined up. We have that. We also need one of the columns with variables to contain opposites or same terms. Neither one of our columns with the variables contain this. We can do a multiplication to the second equation so that the first terms of each are either opposites or sames. It doesn't matter which. I like opposites because you just add the equations together. So I'm going to multiply the second equation by -3.I will rewrite the system with that manipulation:9x-9y=0-9x+12y=-30----------------------Add them up!0+3y=-30 3y=-30 y=-10So now once you find a variable, plug into either equation to find the other one.I'm going to use 9x-9y=0 where y=-10.So we are going to solve for x now.9x-9y=0 where y=-10.9x-9(-10)=0 where I plugged in -10 for y.9x+90=0 where I simplified -9(-10) as +90.9x =-90 where I subtracted 90 on both sides.x= -10 where I divided both sides by 9.The solution is (x,y)=(-10,-10)
How much will a person pay for 6.7 pounds of bananas at a price of $0.64 per pound?
Answer:
Price per pound = $0.64
which means one pound of banana costs 0.64$.
so,If 1 pound costs 0.64$ then 6.7 pounds would cost:
1 pound = $0.64
6.7 pounds= 0.64× 6.7
that is equal to $4.288.
Step-by-step explanation:
Price per pound = $0.64
which means one pound of banana costs 0.64$.
so,If 1 pound costs 0.64$ then 6.7 pounds would cost:
1 pound = $0.64
6.7 pounds= 0.64× 6.7
that is equal to $4.288.
The median home price in the United States over the period 2003-2011 can be approximated by P(t) = −5t2 + 75t − 30 thousand dollars (3 ≤ t ≤ 11), where t is time in years since the start of 2000.†
Find P'(t) and P'(7).
P'(t) =
P'(7) =
What does the answer tell you about home prices? HINT [See Example 2.]
The median price of a home was increasing at a rate of $ ___ per year in ____.
The value of P'(t) will be , P'(t) = -10t + 75 and value of P'(7) will be equal to 5 and rate of change of price will be $5 thousands per year in dollars.
It is given that median home price is represented by P(t) = -5t² + 75t - 30.
We have to find out the values of P'(t) and P'(7).
What is differentiation ?
Differentiation is a method in which the function's rate of change compared to the rate of change of the independent variable is calculated.
As per the question ;
The equation given is ;
P(t) = -5t² + 75t - 30
So ;
Differentiating the above equation with respect to t , we get ;
[tex]\frac{d}{dt}[/tex] (P(t)) = [tex]\frac{d}{dt}[/tex] ( -5t² + 75t - 30 )
[tex]\frac{d}{dt}[/tex] (P(t)) = P'(t) = -5 × 2 × t + 75
⇒ P'(t) = -10t + 75
And
The value of P'(7) will be ;
P'( t = 7) = -10 × 7 + 75
P'(7) = - 70 + 75
P'(7) = 5
Also ;
P(11) = 190
P(3) = 150
And
The rate of change of price will be ;
[tex]\frac{P(11) - P(3)}{11 - 3}[/tex]
= 190 - 150 / 8
= 40 / 8
= 5
Thus , the value of P'(t) will be , P'(t) = -10t + 75 and value of P'(7) will be equal to 5 and rate of change of price will be $5 thousands per year in dollars.
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7th grade question
2. Mateo is draining a pool at a rate of 1/8 of a gallon of water every 1/2 hour.
If he continues to drain the pool at this rate, how much water will be
drained in 1 hour?
A.4 gallons
B.1 gallon
C. 1/2 gallon
D.1/4 gallon
Answer:
1/4 gallon
Step-by-step explanation:
Just multiply the numerator and denominator of the given rate by 2:
2(1/8) gallon 1/4 gallon
------------------ = ---------------- = 1/4 per gallon
2(1/2) hour 1 hour
Bill school is selling tickets to a fall musical. On thefirst day of ticket sales to school sold one adult ticket and 6 children tickets for a total of $52. The school took in $66 on the second day by selling 1 adult ticket and 8 child tickets. Find the price of an adult ticket
The price of an adult ticket is x= $10
What is mathematical expression?
A mathematical expression is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Let the adult ticket be x
Let the children ticket be y
first day,
1*x + 6*y = 52
x + 6y = 52 equation 1
second day,
1*x + 8*y = 66
x + 8y = 66 equation 2
Solving both the equations
Subtracting equation 2 -equation 1, we get
x + 8y - x - 6y = 66 - 52
2y = 14
y = 7
putting the value of y in equation 1, we get
x + 6*7 = 52
x = 52 - 42
x = 10
Therefore, price of an adult ticket is x= $10
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Express the ratio below in its simplest form.
15:10
Answer:
3:2
Step-by-step explanation:
Divide 15 and 10 by 5.
f(x)=3x-3. find f(4)
Answer:
f(4)=9
Step-by-step explanation:
The given function is
F(x)=3x-3
so,
f(4)=3*4-3
f(4)=12-3
f(4)=9
putting 4 in place of x we get answer of the question.
Amelia has 120 ten-dollar bills and 75 twenty-dollar bills. What is the ratio of ten-dollar bills to twenty-dollar bills
Simplify your answer.
Enter your answer as a fraction.
●
please help
Answer:
24/15
Step-by-step explanation:
120 : 75 =
120/5 : 75/5 =
24 : 15 =
24/15
Answer:
8 : 5 or 8/5
Step-by-step explanation:
starts as 120 : 75
divide both by 5
= 24 : 15
divide both by 3
= 8 : 5 or 8/5
you just need to get them down to the smallest they can. in this case it is 8 and 5. for every 8 $10 bills there are, you have 5 $20 bills.
the plane that is defined by a fixed z coordinate and contains the point(1,2,3)
The plane where the point is located and which has a fixed z coordinate (1,2,3) is y=2x.
Given that,
The plane where the point is located and which has a fixed z- coordinate (1,2,3).
The plane has the point (1, 2, 3) and the z-axis.
Any plane that has the z-axis is of the type that (0,0,1)
Therefore, By cross product of the two points we can find the equation.
Let take a matrix with x,y and z
We can se in the picture there is a matrix.
We now find the determination of that matrix .
determination means the scalar value calculated for a given square matrix is the determinant of a matrix. The determinant is a concept in linear algebra, and its components are square matrixes. It can be viewed as the matrix transformation's scaling factor. useful for performing calculus operations, computing the inverse of a matrix, and solving systems of linear equations.
-2x+y=0
y=2x
Therefore, The plane where the point is located and which has a fixed z coordinate (1,2,3) is y=2x.
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The perimeter of a rectangular field is 356 yards. If the width of the field is 79 yards, what is its length?
Answer:
length = 99 yards
Step-by-step explanation:
the perimeter (P) of a rectangle is calculated as
P = 2(l + w) ← l is the length and w the width
given P = 356 and w = 79 , then
356 = 2(l + 79) ← divide both sides by 2
178 = l + 79 ( subtract 79 from both sides )
99 = l
the length of the field is 99 yards
Please help meeee!!!!
Answer:2
Step-by-step explanation: x+1+2x=3x+1 3x+1=7 3x=7-1 x=6/3 x=2
hope it helps
Answer:
3
Step-by-step explanation:
x+1+2x=7
variables should be on the left and the whole number on the right.
something like this: x+2x=7-1
simplify the equation: 3x=6
find x: x= 6/3=2
so u know x is 2
then substitute the 2 into x. so 2+1 =3#
Your answer is three.
What is 0.0001 as a
single digit times a
power of 10.
Answer:
[tex]10^{-4}[/tex]
Step-by-step explanation:
0.0001
1/10,000
1/[tex]10^{4}[/tex]
[tex]10^{-4}[/tex]
If anyone can help me with this I'd appreciate it a ton!
Thank you so much.
Answer:
help this yuo what is yuor problem
which expression is equivalent to 25s^3+12s/5s
Answer:
A
Step-by-step explanation:
1 Factor out the common term ss.
\frac{s(25{s}^{2}+12)}{5s}
5s
s(25s
2
+12)
2 Cancel s.
\frac{25{s}^{2}+12}{5}
5
25s
2
+12
Help please i need it
Answer:
25 spells
Step-by-step explanation:
each colored box is 5 spells so count by five to see how many spells Ginny casts
The number of spells is 25, that the spells Ginny casts.
Here, we have,
from the given information we get,
we have to find that number spells that the spells Ginny casts.
we have,
the tape diagram gives that,
the ratio of luna and ginny is:=> 4 : 5
so, let , Ginny casts = 5x spells
luna casts = 4x spells
now, we have,
ATQ, 4x+5x = 45
=> 9x= 45
=> x = 45/9
=> x = 5
so, we get,
Ginny casts = 5*5 spells
= 25 spells
Hence, The number of spells is 25, that the spells Ginny casts.
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Match the Statement with the property that matches. A. R=R B. 5-P C. If 3 = x and y = -2 Then the expression y = x + b can be written as -2 = 3 + b D. a=4 Then it can be said that a = b E. DEIR then DF = LR F. VW - WY - VT Substitution Property Transitive Property Symmetric Property Reflexive Property Segment Addition Postulate Segment Congruence Postulate
An expression in math is a sentence with a minimum of two numbers or variables and at least one math operation. a) R=R is a reflexive property.
b) 5-P Substitution Property, c) If 3 = x and y = -2 Then the expression y = x + b can be written as -2 = 3 + b is a substitution property, d) a=4 Then it can be said that a = b e)DEIR then DF = LR Segment Addition Postulate
f)VW - WY - VT is a Transitive Property
What is Expression?
An expression in math is a sentence with a minimum of two numbers or variables and at least one math operation.
We have match the given expression with an appropriate property.
a) R=R is a reflexive property.
b) 5-P Substitution Property
c) If 3 = x and y = -2 Then the expression y = x + b can be written as -2 = 3 + b is a substitution property
d) a=4 Then it can be said that a = b
e)DEIR then DF = LR Segment Addition Postulate
f)VW - WY - VT is a Transitive Property
Therefore .a) R=R is a reflexive property. b) 5-P Substitution Property, c) If 3 = x and y = -2 Then the expression y = x + b can be written as -2 = 3 + b is a substitution property, d) a=4 Then it can be said that a = b e)DEIR then DF = LR Segment Addition Postulate f)VW - WY - VT is a Transitive Property.
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Perform the operation and write the result in the standard form.
(1 + 9i)(1 - 9i)
Answer:
10-18i
Step-by-step explanation:
1(1-9i)+9(1-9i)
1-9i+9-9i
1+9-9i-9i
10-18i
Aubrey worked at three part-time jobs last week. At one job, she worked 15 hours at a salary of S15 per hour, at another she worked 10 hours at S14 per hour,
and at the third she worked 11 hours at $21 per hour. What was her mean hourly wage?