Types of Goals in line with Anelisa's story:
Personal Growth Goals: These goals are related to Anelisa's personal development and could include things like increasing her knowledge, learning new skills, and expanding her experiences.Career Goals: These goals are related to Anelisa's professional development and could include things like advancing in her career, increasing her income, or improving her job performance.Benefits of goal setting on your career choiceGoal setting helps individuals focus their efforts, prioritize their time and resources, and measure their progress towards achieving their goals. In the context of career choice, goal setting can help individuals:
Clarify their career aspirations and identify their strengths, weaknesses, opportunities, and threats.Develop a roadmap for their career journey and align their actions with their career vision.Stay motivated and engaged in their work and avoid feeling overwhelmed or discouraged by the challenges they face.Improve their performance and productivity, and increase their chances of achieving their desired outcomes.Challenges Anelisa experienced:
Lack of resources: Anelisa may have lacked the resources needed to achieve her goals, such as financial support, access to training programs, or sufficient time to complete her tasks.Resistance from others: Anelisa may have encountered resistance from others who opposed her goals, either because they disagreed with her vision or because they felt threatened by her ambition.Self-doubt and fear of failure: Anelisa may have struggled with self-doubt and fear of failure, which could have held her back from pursuing her goals and taking calculated risks to achieve them.Anelisa's personal values, such as determination, resilience, and a strong work ethic, likely played a significant role in her success. These values likely gave her the motivation and drive to pursue her goals, despite the challenges she faced. Additionally, her values may have helped her stay focused on her goals, maintain her sense of purpose, and make decisions that aligned with her vision for her life.
Relationships can impact the achievement of goals in both positive and negative ways. On the positive side, supportive relationships can provide individuals with emotional support, encouragement, and practical resources that can help them achieve their goals. On the negative side, relationships can also be a source of conflict and distraction, especially if they are not aligned with the individual's goals or values.
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Answer:
Step-by-step explanation:
lack of financial stability Nd resoouces
What does the notation f(2) mean?
multiply fby 2
the output (y-value) when x = 2
O the value of x when the output is 2
Evaluate f(2)=
[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
y = f(x) represents a function with output value as y and x as an input in the function.
Therefore, f(2) represents value of function (y) at x = 2
so the correct choice here would be : " the output (y - value) when x = 2 "
Use the formula for continuous compounding to compute the balance in the account after 1, 5, and 20 years. Also, find the APY for the account. A $1000 deposit in an account with an APR of 3%. The balance in the account after 1 year is approximately
The APY for the account would be 3.045%.
What is continuous compounding?
The formula for continuous compounding is given by:
A = P[tex]e^{(rt)}[/tex]
where:
A = final amount
P = principal amount
e = the mathematical constant e (approximately 2.71828)
r = annual interest rate
t = time in years
In this case, P = $1000, r = 0.03 (since APR is given), and t = 1 year.
Using the formula, we have:
A = 1000[tex]e^{(0.03*1)}[/tex]
A ≈ $1030.45
So the balance in the account after 1 year is approximately $1030.45.
To find the balance after 5 and 20 years, we simply need to plug in the corresponding values for t:
For t = 5:
A = 1000[tex]e^{(0.03*5)}[/tex]
A ≈ $1160.92
For t = 20:
A = 1000[tex]e^{(0.03*20)}[/tex]
A ≈ $1806.11
To find the APY, we use the formula:
APY = (1 + r/n)ⁿ- 1
where:
n = number of compounding periods per year
Since this is continuous compounding, we take the limit as n approaches infinity:
APY = eʳ - 1
APY = [tex]e^{0.03}[/tex] - 1
APY ≈ 0.03045 or 3.045%
Hence, the APY for the account would be 3.045%.
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15. Which set is an example of like fractions?
A. 7/4 and 4/7
B. 10/10 and 5/5
C. 2/1 and 2/3
D. 1/2 and 3/2
Selma wants to take her money out of the bank account with simple interest rate of 5% annually and put it in another account with compound interest rate of 8% compounded quarterly if the bank says closing the current account has a penalty of 5% at least how many years she let the money stay at the new account in order to recover the penalty paid using extra interest of the second?
Selma needs to leave the money in the new account for at least one year, option A. 1 to recover the penalty paid.
How did we get this assertion?Assuming Selma wants to recover the penalty paid using the extra interest earned in the new account, she needs to calculate the amount of time it will take for the new account's interest to make up for the penalty paid.
Let P be the initial amount of money in Selma's current account. The penalty for closing the account is 5% of P, which is 0.05P.
If Selma leaves the money in the current account for one year, she will earn simple interest of 5% of P, which is 0.05P. So at the end of the year, the balance in the current account will be P + 0.05P = 1.05P.
If Selma transfers the money to the new account, it will earn compound interest of 8% per year, compounded quarterly. The quarterly interest rate is 8%/4 = 2%.
After one quarter, the balance in the new account will be:
P*(1 + 2%/4) = P*1.02
After two quarters, the balance will be:
P1.02(1 + 2%/4) = P*1.02^2
After three quarters, the balance will be:
P1.02^2(1 + 2%/4) = P*1.02^3
After four quarters (i.e., one year), the balance will be:
P1.02^3(1 + 2%/4) = P*1.02^4
So at the end of one year, Selma will have 1.05P in the current account and P*1.02^4 in the new account.
To recover the penalty of 0.05P, Selma needs the balance in the new account to be at least 1.05P. That is,
P*1.02^4 >= 1.05P
Simplifying, we get:
1.02^4 >= 1.05
Taking the fourth root of both sides, we get:
1.02 >= 1.05^(1/4)
Using a calculator, we get:
1.02 >= 1.01298
So Selma needs to leave the money in the new account for at least one year to recover the penalty paid.
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Evaluate the following expressions. a) 10^15/10^12.
b) 10^1/10^0
c) 10^5+10^-3
d) 10^5-10^3
(^ means to the power of)
On solving the provided question we can say that the expressions are
a) 1000
b 10
c) 100000.001
d) 99000
What is expression?In mathematics, it is pοssible to multiply, divide, add, or remove. The construction of an expression is as follows: Expressiοn, number, and mathematical operatοr Numbers, variables, and functions are the building blocks of a mathematical expressiοn (such as addition, subtraction, multiplicatiοn or division etc.)
Expressiοns and phrases can be contrasted. Any mathematical statement with variables, numbers, and an arithmetic οperation between them is called an expression or an algebraic expressiοn. For instance, the expressiοn 4m + 5 has the terms 4m and 5 as well as the variable m of the supplied expressiοn, all of which are separated by the arithmetic sign +.
The expressiοns are
a)10¹⁵/10¹² = 10³ = 1000
b)10¹/10⁰ = 10/1 = 10
c)10⁵ + 10⁻³ = [tex]\dfrac{10^8 +1}{100}[/tex]= 100000.001
d)10⁵ - 10³ = 100000 - 1000 = 99000
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Determine the Domain and Range for the graph below. Write your answer in interval notation and as an inequality.
Domain written in Interval Notation:
Domain written as an Inequality:
Range written in Interval Notation:
Range written as an Inequality:
Points on the graph are:
(0,1) (1,2) (2,3) (3,4) (4,5)
(-1,0) (-2,-1) (-3,-2) (-4,-3)
The domain and the range of the ordered pairs in interval notations are as follows: 1) Domain = [0,4], Range = [1,5] ; (2) Domain [-1,-4], Range =[0,-3]
Domain and Range of a set of Ordered pairsThe domain of a function is the collection of all conceivable values that may be used as inputs, or alternatively, it is the whole array of possible values for independent variables.
The range of a function is the set of every possible value for the dependent variable's outcomes, or the whole set of all possible values when the domain is substituted.
In a set of ordered pairs (x,y), the domain (input) refers to all the x-values and the range refers to all the y-values.
From the points on the graph given:
(0,1) (1,2) (2,3) (3,4) (4,5)
Domain: (0, 1, 2, 3, 4)
In interval notation: = [0,4]Range: (1, 2, 3, 4, 5)
In interval notation: = [1, 5](-1,0) (-2,-1) (-3,-2) (-4,-3)
Domain: ( -1, -2, -3, -4)
In interval notation: = [-1, -4]Range: (0, -1, -2, -3)
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Sharon wants to buy a shirt that costs $50. The sales tax is 5%. How much is the sales tax? What is her total cost for the shirt
Answer: The sales tax is $2.5 and her total cost of the shirt is $47.5.
Step-by-step explanation:
40°
Find the value of x.
X
190°
x = [?]°
The value of x for the given circle will be going to be 50°.
What is a circle?A circle is a geometrical figure which becomes by plotting a point around a fixed point by keeping a constant distance.
In our daily life, we always see circle objects for example our bike wheel.
Area of circle = πr² and the perimeter of circle = 2πr
where r is the radius of the circle.
Angles of Intersecting Chords Theorem
When two chords cross inside of a circle, the resulting angle's measure is equal to the product of the lengths of the arcs it intercepts and its vertical angle, divided by two.
So, angle x = (190° + 40°)/2 = 115°
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systems of equations - substitution method
2a + 7b = 13
8b = 2 - a
By using the substitution method, the value of a is equal to 10 and the value of b is equal to -1.
How to solve this system of equations?In order to solve the given system of equations, we would apply the substitution method. From the information provided in the image attached above, we have the following system of equations:
2a + 7b = 13 .......equation 1.
8b = 2 - a .......equation 2.
From equation 2, we have:
a = 2 - 8b .......equation 3.
By using the substitution method to substitute equation 3 into equation 1, we have the following:
2(2 - 8b) + 7b = 13
4 - 16b + 7b = 13
4 - 9b = 13
9b = 4 - 13
9b = -9
b = -1
a = 2 - 8b
a = 2 - 8(-1)
a = 2 + 8
a = 10.
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the company bought $15000 worth of equipment beginnning of year 2018 the equipment is est. to increase in value at a rate of 15.9% per year how many yers, t , after which the value of the company's new equipment will be less than 7500 what formula would be used
Answer:
To solve this problem, we can use the formula for exponential growth:
V(t) = V0 * (1 + r)^t
where:
V(t) is the value of the equipment at time t
V0 is the initial value of the equipment ($15,000)
r is the annual growth rate (15.9% or 0.159)
t is the time in years
We want to find the time t when the value of the equipment will be less than $7500. So we can set up the following inequality:
V(t) < 7500
Substituting the formula for V(t), we get:
V0 * (1 + r)^t < 7500
Substituting the given values, we get:
15000 * (1 + 0.159)^t < 7500
Simplifying and solving for t,
we get:(1 + 0.159)^t < 0.5
t * log(1 + 0.159) < log(0.5)
t > log(0.5) / log(1 + 0.159)
Using a calculator, we get:
t > 2.64
So the company's new equipment will be worth less than $7500 after about 2.64 years.
Step-by-step explanation:
one night a movie theater sold 124 tickets. an adult ticket cost $12.50 and a child ticket cost $6.50. in all, $1298 was taken in. how many of each kind of ticket were sold?
Using the system of equations, we found the number of adult tickets sold as 82 and the number of child tickets sold as 42.
What is the system of equations?
A finite set of equations for which common solutions are sought is referred to in mathematics as a set of simultaneous equations, often known as a system of equations or an equation system.
Given,
Number of tickets sold = 124
Cost of one adult ticket = $ 12.50
Cost of one child ticket = $ 6.50
Total money received = $ 1298
We can write a system of equations using the above information.
Number of adult tickets sold = x
Number of child tickets sold = y
x + y = 124
12.5x + 6.5y = 1298
Solving the above equations, we can find x and y.
Multiplying the first equation by 6.5
6.5x+ 6.5y = 806
12.5x + 6.5y = 1298
Subtracting the equations,
-6x = -492
x = 82
Substituting the above value in any one of the equation
82 + y = 124
y = 42
Therefore from the system of equations, the number of adult tickets sold is 82 and the number of child tickets sold is 42.
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Which statement is true about f(x) +2=1/6|x-3|?
Answer: The range of f(x) is [tex]f(x)\geq -2[/tex]
Step-by-step explanation:
The given function is
[tex]f(x)+2=\frac{1}{6}|x-3|[/tex]
It can be written as
[tex]f(x)=\frac{1}{6}|x-3|-2[/tex] ...(1)
The function is in the form of
[tex]g(x)=a|x-h|+k[/tex] ...(2)
Where, a is scale factor and (h,k) is vertex of the graph.
On comparing (1) and (2), we get
[tex]a=\frac{1}{6}[/tex]
[tex]h=3[/tex]
[tex]k=-2[/tex]
The value of a is [tex]\frac{1}{6}[/tex] . So, the graph compressed vertically. The value of a is positive, therefore the graph of f(x) opens upward.
We know the absolute value is always greater than or equal to 0.
[tex]|x-3|\geq 0[/tex]
[tex]\frac{1}{6}|x-3|\geq \frac{1}{6}(0)[/tex]
[tex]\frac{1}{6}|x-3|\geq 0[/tex]
[tex]\frac{1}{6}|x-3|-2\geq 0-2[/tex]
[tex]f(x)\geq -2[/tex]
hope you've understood...
A recycling bin is in the shape of a rectangular box. Find the height of the box if its length is 16 ft, its width is 7 ft, and its surface area is 500 ft2. (In the figure, h=height. Assume that the given surface area includes that of the top lid of the box.)
Answer:
Step-by-step explanation:
The surface area of a rectangular box can be calculated as follows:
Surface area = 2lw + 2lh + 2wh
Where l is the length, w is the width, and h is the height.
We are given the surface area (500 ft2) and the length and width (16 ft and 7 ft respectively), so we can use these values to solve for the height:
500 = 2 * 16 * 7 + 2 * 16 * h + 2 * 7 * h
500 = 224 + 32h + 14h
500 = 238h + 224
276 = 238h
h = 276 / 238
h = 1.16 ft
So the height of the box is approximately 1.16 feet.
Use the coordinates below to determine if AABC and ADEF are congruent.
AABC: A(2, -8), B(-5, -2), C(-7, 3); ADEF: D(-9, 7), E(-11, 12), F(-2, 1)
AB=
DE=
BC=
EF=
AC=
DF=
Are the angles congruent? If yes, explain your reasoning and write a congruency statement.
No The triangles' are not congruent
How to determine if the triangles are congruentThe length of the corresponding sides should be equal and the length of sides is gotten using the equation:
The length of line in an ordered pair is calculated using the formula
d = √{(x₂ - x₁)² + (y₂ - y₁)²}
where
d = distance between the points
x₂ and x₁ = points in x coordinates
y₂ and y₁ = points in y coordinates
distance between points A(2, -8), B(-5, -2), C(-7, 3); and ΔDEF: D(-9, 7), E(-11, 12), F(-2, 1) are calculated as follows:
AB =√{(-2 + 5)² + (-8 + 2)²} = 6.71
DE = √{(-9 + 1)² + (7 - 12)²} = 9.43
BC = √{(-5 + 7)² + (-2 - 3)²} = 5.39
EF = √{(-11 + 2)² + (12 - 1)²} = 14.21
AC = √{(2 + 7)² + (-8 - 3)²} = 14.21
DF = √{(-9 + 2)² + (7 - 1)²} = 9.22
Examining the figures showing the side lengths shows that the two triangles are not congruent
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Please help me. I'll give brainliest.
Answer: interest= 670
Step-by-step explanation:
(1+r)^t*principal
Here is a number machine.
input
x 4
a) Work out the output when the input is 6
b) Work out the input when the output is 31
- 5
output
The solution is,
(a) Output = 37
(b) Input = 9
What is simplification?Simplify simply means to make it simple. In mathematics, simply or simplification is reducing the expression/fraction/problem in a simpler form. It makes the problem easy with calculations and solving.
here, we have,
Given
y=5x -3
Solving (a): Output, when input = 8
This means that, we solve for y when x = 8
So, we have:
y=5x -3
=5*8 -3
=40 - 3
= 37
Solving (b): Input, when output = 42
This means that, we solve for x when y = 42
So, we have:
y=5x -3
5x = 42+3
= 45
Collect like terms
Divide both sides by 5
x = 9
Hence, The solution is,
(a) Output = 37
(b) Input = 9
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You are given the yearly interest earned from a total of $18,000 invested in two funds paying the given rates of simple interest. Write and solve a system of equations to find the amount invested at each rate. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.
Yearly Interest: $579
Rate 1: 2.75%
Rate 2: 4.75%
(dollars invested at 2.75%, dollars invested at 4.75%) =
The amount of investment at a rate of 2.75% and 4.75% will be $13,800 and $4,200m, respectively.
What is simple interest?Simple interest is the concept that is used in many companies such as banking, finance, automobile, and so on.
The formula for interest is written as,
I = (PRT)/100
Where P is the principal, R is the rate of interest, and T is the time.
Let 'x' be the amount of investment at a rate of 4.75%. Then the amount of investment at a rate of 2.75% is ($18,000 - x). Then the equation is given as,
(18,000 - x) × 0.0275 + 0.0475x = 579
495 - 0.0275x + 0.0475x = 579
0.02x = 84
x = $4,200
Then the value of ($18,000 - x) is given as,
$18,000 - x = $18,000 - $4,200
$18,000 - x = $13,800
The amount of investment at a rate of 2.75% and 4.75% will be $13,800 and $4,200m, respectively.
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Question is attached below!
Thanks! :)
The parameters for the home mortgage, and the monthly payment, PMT, for the mortgage are as follows;
Part 1;
P = $175,000, APR = 7%, n = 12, Y = 30
Part 2;
The monthly payment for the mortgage, PMT ≈ $1,164.28
What is a mortgage?A mortgage is a loan agreement, with the possession of the collateral presented for the loan such as a house being bought with the loan amount, being relinquished by the debtor, upon default
Part 1
The value of the mortgage = $175,000
The fixed Annual Percentage Rate, APR = 7%
The duration of the payment = 30 years
The principal amount of the mortgage, P = $175,000The annual percentage rate, APR = 7% = 0.07The number of payment (periods), per year, n = Monthly = 12The number of years of the loan, Y = 30Part 2; The monthly payment PMT loan formula can be presented in the following format;
[tex]PMT = \frac{P\times \frac{APR}{n} }{1-\left(1+\frac{APR}{n}\right)^{(-n\cdot Y)} }[/tex]Therefore;
[tex]PMT = \frac{175000\times \frac{0.07}{12} }{1-\left(1+\frac{0.07}{12}\right)^{(-12\times 30)} } \approx 1164.28[/tex]
The monthly payment for the mortgage is, PMT ≈ $1,164.28
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The regular price of an Apple Watch is $199. The watch is on sale for 15% off. Which expression is equal to the sale price in dollars, of the Apple Watch
Answer: $169.15
Step-by-step explanation: 15 percent of 199 is 29.85, and 199-29.85 is 169.15
A 80 kg monkey climbs a 15 meter tree in half a minute. What is the magnitude of power the monkey demonstrated?.
The demonstration of magnitude of power given by 80 kg monkey by climbing a 15 meter tree in half a minute is equal to option b. 392 J/S.
Weight 'm' of the monkey is equal to 80kilogram
Height 'h' of the tree is equal to 15 meter
Time 't' taken by monkey to climb a tree = half a minute
= 30 minutes
g = 9.8 m/s²
Magnitude of power = ( m × g × h )/ t
⇒ Magnitude of power = ( 80 × 9.8 × 15 )/ 30
⇒ Magnitude of power = 784 / 2
⇒ Magnitude of power = 392 J/S
Therefore, the magnitude of the power for the given weight , displacement and time is equal to option b. 392 J/S.
The above question is incomplete, the complete question is:
A 80 Kg monkey climbs a 15 meter tree in half a minute. What is the magnitude of power the monkey demonstrated?
a. 13.1 J/S
b. 392 J/S
c. 784 J/s
d. 11760 J/s
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4) Marla is jumping rope in her school's jump-a-thon to raise money for the school library.
Marla's parents pledge $0.25 for each time Marla jumps rope and write a check for $63.25. This
situation is modeled by the equation 0.25c = 63.25, where c represents the number of times that
Marla jumps rope in the jump-a-thon. What is the value of c in the equation?
A. 16
B. 64
C. 63
D. 253
Step-by-step explanation:
0.25c = 63.25
We would just get c alone by dividing by 0.25 on both sides of the equal sign
c = 253
Answer: C = 253
Step-by-step explanation:
a ferris wheel at a carnival has a diameter of 62 feet. suppose a passenger is traveling 6 miles per hour. Find the angular speed in radians per minute. Find the number of revolutions the wheel makes per hour.
On solving the provided question, we can say that relation between tangential speed and angular speed, v = r * w; v: tangential speed; r: radius; w: angular speed
what is radius?The length of a circle or sphere, in more contemporary use, is the same as its radius in classical geometry, which is one of the line segments from its center to its circumference. The Latin word radius, which also refers to the spokes of a wagon wheel, gave rise to the term. The distance a circle's center is from any point on its perimeter is its radius. Usually, "R" or "r" is used to indicate it. A radius is a line segment that has one endpoint in the center and one on the circumference of a circle. Circular diameter equals radius The diameter of a circle is the segment that traverses its center and has ends that are on the circle.
relation between tangential speed and angular speed
v = r * w
v: tangential speed
r: radius
w: angular speed
the radius is
r = d/2
d is the diameter
[tex]v = (d * w)/2\\w = 2*v/d\\w = (2*9 mile/h)/(58 feet)\\w = (2*9 mile/h)/(0.011 miles) = 1636 rad/h\\w = 1636/60 = 27.2 rad/min\\[/tex]
the angular speed is in radians,
[tex]n = v/(π*d)[/tex]
n = (9 mile/h)/(π*0.011 mile) = 260 rev/h
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North A North 110% B
Answer:
Step-by-step explanation:
I do not know bro. If I was a robot i'd know. I don't. I'm different.
PLS HELP ME MY GRADE DEPENDS ON IT
The constant of proportionality for the given graph is 6.
What is a constant of proportionality?The two sequences of numbers are proportional or directly proportional if the ratio between the corresponding elements of two sequences of numbers, typically experimental data, is constant and is known as the coefficient of proportionality or proportionality constant.
From the given graph the point is (5,30) from which the line is passing.
The equation for the constant of proportionality is written as below,
y = kx
Substitute the points in the above equation,
y = kx
30 = k x 5
k = 30 / 5
k = 6
Therefore, the proportionality constant for the displayed graph is 6.
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Use the scale to help you solve the equation and find the value of x?
? - 85 ÷ 641 = 650
Need help
Answer:
416,`735
Step-by-step explanation:
Do inverse operations
Please help fast!! Which of the following equations represents a linear function? Please show the steps of the answer! Thank you
A group of students stood in a circle to play a game. The circle had a diameter of 22 meters. Which measurement is closest to the area of the circle in meters?
The measurement closest to the area of the circle in meters is 379.94 sq. meters.
What is area?An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure. The square unit, which is frequently expressed as square inches, square feet, etc., is the accepted unit of area.
The word "area" refers to a free space. A shape's length and width are used to compute its area. The majority of forms and things have corners and edges. When computing the area of a certain form, both the length and width of these edges are taken into account.
Given that,
diameter = 22m
radius = 22/2 = 11m
The area of the circle is:
A = πr²
Substituting the value of π = 3.14 and r = 11 we have:
A = (3.14)(11)²
A = 379.94
Hence, the measurement closest to the area of the circle in meters is 379.94 sq. meters.
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The top 5% of applicants on a test will receive a scholarship. If the test scores are normally distributed with a mean of 76 and a standard distribution of 12, what is the lowest test score that still qualifies for a scholarship? Use Excel, and round your answer to the nearest integer.
Answer:
96
Step-by-step explanation:
Z-Scores:z-scores are defined as: [tex]z=\frac{x-\mu }{\sigma}[/tex], and this may look a bit confusing at first but by looking at the numerator and then denominator we can get a more understandable definition.
The numerator is first finding the difference between the statistic and the mean. It then divides by the standard deviation, so essentially it's telling us how far the statistic is from the mean in terms of standard deviation.
We can actually rewrite the equation to express this:
[tex]z=\frac{x-\mu }{\sigma}\\\\z\sigma = x-\mu\\\mu + z\sigma=x[/tex]
So in essence, how many standard deviations the statistic is away from the mean.
Now this may seem very off topic compared to what the problem is asking, but we want to convert the top 5% to a z-score. Now let's first convert this top 5% to a percentile. To be in the top 5% you just need to be in the 95th percentile and using technology we can convert this into a z-score which is approximately 1.645.
So this means the 95th percentile is 1.645 standard deviations away and in this case above (since it's positive) from the mean.
The good thing is we know the standard deviation and mean, now let's just apply it: [tex]76+1.645(12)=95.74[/tex]. We now want to round this to the nearest integer of 96, and now we have our answer!
find the value of each variable provide proofs
Answer:
both sides are equal to: [tex]8\sqrt{2}[/tex]
Step-by-step explanation:
Trigonometric Functions:Trigonometric functions are defined as the ratios between sides based on some angle. There's three main trig functions you want to remember which are defined as:
[tex]sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\\\\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\\\\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}[/tex]
and then there are three other trig functions, which are simply the reciprocals:
[tex]csc(\theta)= \frac{\text{hypotenuse}}{\text{opposite}}=\frac{1}{sin(\theta)}\\\\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}}=\frac{1}{cos(\theta)}\\\\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}=\frac{1}{tan(\theta)}[/tex]
Now in our case we have the angle of 45 degrees and we have the hypotenuse with length 16, and we want to either solve for the opposite (x) or adjacent side (y)
Let's solve for the adjacent side, in which case we want a trig function defined using the hypotenuse and adjacent side, which is our cosine function.
[tex]cos(45)=\frac{y}{16}\implies cos(45)*16=y[/tex] and from here we can use a calculator or the unit circle which gives us an exact value. Using the unit circle, we can determine that: [tex]cos(45)=\frac{\sqrt{2}}{2}[/tex] and plugging that into our equation we get: [tex]y=\frac{\sqrt{2}}{2}*16=8\sqrt{2}[/tex] and we can use a calculator to approximate this: 11.313
Now we can also use the trig functions to find "x", which is the opposite side. In this case we want to use a trig function which is defined using the opposite and hypotenuse side, which is our sine function: [tex]sin(45)=\frac{x}{16}\implies sin(45)*16=x[/tex] and the cool thing about this, is that sin(45) and cos(45) are actually equal, so our "x" and "y' side are exactly equal: [tex]x=8\sqrt{2}[/tex]
We can verify this using the Pythagorean Theorem: [tex](8\sqrt{2})^2+(8\sqrt{2})^2=16^2\\\\(8^2*\sqrt{2}^2)+(8^2+\sqrt{2}^2)=256\\\\(64*2)+(64*2)=256\\\\128+128=256\\\\256=256[/tex]
