The indicated operation is the composition of functions. To perform this operation, we substitute the expression for g(x) into f(x). The composition of f(g(x)) is given by f(g(x)) = (4x - 1)^2 + 2.
To compute f(g(x)), we first evaluate g(x) by substituting x into the expression for g(x): g(x) = 4x - 1. Next, we substitute this result into f(x): f(g(x)) = f(4x - 1).
Now, let's expand and simplify f(g(x)):
f(g(x)) = (4x - 1)^2 + 2
= (4x - 1)(4x - 1) + 2
= 16x^2 - 8x + 1 + 2
= 16x^2 - 8x + 3.
The domain of f(g(x)) is the same as the domain of g(x) since the composition involves g(x). In this case, g(x) is defined for all real numbers. Therefore, the domain of f(g(x)) is also all real numbers.
In summary, the composition of f(g(x)) is given by f(g(x)) = 16x^2 - 8x + 3, and the domain of f(g(x)) is all real numbers.
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Todd noticed that the gym he runs seems less crowded during the summer. He decided to look at customer data to see if his impression was correct.
Week
5/27 to 6/2
6/3 to 6/9
6/10 to 6/16
6/17 to 6/23
6/24 to 6/30
7/1 to 7/7
Use
618 people
624 people
618 people
600 people
570 people
528 people
A: What is the quadratic equation that models this data? Write the equation in vertex form.
B: Use your model to predict how many people Todd should expect at his gym during the week of July 15.
Todd should expect_______people.
Todd should expect approximately 624 people at his gym during the week of July 15.
A: To find the quadratic equation that models the data, we can use the vertex form of a quadratic equation:
[tex]y = a(x - h)^2 + k[/tex] where (h, k) represents the vertex of the parabola.
Let's analyze the data to determine the vertex. We observe that the number of people is highest during the first week and gradually decreases over the following weeks.
This suggests a downward-opening parabola.
From the data, the highest point occurs during the week of 6/3 to 6/9 with 624 people.
Therefore, the vertex is located at (6/3 to 6/9, 624).
Using the vertex form, we have:
[tex]y = a(x - 6/3 to 6/9)^2 + 624[/tex]
Now, we need to find the value of 'a.'
To do this, we can substitute any other point and solve for 'a.' Let's use the data from the week of 5/27 to 6/2:
[tex]618 = a(5/27 to 6/2 - 6/3 to 6/9)^2 + 624[/tex]
Simplifying the equation and solving for 'a,' we find:
[tex]618 - 624 = a(-6/3)^2[/tex]
-6 = 4a
a = -3/2
Therefore, the quadratic equation in vertex form that models the data is:
[tex]y = (-3/2)(x - 6/3 to 6/9)^2 + 624[/tex]
B: To predict the number of people Todd should expect during the week of July 15, we substitute x = 7/15 into the equation and solve for y:
[tex]y = (-3/2)(7/15 - 6/3 to 6/9)^2 + 624[/tex]
Simplifying the equation, we find:
[tex]y = (-3/2)(1/15)^2 + 624[/tex]
y = (-3/2)(1/225) + 624
y = -3/450 + 624
y = -1/150 + 624
y = 623.993
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Statement: Reasons:
DF=EG.
DE
Prove: DE=FG
Statements
DF=EG
DF=DE+EF.
EG=EF+FG
DE+EF=EF+ FG
FL
G
Answer:
az
Step-by-step explanation:
Tamika practiced oboe for 1/4 hour in the morning and 5/6 hour in the afternoon how long did she practice in all write your answer as a mixed number
how do you find out how many positive zeros and negative zeros are in a polynomial based on a graph?
In this context, the term "zeros" refers to the roots or x-intercepts.
You're looking for where the graph either,
a) touches the x axis, or,b) crosses the x axisSee the diagram below. That example shows two positive roots, because each root is to the right of the vertical y axis.
help please its due in 50 minutes ill mark brainliest answer too and no need to show work
The output value f(3) in the functions f( x ) = 3x + 5, f( x ) = [tex]\frac{1}{2}x^2-1.5[/tex] and f( x ) = [tex]\frac{3}{2x}[/tex] is 14, 3 and 1/2 respectively.
What is the output value of f(3) in the given functions?Given the functions in the question:
f( x ) = 3x + 5
f( x ) = [tex]\frac{1}{2}x^2-1.5[/tex]
f( x ) = [tex]\frac{3}{2x}[/tex]
To evaluate each function at f(3), we simply replace the variable x with 3 and simplify.
a)
f( x ) = 3x + 5
Replace x with 3:
f( 3 ) = 3(3) + 5
f( 3 ) = 9 + 5
f( 3 ) = 14
b)
f( x ) = [tex]\frac{1}{2}x^2-1.5[/tex]
Replace x with 3:
[tex]f(3) = \frac{1}{2}(3)^2 - 1.5\\\\f(3) = \frac{1}{2}(9) - 1.5\\\\f(3) = 4.5 - 1.5\\\\f(3) = 3[/tex]
b)
f( x ) = [tex]\frac{3}{2x}[/tex]
Replace x with 3:
[tex]f(3) = \frac{3}{2(3)} \\\\f(3) = \frac{3}{6} \\\\f(3) = \frac{1}{2}[/tex]
Therefore, the output value of f(3) is 1/2.
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Hcf of two expressions is (x + 1) and lcm is (x^3+ x^2 – x – 1). if one expression is (x^2 - 1), then what is the second expression?
After solving by formula the second expression is y = [tex](x^2 + 1)[/tex].
We know that the product of the HCF and LCM of two numbers is equal to the product of the numbers themselves. In this case, we can apply the same principle to expressions:
HCF * LCM = (x + 1) * [tex](x^3+ x^2 - x - 1)[/tex]
the first number is [tex]x^{2} -1\\[/tex] and let the second number is y
Therefore, we can set up the equation:
(x + 1) * [tex](x^3+ x^2 - x - 1)[/tex] = [tex]x^{2} -1\\[/tex] * y
[tex]x^4 + x^3 + x^2 - x^3 - x^2 + x - x - 1 = x^2 - 1 * y[/tex]
Simplifying:
[tex]x^4 - 1 = (x^2 - 1) * y[/tex]
Now, we can divide both sides by [tex](x^2 - 1)[/tex]:
[tex](x^4 - 1) / (x^2 - 1) = y[/tex]
Notice that [tex](x^2 - 1)[/tex]can be factored as (x + 1)(x - 1). Therefore, we can simplify further:
[tex](x^4 - 1) / ((x + 1)(x - 1)) = y[/tex]
The expression [tex](x^4 - 1)[/tex] can be factored using the difference of squares:
[tex](x^4 - 1) = (x^2 + 1)(x^2 - 1)[/tex]
[tex][(x^2 + 1)(x^2 - 1)] / ((x + 1)(x - 1)) = y[/tex]
Now, we can cancel out the common factor [tex](x^2 - 1)[/tex] from the numerator and denominator:
[tex]y =(x^2 + 1)[/tex]
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In December 2016 the average price of unleaded
A sixth-grade class recorded the number of letters in each student's first name.
The results are shown in the dot plot.
A dot plot titled lengths of student names show the number of students with a certain number of letters in their name. The data is as follows. 1 dot above 3, 2 dots above 4, 4 dots above 5, 7 dots above 6 and 7, 3 dots above 8, 1 dot above 9, 2 dots above 10, and 3 dots above 11.
Which is the best representation of the center of this data set?
A. 8
B. 5
C. 7
D. 6
What is the distance between points R (5, 7) and S(-2,3)?
Answer:
d ≈ 8.1
Step-by-step explanation:
calculate the distance d using the distance formula
d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]
with (x₁, y₁ ) = R (5, 7 ) and (x₂, y₂ ) = S (- 2, 3 )
d = [tex]\sqrt{(-2-5)^2+(3-7)^2}[/tex]
= [tex]\sqrt{(-7)^2+(-4)^2}[/tex]
= [tex]\sqrt{49+16}[/tex]
= [tex]\sqrt{65}[/tex]
≈ 8.1 ( to 1 decimal place )
You own a portfolio that has $3,000 invested in Stock A and $4,100 invested in Stock B. Assume the expected returns on these stocks are 10 percent and 16 percent, respectively. What is the expected return on the portfolio?
The expected return on the portfolio is approximately 13.465%.
To calculate the expected return on the portfolio, we need to consider the weights of each stock in the portfolio.
Let's denote the weight of Stock A as wA and the weight of Stock B as wB. The weight of a stock is the proportion of the total portfolio value that is invested in that stock.
Given that $3,000 is invested in Stock A and $4,100 is invested in Stock B, we can calculate the weights as follows:
wA = $3,000 / ($3,000 + $4,100) = $3,000 / $7,100
wB = $4,100 / ($3,000 + $4,100) = $4,100 / $7,100
Next, we need to calculate the weighted average of the expected returns of the two stocks using their respective weights:
Expected return on the portfolio = (wA * Return on Stock A) + (wB * Return on Stock B)
Expected return on the portfolio = (wA * 10%) + (wB * 16%)
Substituting the calculated weights into the equation:
Expected return on the portfolio = ($3,000 / $7,100 * 10%) + ($4,100 / $7,100 * 16%)
Simplifying the equation:
Expected return on the portfolio = (0.4225 * 10%) + (0.5775 * 16%)
Expected return on the portfolio = 0.04225 + 0.0924
Expected return on the portfolio = 0.13465 or 13.465%
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What is the graph of f(x) = 0.5(4)x (x is an exponent)
It should be a positive graph .
Jalen bought a new iPad. The screen has a perimeter of inches 36 inches and an area of 80 square inches. What are the dimensions of the iPad’s screen?
Let's assume the dimensions of the iPad's screen are represented by length (L) and width (W). We can use the given information to set up two equations.
Perimeter equation:
The perimeter of a rectangle is given by the formula: 2(L + W). In this case, the perimeter is given as 36 inches.
So, 2(L + W) = 36.
Area equation:
The area of a rectangle is given by the formula: L * W. In this case, the area is given as 80 square inches.
So, L * W = 80.
We now have a system of two equations:
2(L + W) = 36
L * W = 80
From equation 1, we can simplify it to L + W = 18 and rewrite it as L = 18 - W.
Now substitute this value of L into equation 2:
(18 - W) * W = 80
Expanding the equation:
18W - W^2 = 80
Rearranging the equation to quadratic form:
W^2 - 18W + 80 = 0
Factoring or using the quadratic formula, we find the two possible values for W.
The solutions are W = 8 and W = 10.
Now we can substitute these values back into the equation L = 18 - W to find the corresponding values of L.
For W = 8:
L = 18 - 8 = 10
For W = 10:
L = 18 - 10 = 8
Therefore, the dimensions of the iPad's screen can be either 8 inches by 10 inches or 10 inches by 8 inches.
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Determine the surface area and volume Note: The base is a square.
The surface area and volume of the square pyramid is 96 squared centimeter and 48 cubic centimeters respectively.
What is the surface area and volume of the square pyramid?The surface area of a square pyramid is expressed as:
SA = [tex]a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }[/tex]
The volume of a square pyramid is expressed as:
Volume = [tex]a^2*\frac{h}{3}[/tex]
Where a is the base edge and h is the height.
From the figure a = 6cm
First, we determine the h, using pythagorean theorem:
h² = 5² - (6/2)²
h² = 5² - 3²
h² = 25 - 9
h² = 16
h = √16
h = 4 cm
Solving for surface area:
SA = [tex]a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }[/tex]
[tex]= a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }\\\\= 6^2 + 2*6 \sqrt{\frac{6^2}{4}+4^2 }\\\\= 36 + 12 \sqrt{\frac{36}{4}+16 }\\\\= 36 + 12 (5)\\\\= 36 + 60\\\\= 96 cm^2[/tex]
Solving for the volume:
Volume = [tex]a^2*\frac{h}{3}[/tex]
[tex]= a^2*\frac{h}{3}\\\\= 6^2*\frac{4}{3}\\\\= 36*\frac{4}{3}\\\\=\frac{144}{3}\\\\= 48 cm^3[/tex]
Therefore, the volume is 48 cubic centimeters.
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Which exponential equation is e
quivalent to the logarithmic equation below? log=200 a
A. 200 = 10
B. 200¹0 = a
C. a¹0 = 200
D. 10 = 200 SUBMIT
The exponential equation a¹⁰ = 200 is equivalent to the logarithmic equation Log = 200 a.
Which rule of logarithms should we use here?The rule of logarithms that we should use here is given below:
[tex]\log \text{x} = \text{a} \iff 10^{\text{a}} = \text{x}[/tex]
We can find the equivalent exponential equation below:The given expression is Log = 200 a.
We can follow the rule log x = a ⇔ 10^a = x to convert this logarithmic equation to an exponential one.
Log = 200 a can be rewritten as a¹⁰ = 200
Therefore, we have found that the exponential equation a¹⁰ = 200 is equivalent to the logarithmic equation Log = 200 a.
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two points A and B, due to two spheres X and Y 4.0m apart, that are carrying charges of 72mC and -72mC respectively. Assume constant of proportionality as 9×10^9Nm²/C². Find the electric field strength at points A and B due to each spheres presence
Due to the presence of spheres X and Y, the electric field strength at point B is [tex]1.01 * 10^6 N/C[/tex] and [tex]-4.05 * 10^6 N/C[/tex], respectively.
Given that two spheres X and Y are carrying charges of 72mC and -72mC respectively, and they are located 4.0 m apart from each other. The electric field strength at points A and B due to the presence of each sphere is to be determined.
Let's begin by calculating the electric field strength at point A due to sphere X. Electric field strength is given by E=kq/r², where k is Coulomb's constant, q is the charge and r is the distance between the two charges. The electric field strength at point A due to sphere X, E₁=kq₁/r₁² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (4.0m)^2 = 4.05 * 10^6 N/C[/tex] (approx.)
Similarly, the electric field strength at point A due to sphere Y can be calculated as follows, E₂=kq₂/r₂² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (4.0m)^2 = 4.05 * 10^6 N/C[/tex] (approx.). Here, the negative sign indicates that the electric field due to sphere Y is in the opposite direction to the electric field due to sphere X. Now, let's calculate the electric field strength at point B. The electric field strength at point B due to sphere X, E₁=kq₁/r₁² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (8.0m)^2 = 1.01 * 10^6 N/C[/tex] (approx.)
Similarly, the electric field strength at point B due to sphere Y can be calculated as follows, E₂=kq₂/r₂² [tex]= (9*10^9Nm^2/C^2) * (-72mC) / (4.0m)^2 = -4.05 * 10^6 N/C[/tex] (approx.). Therefore, the electric field strength at point A due to the presence of sphere X is [tex]4.05 * 10^6 N/C[/tex] and due to the presence of sphere Y is [tex]-4.05 * 10^6 N/C[/tex]. The electric field strength at point B due to the presence of sphere X is [tex]1.01 * 10^6 N/C[/tex] and due to the presence of sphere Y is [tex]-4.05 * 10^6 N/C[/tex].
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Of the books in a personal library, 4/7 are fiction. Of these books, 1/3 are paperback. What fraction of the books in the library are fiction and paperbacks?
4/21 of the books in the library are both fiction and paperbacks.
To determine the fraction of books in the library that are both fiction and paperback, we need to multiply the fractions representing each condition.
Let's start with the fraction of books in the library that are fiction. If 4/7 of the books are fiction, then this fraction represents the number of fiction books.
Next, we want to find the fraction of fiction books that are also paperbacks. Since 1/3 of the fiction books are paperbacks, we multiply 4/7 (fiction books) by 1/3 (paperback fraction).
Multiplying fractions is done by multiplying the numerators together to get the new numerator and multiplying the denominators together to get the new denominator.
Thus, the fraction of books in the library that are both fiction and paperbacks is:
(4/7) * (1/3) = (4 * 1) / (7 * 3) = 4/21
Therefore, 4/21 of the books in the library are both fiction and paperbacks.
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Question #5
Find the measure of the indicated arc.
OOOO
90°
80°
100°
70°
G
H
40°
F
The value of the required arc in the figure is solved to be
80°How to find the value of the arcThe inscribed angle is given in the problem as 40 degrees. This is the angle formed at the circumference of the circle
The relationship between inscribed angle and the intercepted arc is
intercepted arc = 2 * inscribed angle
in the problem, we have that
intercepted arc = ?
inscribed angle = 40
plugging in the values
intercepted arc = 2 * 40
intercepted arc = 80 degrees
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Algebra
Solve for k: 10-10|-8k+4|=10
Write your answer in set notation.
The solution for k in the equation 10 - 10|-8k + 4| = 10, expressed in set notation, is {1/2}.
1. Start with the equation: 10 - 10|-8k + 4| = 10.
2. Simplify the expression inside the absolute value brackets: -8k + 4.
3. Remove the absolute value brackets by considering two cases:
Case 1: -8k + 4 ≥ 0 (positive case):
-8k + 4 = -(-8k + 4) [Removing the absolute value]
-8k + 4 = 8k - 4 [Distributive property]
-8k - 8k = -4 + 4 [Group like terms]
-16k = 0 [Combine like terms]
k = 0 [Divide both sides by -16]
Case 2: -8k + 4 < 0 (negative case):
-8k + 4 = -(-8k + 4) [Removing the absolute value and changing the sign]
-8k + 4 = -8k + 4 [Simplifying the expression]
0 = 0 [True statement]
4. Combine the solutions from both cases: {0}.
5. Check if the solution satisfies the original equation:
For k = 0: 10 - 10|-8(0) + 4| = 10
10 - 10|4| = 10
10 - 10(4) = 10
10 - 40 = 10
-30 = 10 [False statement]
6. Since k = 0 does not satisfy the equation, it is not a valid solution.
7. Therefore, the final solution expressed in set notation is {1/2}.
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6/7 .r = 3/4 write it as a fraction or as a whole or a mixed number
The solution for "r" is 21/24, which can be simplified to 7/8. The answer can be written as a fraction of 7/8.
To solve for "r" in the given equation, we can use algebraic manipulation.
First, we can multiply both sides of the equation by the reciprocal of 6/7, which is:
7/6: 6/7 · r = 3/4
7/6 · 6/7 · r = 7/6 · 3/4
r = 21/24.
Thus, the solution for "r" is 21/24, which can be simplified to 7/8.
Therefore, the answer can be written as a fraction of 7/8.
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Please answer ASAP I will brainlist
Answer:
A) The y-intercept(s) is/are 2
Step-by-step explanation:
Y-intercepts are where the graph of a function cross over the y-axis. In this case, the line passes through y=2, which is the y-intercept.
21. An RSTU rectangle is drawn on the coordinate plane with coordinates R(-1, 5), S(4, 5), T(4, 9) and then translated by T(2,-3), then the image coordinates of point U are
The image coordinates of point U, after translating the RSTU rectangle by T(2,-3), would be U(6, 6).
To find the image coordinates of U, we need to apply the translation vector T(2,-3) to each of the original coordinates.
The translation vector represents the horizontal and vertical distances by which each point is moved.
Starting with the original coordinates of point U, which are (4, 9), we add the horizontal distance of 2 to the x-coordinate and subtract the vertical distance of 3 from the y-coordinate.
Therefore, the new x-coordinate of U is 4 + 2 = 6, and the new y-coordinate is 9 - 3 = 6.
Thus, the image coordinates of point U after the translation are (6, 6). This means that U has been moved 2 units to the right and 3 units downward from its original position.
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You just inherited a sum of money from a distant uncle. The only stipulation is that you need to save it for 10 years and then you can do whatever you want with it. The amount of the inheritance is $25,000.
Option one: You can put it into a saving account that earns 6% compounded quarterly.
Option two: You can put it into a checking account that earns 4% compounded monthly.
Option three: You can place it into a money market account that earns 3% compounded daily.
Which option is best for you? Why?
Submit a report detailing the reasons you have for the decision you make.
This flexibility of a money market account makes it an ideal option for people who need to save money without risking their inheritance.
After inheriting $25,000 from a distant uncle, you would like to save it for ten years before doing anything with it. Since you want to save the money, there are several options for keeping it safe and earning interest, including a savings account, a certificate of deposit (CD), and a money market account.
Money market accounts, in my opinion, would be the best place to keep the inheritance. The money market account is a low-risk account with high-interest rates, making it an attractive option for someone who wants to save their money. As a result, it would be reasonable to place the inheritance into a money market account that pays a daily compounded rate of 3%.There are several reasons for choosing this option.
Firstly, the daily compounded interest will generate a higher return over the ten-year period than the simple interest or monthly compounded interest offered by other accounts. Second, the account is FDIC-insured, which means that the account holder is guaranteed to receive their money in the event of a bank failure.
Furthermore, the money market account provides easy access to the account holder's money while still earning interest. Most money market accounts have a limit on the number of withdrawals a person can make per month. Still, the account holder can easily transfer funds into a checking account or withdraw money from an ATM if needed.
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Given the piecewise functions shown below, select all of the statements that are true.
The true statements are:
a. f(-1) = 2
c. f(1) = 0
Let's evaluate each statement using the given piecewise function f(x):
a. f(-1) = -(-1) + 1 = 2
b. f(-2) = -(-2) + 1 = 3 (Not 0, so this statement is false)
c. f(1) = (1)^2 - 1 = 0
d. f(4) = (4)^2 - 1 = 16 - 1 = 15 (Not 7, so this statement is false)
Therefore, the correct statements are:
a. f(-1) = 2
c. f(1) = 0
Statement a is true because when x = -1, we use the first piece of the piecewise function, which gives us -(-1) + 1 = 2.
Statement c is true because when x = 1, we use the third piece of the piecewise function, which gives us (1)^2 - 1 = 0.
Statements b and d are false because they do not match the corresponding values obtained from evaluating the piecewise function at the given inputs.
Therefore, the true statements are:
a. f(-1) = 2
c. f(1) = 0
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Solve |5x - 1| < 1
please help
Answer:
|5x - 1| < 1
-1 < 5x - 1 < 1
0 < 5x < 2
0 < x < 2/5
Solve a triangle with a = 4. b = 5, and c = 7."
a. A=42.3°; B = 42.5⁰; C = 101.5⁰
b. A= 34.1°; B = 44.4°; C= 99.5⁰
C.
d.
OA
OB
C
OD
A = 34.1°: B=42.5°: C= 101.5°
A = 34.1°: B= 44.4°: C= 101.5°
Please select the best answer from the choices provided
Angle C can be found by subtracting the sum of angles A and B from 180 degrees:
b. A = 34.1°; B = 44.4°; C = 101.5°
To solve a triangle with side lengths a = 4, b = 5, and c = 7, we can use the law of cosines and the law of sines.
First, let's find angle A using the law of cosines:
[tex]cos(A) = (b^2 + c^2 - a^2) / (2\times b \times c)[/tex]
[tex]cos(A) = (5^2 + 7^2 - 4^2) / (2 \times 5 \times 7)[/tex]
cos(A) = (25 + 49 - 16) / 70
cos(A) = 58 / 70
cos(A) ≈ 0.829
A ≈ arccos(0.829)
A ≈ 34.1°
Next, let's find angle B using the law of sines:
sin(B) / b = sin(A) / a
sin(B) = (sin(A) [tex]\times[/tex] b) / a
sin(B) = (sin(34.1°) [tex]\times[/tex] 5) / 4
sin(B) ≈ 0.822
B ≈ arcsin(0.822)
B ≈ 53.4°
Finally, angle C can be found by subtracting the sum of angles A and B from 180 degrees:
C = 180° - A - B
C = 180° - 34.1° - 53.4°
C ≈ 92.5°.
b. A = 34.1°; B = 44.4°; C = 101.5°
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Please help! :')
Prove that the two circles shown below are similar. (10 points)
Circle X is shown with a center at negative 2, 8 and a radius of 6. Circle Y is shown with a center of 4, 2 and a radius of 3.
NO LINKS!! URGENT HELP PLEASE!!
25. Use the relationship in the diagrams below to solve for the given variable.
Justify your solution with a definition or theorem.
Answer:
x = 110°
Step-by-step explanation:
The opposite angles are equal in a parallelogram
3x - 60 = 2x + 50
⇒ 3x - 2x = 60 + 50
⇒ x = 110°
Answer:
x = 110°
Step-by-step explanation:
As the top and bottom line segments of the given shape are the same length and parallel (indicated by the tick marks and arrows), the shape is a parallelogram.
As the opposite angles of a parallelogram are equal, to find the value of the variable x, equate the two angle expressions and solve for x:
[tex]\begin{aligned}3x-60^{\circ}&=2x+50^{\circ}\\3x-60^{\circ}-2x&=2x+50^{\circ}-2x\\x-60^{\circ}&=50^{\circ}\\x-60^{\circ}+60^{\circ}&=50^{\circ}+60^{\circ}\\x&=110^{\circ}\end{aligned}[/tex]
Therefore, the value of x is 110°.
Note: There must be an error in the question. If x = 110°, each angle measures 270°, which is impossible since the sum of the interior angles of a quadrilateral is 360°.
Circle 1 is centered at (−4,−2) and has a radius of 3 centimeters. Circle 2 is centered at (5,3) and has a radius of 6 centimeters.
What transformations can be applied to Circle 1 to prove that the circles are similar?
Enter your answers in the boxes.
The circles are similar because you can translate Circle 1 using the transformation rule ( , ) and then dilate it using a scale factor of .
The circles are similar because you can translate Circle 1 using the transformation rule (9, 5) and then dilate it using a scale factor of 2.
To prove that Circle 1 and Circle 2 are similar, we need to identify the transformations that can be applied to Circle 1 to obtain Circle 2.
First, let's consider the translation of Circle 1. The translation rule is given by (a, b), where a represents the horizontal shift and b represents the vertical shift.
In this case, to translate Circle 1 to align with Circle 2, we need to shift it 9 units to the right and 5 units up. Therefore, the translation rule for Circle 1 is (9, 5).
Next, let's consider the dilation. A dilation is a transformation that changes the size of the figure but preserves its shape. The scale factor, denoted by k, determines the amount of scaling. In this case, Circle 1 needs to be dilated to match the size of Circle 2.
The scale factor can be determined by comparing the radii of the two circles. The radius of Circle 1 is 3 centimeters, while the radius of Circle 2 is 6 centimeters. The scale factor is obtained by dividing the radius of Circle 2 by the radius of Circle 1: 6/3 = 2.
Therefore, the transformation applied to Circle 1 to prove that the circles are similar is a translation by (9, 5) followed by a dilation with a scale factor of 2.
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Calculate:
1+2-3+4+5-6+7+8-9+…+97+98-99
The value of the given expression is 1370.
To calculate the given expression, we can group the terms in pairs and simplify them.
We have the following pattern:
1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 + ... + 97 + 98 - 99
Grouping the terms in pairs, we can see that each pair consists of a positive and a negative term. The positive term increases by 1 each time, and the negative term decreases by 1 each time. Therefore, we can rewrite the expression as:
(1 - 3) + (2 + 4) + (5 - 6) + (7 + 8) + ... + (97 + 98) - 99
The sum of each pair in parentheses simplifies to a single term:
-2 + 6 - 1 + 15 + ... + 195 - 99
Now, we can add up all the terms:
-2 + 6 - 1 + 15 + ... + 195 - 99 = 1370
As a result, the supplied expression has a value of 1370.
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Consider a graph of the function y = x² in xy-plane. The minimum distance between point (0, 4) on the y-axis and points on the graph is [1-2] You should rationalize the denominator in the answer. PLEASE HELP ME
The minimum distance between the point (0, 4) on the y-axis and points on the graph is 4.
To find the minimum distance between the point (0, 4) on the y-axis and points on the graph of the function y = x², we can use the concept of perpendicular distance.
The distance between a point (x, y) on the graph and the point (0, 4) is given by the formula:
distance = √((x - 0)² + (y - 4)²) = √(x² + (y - 4)²)
Substituting the function y = x² into the distance formula, we get:
distance = √(x² + (x² - 4)²) = √(x² + (x⁴ - 8x² + 16))
Simplifying further, we have:
distance = √(x⁴ + x² - 8x² + 16) = √(x⁴ - 7x² + 16)
To find the minimum distance, we need to minimize the expression x⁴ - 7x² + 16. Since this is a quadratic-like expression, we can use calculus to find the minimum.
Taking the derivative of x⁴ - 7x² + 16 with respect to x, we get:
d/dx (x⁴ - 7x² + 16) = 4x³ - 14x
Setting the derivative equal to zero to find critical points:
4x³ - 14x = 0
Factorizing, we have:
2x(2x² - 7) = 0
This gives us two critical points: x = 0 and x = ±√(7/2).
Next, we evaluate the expression x⁴ - 7x² + 16 at these critical points and the endpoints of the interval:
f(0) = 0⁴ - 7(0)² + 16 = 16
f(±√(7/2)) = (√(7/2))⁴ - 7(√(7/2))² + 16 ≈ 4.157
Comparing these values, we find that the minimum distance occurs at x = 0, giving us a minimum distance of √(0⁴ - 7(0)² + 16) = √16 = 4.
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