Therefore , the solution of the given problem of equation comes out to be f(x) = sin(/2 x) (when x = 1 or 2).
What is an equation?Variable words are commonly used in complex algorithms to show consistency between two statements that are not compatible. Academic expressions called equations are used to show the equality of various academic numbers. In this case, normalization yields b + 7 as opposed to a single formula the fact that split 12 into two parts, which can be used with data collected from y + 7.
Here,
The goal is to construct a trigonometric equation with only answers at
x = 1 and x = 2.
f(x) = A sin(Bx+C+D)
=> B + C + D + A = 0 (equation 1)
=> Equation 2: A sin(2B + C + D) = 0
=> B + C sin A = 0 (equation 3)
=> Sin(2B+C) = A = 0 (equation 4)
Since sin(x) = 0 when x is a multiple of, formulae 3 and 4 can be rewritten as follows:
=> B + C = nπ (equation 5)
=> 2B + C = mπ (equation 6)
with numbers n and m. When we simultaneously solve equations 5 and 6 for B and C, we obtain:
=> B = (m - n)π/2 (equation 7)
=> C = nπ - B (equation 8)
=> When x = 1, C = /2 - /2 = 0.
=> When x = 2, C = − /2 = 3/2.
Thus, the result is as follows:
when f(x) = sin(/2 x) (when x = 1 or 2)
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The ratio of James ' age to Andy's age is 2 : 7. In 8 years' time, the ratio will become
2 : 5, Find James' present age.
Answer: James' age now is 12 years old.
Jim worked 45 hours this week. He earns time and a half for overtime. He is paid $12.59 per/hour, how much will he earn this week?
Answer: 566.55 US dollars
Step-by-step explanation: i think, please give 5 stars
what led to the development of the aviation industry
a. the development of texas’ wwII aircraft training facilities
b. the placement of the johnson space center
c. the petrochemicals industry
d. texas’ location and climate
Texas' location and climate led to the development of the aviation industry.
What is the significance of Texas in the development of the aviation industry?
The development of the aviation industry in Texas can be attributed to its favorable geographical location and climate. Texas is located in the southern part of the United States, making it a strategic location for air travel between the east and west coasts of the country. Additionally, its warm and dry climate provided ideal conditions for flight testing and training.
Reason what led to the development of the aviation industry :
Texas has a long history of aviation, with the Wright brothers conducting one of their earliest flight demonstrations in the state in 1910. The establishment of military bases during World War II and the subsequent development of aircraft training facilities in Texas further fueled the growth of the aviation industry in the state. Additionally, the placement of the Johnson Space Center in Houston helped to establish Texas as a hub for aerospace research and development. However, it is Texas' favorable location and climate that truly made it a prime location for the aviation industry. Its central location made it a strategic location for air travel and logistics, while its warm and dry climate provided ideal conditions for flight testing and training. Today, Texas remains one of the top states for aviation and aerospace industries, with major airports, aircraft manufacturers, and research institutions located throughout the state.
Therefore, option (d) is correct.
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Rose has made this scale drawing of her house. If her house actually 56 feet wide, then scale of her drawing is
If her hοuse actually 56 feet wide, then scale οf her drawing is scale = 56 feet / (drawing width)
What is the slοpe?The slοpe is a mathematical cοncept that refers tο the steepness οf a line. It is cοmmοnly denοted by the letter "m" and can be calculated using the fοrmula:
slοpe (m) = (change in y) / (change in x)
Withοut having the actual measurements οf the drawing, we cannοt determine the scale.
Tο find the scale οf a drawing, yοu need tο knοw the actual measurements οf the οbject being drawn as well as the cοrrespοnding measurements in the drawing. Fοr example, if yοu knοw that the actual width οf Rοse's hοuse is 56 feet and the width οf the hοuse in the drawing is 8 inches, yοu can determine the scale by using the fοllοwing fοrmula:
scale = actual width/drawing width
Using this fοrmula with the given infοrmatiοn, we get:
scale = 56 feet / (drawing width)
Therefοre, we dο nοt have the drawing width, we cannοt calculate the scale.
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The Hillmans have $12,000 in a savings account. The bank pays 1.25% interest on the savings account, compounded continuously. Find the total balance after four years.
The total balance after four years when compounded continuously is $12,615.24.
What is continuous compounding?A technique of calculating interest on a loan or investment where interest is compounded an endless number of times annually is called continuous compounding. Continuous compounding is a method of growing an investment or debt over time by continually adding interest earned over a relatively brief period of time to the principal. In financial computations where interest is earned or charged continuously, such in the bond or mortgage markets, continuous compounding is frequently utilised.
The compound interest is given as:
[tex]A = Pe^{(rt)}[/tex]
Substituting the values:
[tex]A = 12000 * e^{(0.0125 * 4)}\\A = 12000 * e^{(0.05)}[/tex]
A = 12000 * 1.05127
A = 12615.24
Therefore, the total balance after four years is $12,615.24.
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Look at the picture, I'm really confused
Answer:
Option (C)
-0.46 brother
Answer:
Option (C)
-0.46
Step-by-step explanation:
Which of the following are true of
all linear functions?
A. The rate of change is
non-zero.
B. The domain is all real
numbers.
C. The range is all real
numbers.
D. There is one x-intercept.
E. There is one y-intercept.
The domain is all real numbers, The range is all real numbers and There is one y-intercept is true from the given option in context of all linear functions. option B. The domain is all real numbers, C. The range is all real numbers , and E. There is one y-intercept is right choice.
All linear functions have a domain that includes all real numbers, a range that includes all real numbers, and exactly
one y-intercept.
The rate of change of a linear function is non-zero only if the function is not constant (i.e. its slope is not zero).
However, there are linear functions that have a slope of zero, such as the function f(x) = 2, which is a horizontal line.
This means that statement A is not true of all linear functions.
Furthermore, there are linear functions that have no x-intercept (if the line is parallel to the x-axis) or that have more
than one x-intercept (if the line intersects the x-axis more than once).
Therefore, statement D is also not true of all linear functions.
The correct answers to this question are B. The domain is all real numbers, C. The range is all real numbers , and E. There is one y-intercept.
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3. Factor puzzle. *Be sure to show work!
SHARED FACTORS
Each side of the square shares a factor with each of its neighboring sides.
Determine the missing values that make this statement true.
The missing values that make this statement true are as
x² + 9x + 14, x² - 3x - 10
x² - 2x - 15, x² + 10x + 21
What is the quadratic equation?
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and x is the variable. The term "quadratic" comes from the Latin word "quadratus," which means "square."
First, let Xa, Xb, Xc, Xd = each factor
Xd = each factor which shows In the graph
Then, we can know that
Xa, Xb are the roots of x² + ___x + 14
Xb, Xd are the roots of x²= 3x - __
Xc, Xd are the roots of x² - __x - 15
Xa, Xc are the roots of x² + 10x + __
Xa * Xb = 14 Xd = 15/10 +Xa
Xb+ Xd = 3 14/Xa + 15/10 +Xa = 3
Xc* Xd = - 15 140 + 29Xa/Xa(10+Xa)
Xa + Xc = -10x 140 + 29Xa = 30Xₐ + 3Xₐ²
Xb = 14/Xa 3Xₐ² + Xₐ² - 140 = 0
14/Xa + Xd = 3 Xa1 = -7 Xa2 = 20/3
Xd = -15/-3 Xb1 = -2 Xb2 = 21/10
Xc = -10 -X1 Xc1 = -3 Xc2 = -30/3
Xc1 = -3 Xc2 = -30/3
Xd1 = 5 Xd2=9/10
There are two kinds of answers.
1) x² + 9x + 1x Xa1 + Xb1 = -9
x² - 3x + 10 Xb1 * Xd1 = -10
x² - 2x - 15 Xc1 + Xd1 = 2
x² + 10x + 21 Xa1 * Xc1 = 21
2)
x² - (263/30)x + 14 Xa2 * Xb2 = 263/30
x² - 3x - (-189/100) Xb2 * Xd2 = 189/100
x² - (413/30)x - 15 Xc2 + Xd2 = 413/30
x² + 10x + (1000/9) Xa2 * Xc2 = 1000/9
Hence, the missing values that make this statement true are as
x² + 9x + 14, x² - 3x - 10
x² - 2x - 15, x² + 10x + 21
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can you solve this question?
y'=?
The differentiation of the variable y is equal to [tex]\frac{1}{2} ( \frac{y^{2}-4x^{3}-4xy^{2} }{2x^{2} y+2y^{3}-xy } )[/tex] for the differential equation.
Given equation: [tex](x^{2} +y^{2} )^{2} = 2xy^{2}[/tex]
differentiate with respect to x
2 [tex](x^{2} + y^{2})[/tex] [ [tex]2x+2y.\frac{dy}{dx}[/tex] ] =[ (1).[tex]y^{2}[/tex] + [tex]x (2y) + \frac{dy}{dx}[/tex] ]
4 [tex](x^{2} + y^{2})[/tex] [ [tex]x+y \frac{dy}{dx}[/tex] ] = [tex]y^{2}[/tex] + [tex]2xy \frac{dy}{dx}[/tex]
4( [tex]x^{3} + x^{2} y \frac{dy}{dx} + xy^{2} + y^{3} \frac{dy}{dx}[/tex] ) = [tex]y^{2}[/tex] + [tex]2xy \frac{dy}{dx}[/tex]
[tex]4x^{3} +4 x^{2} y \frac{dy}{dx} +4 xy^{2} +4 y^{3} \frac{dy}{dx}[/tex] = [tex]y^{2}[/tex] + [tex]2xy \frac{dy}{dx}[/tex]
[tex]4x^{2}y \frac{dy}{dx}[/tex] + [tex]4y^{3}[/tex] [tex]\frac{dy}{dx}[/tex] - [tex]2xy\frac{dy}{dx}[/tex] = [tex]y^{2} - 4x^{3} - 4xy^{2}[/tex]
[tex](4x^{2}y + 4y^{3} - 2xy )[/tex] [tex]\frac{dy}{dx}[/tex] = [tex]y^{2} - 4x^{3} - 4xy^{2}[/tex]
[tex]\frac{dy}{dx}[/tex] = [tex]y^{2} - 4x^{3} - 4xy^{2}[/tex] / [tex](4x^{2}y + 4y^{3} - 2xy )[/tex]
[tex]\frac{dy}{dx}[/tex] = [tex]\frac{1}{2} ( \frac{y^{2}-4x^{3}-4xy^{2} }{2x^{2} y+2y^{3}-xy } )[/tex]
Hence solved. The differentiation for the given differential equation is done using the technique of implicit differentiation. The differentiation of the variable y is equal to [tex]\frac{1}{2} ( \frac{y^{2}-4x^{3}-4xy^{2} }{2x^{2} y+2y^{3}-xy } )[/tex] for the differential equation.
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2-3(x+4)=8
-2/3
-6
2/3
6
Answer:
x = -6
Step-by-step explanation:
2-3(x+4) = 8
2 - 3x - 12 = 8
-3x - 10 = 8
-3x = 18
x = -6
Answer:
x = -6
Step-by-step explanation:
2 - 3 (x + 4) = 8
2 - 3x - 12 = 8
- 10 - 3x = 8
- 3x = 8 + 10
- 3x = 18
x = -18/3
x = -6
___________
hope this helps!
Your sister has R375 000 and wants to retire. She expects to live for another 25 years, and she also expects to earn 8% on her invested funds. How much could she withdraw at the beginning of each of the next 25 years, and end up with zero in the account?
Answer:
Step-by-step explanation:
Your sister can use the formula for calculating the present value of an annuity to determine how much she can withdraw each year.
The formula for the present value of an annuity is:
PV = Payment x (1 - (1 + r)^-n) / r
Where:
- PV is the present value of the annuity
- Payment is the amount of each withdrawal
- r is the annual interest rate
- n is the number of periods (in this case, 25 years)
We can rearrange this formula to solve for Payment:
Payment = PV x r / (1 - (1 + r)^-n)
We know that your sister has R375 000 to start with, and she wants to end up with zero in 25 years. So, her present value (PV) is R375 000, and we can assume her future value (FV) is zero.
Using the future value formula, we can calculate the interest rate she needs to earn in order to end up with zero in 25 years:
FV = PV x (1 + r)^n
0 = 375000 x (1 + r)^25
(1 + r)^25 = 1
1 + r = (1)^1/25
r = 0
This means that your sister needs to withdraw all of her money over the 25 years in order to end up with zero at the end.
Her withdrawal each year would be:
Payment = PV x r / (1 - (1 + r)^-n)
Payment = 375000 x 0.08 / (1 - (1 + 0.08)^-25)
Payment = R34,028.82 per year
Your sister can withdraw R34,028.82 at the beginning of each year for the next 25 years, and she will end up with zero at the end, assuming she earns an 8% return on her invested funds.
Need help due soon.
Answer:
The correct equation that represents the relationship between b and f is:
Ob = (3/2)f
Explanation:
We know that Priya uses 4 cups of flour to make 3 loaves of banana bread.
So, the amount of flour required per loaf of bread is 4/3 cups.
Now, if Andre makes b loaves of bread using f cups of flour, then the amount of flour required per loaf of bread is f/b cups.
We can set up a proportion as follows:
4/3 = f/b
Cross multiplying, we get:
4b = 3f
Dividing both sides by 3, we get:
b = (3/4)f
This means that Andre can make (3/4) loaves of bread per cup of flour.
But the question asks for the relationship between b and f, so we need to rearrange the equation:
b = (3/4)f
Multiplying both sides by (2/3), we get:
(2/3)b = (1/2)f
Simplifying, we get:
Ob = (3/2)f
Therefore, the correct equation that represents the relationship between b and f is Ob = (3/2)f.
Reflections; rotations and translations are transformations that change the what?
Reflections, rotations, and translations are all transformations that change the position and/or orientation of a geometric figure.
What is reflection?In mathematics, reflection is a transformation that flips a figure over a line called the line of reflection. This line acts like a mirror, reflecting the original figure onto the opposite side of the line.
Reflections, rotations, and translations are all transformations that change the position and/or orientation of a geometric figure.
Reflections (also known as flips) change the orientation of a figure by flipping it across a line of reflection, which acts like a mirror.
Rotations change the orientation of a figure by rotating it around a fixed point. The figure stays the same shape and size, but its position and orientation in space changes.
Translations (also known as slides) change the position of a figure by sliding it along a straight line without changing its orientation or shape.
All of these transformations are important in geometry and other fields, such as physics and computer graphics, and can be used to describe the motion and properties of geometric objects.
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An appliance store decreases the price of a 19-in. television set 28% to a sale price of $435.60.What was the original price?
Answer:
Step-by-step explanation:
Let x be the original price of the television set.
The sale price is 28% off the original price, which means that the sale price is equal to 100% - 28% = 72% of the original price.
So we can write an equation based on this information:
0.72x = 435.60
Solving for x, we get:
x = 435.60 / 0.72
x ≈ $605
Therefore, the original price of the television set was approximately $605.
Figure ABCD is a parallelogram. Is it also a rhombus? Why or why not?
D
5m
B
5 m
C
OA. It cannot be determined, because a parallelogram with congruent
adjacent sides may or may not be a rhombus.
B. No, because all four sides are congruent.
C. Yes, because adjacent sides are congruent.
D. No, because adjacent sides are congruent.
The true statement about the parallelogram is (c) Yes, because adjacent sides are congruent.
How to determine the true statement about the parallelogramGiven that
Figure ABCD is a parallelogram
Such that
Side lengths = 5 cm
This means that
The figure is a square
As a general rule
All squares can be classified as rhombus
This is because a rhombus is a quadrilateral with all four sides of equal length and square is a type of rhombus in which all four sides are equal in length
Hence, the true statement is (c)
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The sum of the measures of the angles of a triangle is 180. The sum of the measures of
the second and third angles is five times the measure of the first angle. The third angle
is 26 more than the second. Let x, y, and z represent the measures of the first, second,
and third angles, respectively. Find the measures of the three angles.
The measures οf the three angIes are x = 51.43 degrees, y = 92.85 degrees, and z = 118.85 degrees.
What is Iinear equatiοn?A Iinear equatiοn is a mathematicaI equatiοn that describes a straight Iine in a twο-dimensiοnaI pIane.
We can use the infοrmatiοn given in the prοbIem tο fοrm a system οf three equatiοns with three variabIes. Let x, y, and z represent the measures οf the first, secοnd, and third angIes, respectiveIy.
Frοm the first piece οf infοrmatiοn, we knοw that: x + y + z = 180
Frοm the secοnd piece οf infοrmatiοn, we knοw that: y + z = 5x
Frοm the third piece οf infοrmatiοn, we knοw that: z = y + 26
We can substitute the third equatiοn intο the secοnd equatiοn tο eIiminate z:
y + (y + 26) = 5x
2y + 26 = 5x
2y = 5x - 26
y = (5x - 26)/2
We can substitute this expressiοn fοr y intο the first equatiοn tο eIiminate y and z:
x + (5x - 26)/2 + (5x - 26)/2 + 26 = 180
2x + 5x - 26 + 26 = 360
7x = 360
x = 51.43
We can substitute this vaIue οf x back intο the expressiοn fοr y tο find y:
y = (5x - 26)/2
y = (5(51.43) - 26)/2
y = 92.85
FinaIIy, we can use the equatiοn z = y + 26 tο find z:
z = y + 26
z = 92.85 + 26
z = 118.85
Therefοre, the measures οf the three angIes are x = 51.43 degrees, y = 92.85 degrees, and z = 118.85 degrees.
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needed help on this one too
Answer:
1st -pentagon -regular
2nd -hexagon -irregular
3rd -octagon -irregular
5/12x6/15=30/180
How do I simplify?
Answer:
0.167
Step-by-step explanation:
if you divide 30/180 it will get you a decimal of 0.166666667 but if you were to divide 180 by 30 you will get 6
technecaly your answer would be 0.166666667 but you only take the first number in the decimal which is 0.1 then take six 0.16 then add the 7 0.167
that should end up being your answer if i did the math right
Answer:
Step-by-step explanation:
5/12 * 6/15 =30/180
1/6=1/6
which is true, Right-hand side is equal to left-hand side
If -26+z=15,what is the value is z?
Answer: 41
Step-by-step explanation:
-26+z =1 5
z= 15+26
z= 41
Answer:
z = 41
Step-by-step explanation:
-26 + z = 15. | + 26
-26 + z + 26 = 15 + 26
z = 41
ms walker has 15 kilograms of clay she wants to give 3 students an equal amount of clay what is the mass of the clay that each student will get
Answer: 5kg
Step-by-step explanation:
If Ms. Walker has 15 kilograms of clay that she wants to distribute evenly among 3 students, all you need to do is divide 15 by 3. Doing so, we get:
15 kg / 3 = 5 kg
Therefore, each student will receive 5 kilograms of clay.
The mass of clay that each student will get is 5kg.
Here, the given questions ask for a solution that deals with the concept of the basic principle of division. hence implementing the principle we can see that,
The given total amount of clay by Ms. walker is 15 kilograms(kg).
Furthermore, this total amount of clay needs to be divided equally between the 3 students.
Then, let the mass of clay given to each student equally be denoted by M in the equation to find out the solution can be written as
M = Total amount of clay ÷ Total number of students
Then,
M = 15 ÷ 3
M = 5 kg
Therefore, The mass of clay that each student will get is 5kg.
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determine what type of transformation is represented
The type of transformation represented in the figure is the translation transformation i.e. (c) none of these
Identifying the type of transformation representedGiven the triangles ABC and A'B'C'
The transformation between the triangles is translation
The translation transformation is a type of transformation that moves an object without changing its size, shape, or orientation.
This transformation involves sliding an object in a particular direction by a certain distance, either horizontally or vertically.
In this case, the direction is horizontally and vertically
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Given (x – 7)2 = 36, select the values of x. x = 13 x = 1 x = –29 x = 42
Answer:
1x
Step-by-step explanation:
Firstly lets expand the brackets for the equation
(x - 7 )2 = 36
If we multiply what's in the brackets by 2 we get this:
2x - 14 = 36
Add 14 to both sides:
2x = 50
Divide both sides by 2:
x = 25
Answer = 1x (Only possible solution
Answer:
The two solutions to the given equation are x = 13 and x = 1.
Step-by-step explanation:
To solve the given equation (x - 7)² = 36, begin by square rooting both sides:
[tex]\implies \sqrt{(x-7)^2}=\sqrt{36}[/tex]
[tex]\implies x-7=\pm6[/tex]
Now add 7 to both sides of the equation:
[tex]\implies x-7+7=\pm6+7[/tex]
[tex]\implies x=7\pm6[/tex]
Therefore, the two solutions are:
[tex]\implies x=7+6=13[/tex]
[tex]\implies x=7-6=1[/tex]
Nathaniel would like to establish a trust fund that will provide R380 000 a year, forever, for his descendants. The trust fund will be invested very conservatively, so the expected rate of return is only 6,45%. How much money must he deposit today, to fund this gift for his descendants?
Answer:
Step-by-step explanation:
To calculate the amount of money Nathaniel needs to deposit today to fund this gift for his descendants, we can use the present value formula for a perpetuity:
Present Value = Annual Payment / Discount Rate
In this case, the annual payment is R380 000 and the discount rate (expected rate of return) is 6.45% in decimals, or 0.0645 as a fraction. So we can plug these values into the formula:
Present Value = R380 000 / 0.0645
This gives us a present value of approximately R5,899,225.68. Therefore, Nathaniel would need to deposit R5,899,225.68 today into the trust fund to provide R380 000 a year forever for his descendants, assuming a 6.45% expected rate of return.
Use a number line to find equivalent fractions to 2/3
The equivalent fraction for 2/3 is 8/12.
Equivalent fraction of a given fraction is got by multiplying or dividing its numerator and denominator by the same whole number. For example, if we multiply the numerator and denominator of 2/3 by 4 we get. 2/3 = 2×4 / 3×4 = 8/12 which is an equivalent fraction of 2/3.
The point (7,8) in the coordinates plane represents a ratio. Adela claims that you can find an equivalent ratio by adding the same number to both coordinates of the point. Is Adela correct?
Answer:
It depends on the context of the problem and the interpretation of the ratio represented by the point (7, 8).
If we interpret the point (7, 8) as representing the ratio 7:8, then Adela's claim is incorrect. Adding the same number to both coordinates of the point would change the value of the ratio. For example, adding 1 to both coordinates would give the point (8, 9), which represents the ratio 8:9, which is not equivalent to 7:8.
However, if we interpret the point (7, 8) as representing a different type of ratio, such as the ratio of the distances from the point to two fixed points or the ratio of the areas of two shapes, then it may be possible to find an equivalent ratio by adding the same number to both coordinates of the point. In this case, Adela's claim could be correct.
Without more information about the context and interpretation of the ratio represented by the point (7, 8), it is difficult to determine the correctness of Adela's claim.
Round 7.19 to the nearest tenth
Answer:
7.2
Step-by-step explanation:
The 1 looks at the number to the right of it. If the number to the right is 5 or more, the 1 moves up to a 2. Hope this helps!
Can anyone show how to solve these two questions. Thank you!
according the given question the exact value of given expression is [tex]$\cos\frac{x}{2} = -\sqrt{\frac{1}{2(1 - \left(-\frac{160}{81}\right)^2)}} = -\sqrt{\frac{81^2}{2(81^2 - 160^2)}} = \boxed{-\frac{81\sqrt{239}}{319}}$[/tex]
First, we need to find [tex]$\sin x$[/tex] using the identity[tex]$\cos^2x + \sin^2x = 1$:$\sin^2x = 1 - \cos^2x = 1 - \left(-\frac{4}{5}\right)^2 = \frac{9}{25}$[/tex]
Since [tex]$\frac{\pi}{2} < x < \pi$[/tex], we know that [tex]$\frac{\pi}{4} < \frac{x}{2} < \frac{\pi}{2}$[/tex]. Therefore, we can use the
identity [tex]$\tan\frac{x}{2} = \frac{\sin x}{1 + \cos x}$[/tex]:
[tex]$\tan\frac{x}{2} = \frac{\sqrt{\frac{9}{25}}}{1 - \frac{4}{5}} = \frac{\frac{3}{5}}{\frac{1}{5}} = \boxed{3}$[/tex]
[tex]If $\tan x = \frac{40}{9}$ and $\pi < x < \frac{3\pi}{2}$, find $\cos\frac{x}{2}$.[/tex]
First, we need to find [tex]$\sin x$[/tex] using the identity [tex]$\tan^2x + 1 = \sec^2x$[/tex]:
[tex]$\sin x = \frac{\tan x}{\sec x} = \frac{\frac{40}{9}}{-\frac{9}{40}} = -\frac{160}{81}$[/tex]
[tex]Since $\pi < x < \frac{3\pi}{2}$, we know that $\frac{\pi}{2} < \frac{x}{2} < \frac{3\pi}{4}$[/tex]. Therefore, we can use the identity [tex]$\cos\frac{x}{2} = \pm\sqrt{\frac{1 + \cos x}{2}}$[/tex]:
[tex]$\cos\frac{x}{2} = -\sqrt{\frac{1 + \cos x}{2}} = -\sqrt{\frac{1 + \frac{\cos^2x}{\sin^2x}}{2}} = -\sqrt{\frac{\sin^2x + \cos^2x}{2\sin^2x}} = -\sqrt{\frac{1}{2(1 - \sin^2x)}}$[/tex]
Plugging in [tex]$\sin x = -\frac{160}{81}$[/tex] , we get:
[tex]$\cos\frac{x}{2} = -\sqrt{\frac{1}{2(1 - \left(-\frac{160}{81}\right)^2)}} = -\sqrt{\frac{81^2}{2(81^2 - 160^2)}} = \boxed{-\frac{81\sqrt{239}}{319}}$[/tex]
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Determine the intercepts of the line.
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A coordinate plane. The x- and y-axes each scale by one. A graph of a line intersects the points zero, negative ten and three, zero.
The intercepts of the line are:
x-intercept: (3, 0)
y-intercept: (0, 1)
What are the intercepts?
To find the x-intercept, we need to find the point where the line intersects the x-axis. This occurs when the y-coordinate of the point is zero. From the given information, we know that the line passes through the point (3, 0), so this point must be the x-intercept.
Therefore, the x-intercept is (3, 0).
To find the y-intercept, we need to find the point where the line intersects the y-axis. This occurs when the x-coordinate of the point is zero. We can find the equation of the line using the two given points (0, y) and (3, 0):
Slope of the line = (y2 - y1) / (x2 - x1)
= (0 - y) / (3 - 0)
= -y / 3
Using the point-slope form of the equation of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, we can write:
y - 0 = (-y/3)(x - 3)
Simplifying this equation, we get:
y = (-y/3)x + 1
Multiplying both sides by 3, we get:
3y = -yx + 3
Adding yx to both sides, we get:
yx + 3y = 3
This is the equation of the line in standard form. To find the y-intercept, we set x = 0:
0 + 3y = 3
Solving for y, we get:
y = 1
Therefore, the y-intercept is (0, 1).
Hence, the intercepts of the line are:
x-intercept: (3, 0)
y-intercept: (0, 1)
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Solve the following quadratic-like equation.
[tex]y^\frac{1}{2} -6y^\frac{1}{4} +8=0[/tex]
Step-by-step explanation:
given quadratic equation is
simplification,
by using,
[tex]y^\frac{1}{2} -6y^\frac{1}{4} +8=0
\\ y^\frac{1}{2} -6y^\frac{1}{4} = - 8 \\
[/tex]
[tex]square[/tex]
[tex] \\ {(y^\frac{1}{2} -6y^\frac{1}{4})}^{2} = {(- 8)}^{2}[/tex]
[tex]y + 36 {y}^{ \frac{1}{2}} \: [/tex]
A sphere has a surface area of 60 square feet. Which choice is the best approximation of its radius? Use 3.14 to approximate pi.\
Answer:
Step-by-step explanation:
The surface area of a sphere is given by the formula:
S = 4πr^2
where S is the surface area and r is the radius of the sphere.
We are given that the surface area of the sphere is 60 square feet. Using the formula above, we can solve for the radius:
60 = 4πr^2
Dividing both sides by 4π, we get:
15/π = r^2
Taking the square root of both sides, we get:
r = sqrt(15/π)
Using 3.14 as an approximation for π, we can evaluate this expression:
r ≈ sqrt(15/3.14)
r ≈ 2.20
Therefore, the best approximation for the radius of the sphere is 2.20 feet.
Answer:
Radius is 2.18
Step-by-step explanation:
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