The value of [tex]a^{2} +\frac{1}{a^{2} }[/tex] if [tex]a=\frac{2+\sqrt{5} }{2}[/tex] is solved to be
5.1How to determine the equationThe equation is solved by substituting for a in the given equation [tex]a^{2} +\frac{1}{a^{2} }[/tex]
[tex]a^{2} +\frac{1}{a^{2} }[/tex]
substituting for a
[tex](\frac{2+\sqrt{5} }{2})=(1+\frac{\sqrt{5} }{2})[/tex]
[tex](1+\frac{\sqrt{5} }{2})^{2} =(1+\frac{\sqrt{5} }{2})*(1+\frac{\sqrt{5} }{2})=1+2\frac{\sqrt{5} }{2} +\frac{5}{4}=\sqrt{5} +\frac{9}{4}=4.48607[/tex]
= 4.8607 + 1/4.48607
= 5.0664
Substituting the value of a in the equation and solving resulted to 5.1
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14 in 8 in 5 in 7 in 9 in 15 in
To find the area of the whole figure, we can divide it into two rectangles, the smaller rectangle has dimensions of 5 inches by 8 inches.
[tex]A_{small}=5in\times8in=40in^2[/tex]The bigger rectangle has dimensions of 9 inches by 15 inches.
[tex]A_{bigger}=9in\times15in=135in^2[/tex]Then, we add
[tex]A=40+135=175in^2[/tex]Hence, the area of the figure is 175 square inches.Find the length of RS.A.73 unitsB.11 unitsC.About 8.5 unitsD About 3.3 units
Let,
[tex]\begin{gathered} R=(x_1,y_1)=(-4,1) \\ S=(x_2,y_2)=(-1,9) \end{gathered}[/tex]The expression to calculate the distance between two points is,
[tex]\begin{gathered} D=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ D=\sqrt[]{(-1-(-4))^2+(9-1)^2} \\ D=\sqrt[]{(3)^2+(8)^2} \\ D=\sqrt[]{9+64} \\ D=\sqrt[]{73} \\ D=8.5\text{ units} \end{gathered}[/tex]Thus, the length of the line RS is 8.5 units, and the correct option is option C.
Solve for a side in right triangles
The length of AC is 2.96 units using trigonometric ratio.
What are the trigonometric ratios?
Sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant are the six trigonometric ratios. A branch of mathematics called trigonometry in geometry deals with the sides and angles of a right-angled triangle.
Given that angle A = 65° and AB = 7 units.
The base of △ABC is AC, height is BC, and hypotenuses is AB.
Cosine of angle is the ratio of base to hypotenuses.
Thus cos 65° = AC/AB
Substitute AB = 7
cos 65° = AC/7
Multiply both sides by 7:
7 × cos 65° = AC
Putting cos 65° = 0.42261826174
7 × 0.42261826174 =AC
AC = 2.95832783218
AC = 2.96 (nearest hundredth)
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A bakery is selling breakfast platters. A platter of 12 bagels costs $10.80. Additionally, it costs $6.50 for a gallon of coffee.
What is the slope of this situation?
0.11
0.90
6.50
10.80
The slope of the given situation of selling breakfast platters is 0.90.
Given, the platter of 12 bagels costs $10.80.
The extra cost of $6.50 for a gallon of coffee.
To find the slope of the situation, we convert the situation into the equation of the line.
y=mx+c
here c is the extra cost that is 6.50.
12m=10.80
For 12 bagels , the cost is 10.80, then the slope will be the cost of each bagel.
m=10.80/12
m=0.90
m=$0.90
Therefore, option c 0.90 is the slope of the given situation of the bakery selling the breakfast platter of 12 bagels and a gallon of coffee.
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can you please help me
The height of the actual building, given the height of the model, can be found to be 1, 455 feet
How to find the height of the actual building?The model is such that every inch of the building on the model is 75 feet of the actual building.
This means that if the height of the building in the model is 19 ² / ₅ inches, then the height of the actual building would be:
= Height of building in model x Scale factor per inch
= 19 ² / ₅ x 75
= 97 / 5 x 75
= 1, 455 feet
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Given f(x) = 5x − 7 and g(x) = −4x + 2, what is (f − g)(x)? x − 9 x − 5 9x − 9 9x − 5
To get (f-g)(x), we're essentially just factoring out x from f(x) and g(x).
-> f(x) - g(x) = x(f-g), and by the properties , x(f-g) is the same as (f-g)(x).
Therefore, (f-g)(x) = 5x - 7 - (-4x + 2) = 5x - 7 + 4x - 2 = 9x - 9.
Answer:
[tex](f-g)(x)=9x - 9[/tex]
Step-by-step explanation:
Given functions:
[tex]\begin{cases}f(x) = 5x-7 \\ g(x) = -4x + 2\end{cases}[/tex]
To find (f - g)(x), subtract function g(x) from function f(x):
[tex]\begin{aligned}(f-g)(x)&=f(x)-g(x)\\&=(5x-7)-(-4x+2)\\&=5x-7+4x-2\\&=5x+4x-7-2\\&=9x-9\end{aligned}[/tex]
Therefore, the solution to the given composite function is:
[tex]9x-9[/tex]Write the equation of the line parallel to x+7y=21 that passes through the point (–14, 4).
Answer:
Equation of line is given as y = mx + c, where m is the gradient and c is the y-intercept.
First, rearrange x + 7y = 21 into y = mx + c form.
x + 7y = 21
7y = - x + 21
y = -1/7x + 3
Parallel lines have the same gradient so gradient of line is -1/7.
Equation of line is now y = -1/7x + c.
Substitute (-14,4) into the equation to find c.
4 = -1/7(-14) + c
4 = 2 + c
c = 2
Hence, equation of line is y = -1/7x + 2.
A restaurant offers a 20% off coupon. The tax rate is 7.5% . If an item is listed at $15 , how much does it cost after the coupon and tax have been applied?
Answer:
11.10 or 11.1
Step-by-step explanation:
1. 15 x 0.075= 1.125 or 1.13
2. 15-1.13=13.87
3. 13.87 x 0.20= 2.774 or 2.77
4. 13.87 - 2.77 = 11.10 or 11.1
Geometry Pls Help, I need hthis very fast
The translation rule is (x, y) ---> (x+8)(x-4).
What is translation rule?
A translation is a type of transformation that moves each point in a figure the same distance in the same direction. The second notation is a mapping rule of the form (x,y) → (x−7,y+5).
A translation in math moves a shape left or right and/or up or down. The translated shapes look exactly the same size as the original shape, and hence the shapes are congruent to each other.
Given,
let A =(x, y) ----> (x+9,y-2) and B =(x, y) ----> (x-1,y-2)
which means A is translated 9 units right and 2 units down
and B is translated 1 unit left and 2 units down.
Therefore, rule for composition is A+B
= (x, y) ------> (x+9-1, y-2-2)
(x,y) -----> (x+8,y-4)
which means translation is 8 units right and 4 units down.
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(8u^2+3u-4)+(8u^2+3u-1)-(3u^2-3u-8)
Simplify the expression
[tex]\mleft(8u^2+3u-4\mright)+\mleft(8u^2+3u-1\mright)-\mleft(3u^2-3u-8\mright)[/tex]Removing all the parentheses, taking special care to change the signs of the last three terms:
[tex]8u^2+3u-4+8u^2+3u-1-3u^2+3u+8[/tex]Now collect like terms:
[tex]8u^2+8u^2-3u^2+3u+3u+3u-4-1+8[/tex][tex]13u^2+9u+3[/tex]solve for x……………………..
Answer:
7.5/3
Step-by-step explanation:
7x-7.5 = 4x
-7.5 = -3x
x= 7.5/3
Write and equation for a line with a slope of 3 that passes through the point P(2,−5)
Answer:
y=3x-11
Step-by-step explanation:
In this question, we will use the point slope form:
Which has the equation:
[tex]y-y_1=m(x-x_1)\\subsitute:\\y+5 = 3(x-2) , (y--5=+5)\\y+5=3x-6\\y=3x-11[/tex]
Hope this helps!
A line passes through the point (-4, -8) and has a slope of 4.
Write an equation in slope-intercept form for this line.
[tex](\stackrel{x_1}{-4}~,~\stackrel{y_1}{-8})\hspace{10em} \stackrel{slope}{m} ~=~ 4 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-8)}=\stackrel{m}{ 4}(x-\stackrel{x_1}{(-4)}) \implies y +8= 4 (x +4) \\\\\\ y+8=4x+16\implies {\Large \begin{array}{llll} y=4x+8 \end{array}}[/tex]
An element with a mass of 310 grams decays by 6.4% per minute. To the nearest tenth of a minute, how long will it be until there are 100 grams of the element remaining?
Answer:
17.1
Step-by-step explanation:
Exponential Functions:
y=ab^x
a=starting value = 310
r=rate = 6.4%=0.064
Exponential Decay:
b=1−r=1−0.064=0.936
Write Exponential Function:
y=310(0.936)^x
Put it all together
Plug in y-value:
100=310(0.936)^x
100/310 = 310(0.936)^x/310
Divide both sides by 310
0.322581=0.936^x
\log 0.322581=\log 0.936^x
Take the log of both sides
log0.322581=xlog0.936
Use power rule to bring x to the front
log0.322581/log0.936 = xlog0.936/log0.936
Divide both sides by log(0.936)
17.106221=x
x≈17.1
17.12 will it take until there are 100 grams of the element remaining
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Given that element with a mass of 310 grams decays by 6.4% per minute.
we need to find the time will it be until there are 100 grams of the element remaining
y=abˣ
a=starting value = 310
r=rate = 6.4%=0.064
Exponential Decay:
b=1−r
=1−0.064
=0.936
Plug in these values in the formula
y=310(0.936)ˣ
100=310(0.936)ˣ
Divide both sides by 310
100/310 = (0.936)ˣ
0.322581=0.936ˣ
Apply log on both sides
log 0.322581=xlog0.936
Divide both sides by log0.936
log0.322581/log0.936 = xlog0.936/log0.936
17.106221=x
Hence, 17.12 will it take until there are 100 grams of the element remaining
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Solve. Round the answer to the nearest cent, if necessary.Last year the profit for a company was $483,000. This year's profit decreased by 2.1%. Find this year's profit.
year's
this year´s profit is $ 472857
Explanation
we can easily solve this by using a rule of three
so
Step 1
a)let x represents this year's profit
so
if it is decreases by 2.1 %
[tex]\begin{gathered} x=100-2.1\text{ = 97}.9 \\ x=97.9\text{\%} \end{gathered}[/tex]so, this year's profit is 97.9% of the last year profit
hence
[tex]\begin{gathered} \text{if} \\ 483000\rightarrow100\text{ \%} \\ x\rightarrow97.9\text{ \%} \end{gathered}[/tex]we can make a proportion
[tex]\frac{483000}{100}=\frac{x}{79}[/tex]finally, solve for x,
[tex]\begin{gathered} \frac{483000}{100}=\frac{x}{97.9} \\ \text{Multiply both sides by 9}7.9 \\ \frac{483000}{100}\cdot97.9=\frac{x}{97.7}\cdot97.9 \\ 472857=x \end{gathered}[/tex]therefore
this year´s profit is $ 472857
I hope this helps you
find the unit rate. round to the nearest hundredth, if necessary $200 for 15 ft
EXPLANATION
The unit rate is given by the following relationship:
Unit rate= $200/15ft = $13.3333.../ft
Then, rounding to the nearest hundreth:
Unit rate = $13.33/ft
Which phrase best describes the transformation(s) of the following graph from the parent quadratic function? f(x)=(x-2)^2+3
The parent quadratic function is 2 unit right and 3 units up.
Given function is
f(x) = (x-2)² +3
= x² - 2² - 2×x×2 + 3
= x2 - 4 - 4X + 3
= X² - 4X -1
Parent function g(x) = x²
The parent function is shifted to 2 units in right and 3 units up.
Parent function transformations are shown in blue. This is a 1 unit shift down. Mirroring of the x-axis is done functionally by multiplying the superordinate function by the negative function.
A parent function is the simplest function that satisfies the definition of a particular function type. For example, consider linear functions that form a family of functions.
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Disclaimer:- your question is incomplete, Please see below for the complete question.
Which phrase best describes the transformation(s) of the following graph from the parent quadratic function? f(x)=(x-2)^2+3
What is the Value of log7343?
Answer: 3.86587352821
Step-by-step explanation: This cannot be solved by hand, you need a calculator for this!
FreshmenSophomoreJuniors464Below, the two-way table is given for aclass of students.Seniors TotalMale2 2Female 36 3TotalIf a student is selected at random, find theprobability the student is a junior given that it'smale. Round to the nearest whole percent.[?]%
Solution
Step 1
Write out the expression for the probability of an event occurring
[tex]Pr(\text{ event occurring)}=\frac{Number\text{ of required events}}{\text{Total number of events}}[/tex]For this question,
The number of required events = The number of students that are male and Junior= 2
The total number of events = The total number of male students= 4+6+2+2 = 14
Step 2
Find the required probability after substitution
[tex]Pr(\text{student }is\text{ a junior given its male) =}\frac{2}{14}=\frac{1}{7}[/tex]Hence the probability the student is a junior given its a male = 1/7
In percentage, the probability will be
[tex]\begin{gathered} \frac{1}{\frac{1}{7}}=\frac{100}{x} \\ \text{x =}\frac{1}{7}\times100 \\ \text{x = 14.29\%} \end{gathered}[/tex]
Where x is the required percentage, to the nearest whole percent, the final answer is 14%
Calculate the following trig ratios. Round to the nearest fourth place. 25 and 45
Apply the trigonometric functions:
Sin B = opposite side / hypotenuse:
Where:
B : angle
Opposite sid
The water bottle sale for a music festival started at 5:00 P.M. The table shows the linear relationship between the number of water bottles remaining and the number of hours since 5:00 P.M.
Which equation can be used to determine y, the number of water bottles remaining at the music festival x hours after 5:00 P.M.?
The equation used to determine y, the number of water bottles remaining at the music festival x hours after 5:00 P.M will be;
⇒ y = - 500x + 5500
What is Equation of line?
The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
The table shows the linear relationship between the number of water bottles remaining and the number of hours since 5:00 P.M.
Now,
Find the linear equation for the given table as;
Take number of hours = x
And, Number of remaining bottles = y
So, Two points for the linear equation as;
⇒ (x, y) = (1, 5000) , (2, 4500)
Thus, The linear equation for the table is,
⇒ y - y₁ = (y₂ - y₁) / (x₂ - x₁) (x - x₁)
⇒ y - 5000 = (4500 - 5000) / (2 - 1) (x - 1)
⇒ y - 5000 = - 500 (x - 1)
⇒ y - 5000 = - 500x + 500
⇒ y = - 500x + 500 + 5000
⇒ y = - 500x + 5500
Thus, The equation used to determine y, the number of water bottles remaining at the music festival x hours after 5:00 P.M will be;
⇒ y = - 500x + 5500
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PLEASE HELP!! ILL GIVE YOU BRAINLIEST
Using your equation from step 2d, estimate the GPA of a student who studies for 15 hours a week. Justify your answer.
hours on left side (x) , gpa on right side (y)
The GPA of a student who studies for 15 hours a week is 2.935.
What is a GPA?
Grade point average, or GPA, is a conventional method of evaluating academic performance in the United States on a scale of 0 to 4.
How to calculate GPA?
To calculate your GPA, divide the total number of grade points earned by the total number of letter-graded units undertaken.
Here,
We have an equation y = 0.164x + 0.475
put x= 15
we get,
y = 0.164* 15 + 0.475
y = 2.935
Hence, The GPA of a student who studies for 15 hours a week is 2.935
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Find the equation of the line shown. Enter your answer in point-slope form.AV15110SX15-10-51015-5-10151
From the graph given, we are asked to find the equation of line, with our answer in point slope form.
from the graph:
y = 9, x = 10
y1 = 5, x1 = 6
recall, the equation of a line is:
y = mx + b
slope m = y - y1/x - x1
m = 9 - 5/10 - 6
m = 4/4
m = 1
y = 1x + b
y = x + b
using the point slope form:
y - y1 = m(x - x1)
Where:
y = y coordinate of second point
y1 = y coordinate of point one
m = slope
x = x coordinate of second point
x2 = x coordinate of point one.
Therefore, the point slope equation is:
y - 5 = 1(x - 6)
y - 5 = x - 6
Could you help me with these two questions? 20 points !
1 ) Create an equation in point-slope form that has a slop of 5 passing through the points (3,5)
2) Create an equation in point slop form that has a slop of 2 and passes through the points (-1, 6)
The equation of line are given as y = 5x - 10 and y = 2x + 8 respectively
Point-Slope FormulaThe point slope form is used to find the equation of the straight line which is inclined at a given angle to the x-axis and passes through a given point. The equation of a line is an equation that is satisfied by each and every point on the line. This means that a linear equation in two variables represents a line. The equation of a line can be found through various methods depending on the available information.
In this case, we have can proceed to write the equation of a straight line which is given as y = mx + c
m = slopec = y-interceptLet's find the y - intercept and write the equation.
y = mx + c
5 = 5(3) + c; c = 5 - 15 = -10
y = 5x - 10
2) We have the slope as 2 and the points are (-1, 6)
The equation of the line can be written in form of y = mx + c
6 = 2(-1) + c
6 = -2 + c; c = 8
The equation is given as y = 2x + 8
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A club currently has 360 members who pay $350 per month for
membership dues. The club's board members want to increase
monthly revenue by lowering the monthly dues in hopes of
attracting new members. A market research study has shown that
for each $1 decrease in the monthly membership price, an
additional 3 people will join the club. What price should the club
charge to maximize revenue? What is the maximum revenue?
The price charge is $235 and the maximum revenue is $165675.
What is a parabola?It is a plane curve that is mirror-symmetrical. It is approximately U-shaped. It is a plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed-line. It is the locus of a fixed point that moves on a plane. It has many important applications. In Mathematics, the parabola plays a crucial role.
Number of members in club = 360
Monthly membership is of $350
For $ 1 price decrease and 3 people joinining
Membership charge will be = 350-x
Members will be 360+3x
Revenue equation will be
R (x) (350-1x)(360+3x)=0
126,000-360x+1050x-3x² =0
126,000+690x- 3x²-=0
This is a parabolic equation. Maximize the function to get
Differentiate to get the parabola vertex
R'(x) = 690-6x
So, 690-6x =0
x =115
Therefore, the price charge is
350-115 = $235
The number of members is
3360+115 = 705
The maximum revenue is given as
$235 × 705=$165675
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The amount of stain needed to cover a wood surface varies directly as the area of the surface. if 3 pints of stain are needed to cover a square deck with sides 6 feet 8 inches, how many pints are needed to cover a rectangular deck of length 10 feet 8 inches and width 8 feet 3 inches?
In linear equation, 10 feet and 8 inches will require 4.6 pints.
What are instances of linear equations?
Ax+By=C represents a two-variable linear equation in its standard form. A standard form linear equation is, for instance, 2x+3y=5.Finding both intercepts of an equation in this form is rather simple (x and y).When resolving systems of two linear equations, this form is also incredibly helpful.For 10 feet and 8 inches,
12 inches = 1 foot,
therefore 8 inches = 8/12 foot = 0.6 foot
10 feet 8 inches = 10 + 0.6 = 10.6 feet
6 feet 8 inches required 3 pints
1 foot would have required (3/6.8) pints
10.6 feet will then require (3/6.8)*10.6 = 4.6 pints.
Therefore 10 feet and 8 inches will require 4.6 pints.
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ions
aic
. Find the algebraic
expression:
3 more than the product of 2
and k
•
•
A) 3(2) + k
B) 3k + 2
C) 3 + 2k
D) 2k - 3
please help me with this algebra question
Answer: D.
Hope it helps
Solve this equation for N. 3N + 5 = 38
ANSWER
N = 11
EXPLANATION
Given:
3N + 5 = 38
Desired Outcome:
Value of N
Solve for N
[tex]\begin{gathered} 3N+5=38 \\ subtract\text{ 5 from both sides} \\ 3N+5-5=38-5 \\ 3N=33 \\ divide\text{ both sides by 3} \\ \frac{3N}{3}=\frac{33}{3} \\ N=11 \end{gathered}[/tex]Hence, the value of N is 11.
If 180° ≤ ≤ 270 and S(A) = −4 then determine the exact values of cos(A) and tan(A)
Recall the definition of the sine of an angle on a right triangle:
[tex]\sin (A)=\frac{a}{c}[/tex]On the other hand, according to this diagram, the values for tan(A) and cos(A) are given by:
[tex]\begin{gathered} \cos (A)=\frac{b}{c} \\ \tan (A)=\frac{a}{b} \end{gathered}[/tex]Since 180≤A≤270, the right triangle that corresponds to the angle A on the coordinate plane looks as follows:
Where a and b are negative distances.
Since sin(A)=-4/7, we can assume that a=-4 and c=7. Use the Pythagorean Theorem to find the exact value of b:
[tex]\begin{gathered} a^2+b^2=c^2 \\ \Rightarrow(-4)^2+b^2=7^2 \\ \Rightarrow16+b^2=49 \\ \Rightarrow b^2=49-16 \\ \Rightarrow b^2=33 \\ \Rightarrow|b|=\sqrt[]{33} \end{gathered}[/tex]We know that b should be negative. Then:
[tex]b=-\sqrt[]{33}[/tex]Substitute b=-sqrt(33) and c=7 to find the exact values for cos(A) and tan(A):
[tex]\begin{gathered} \cos (A)=-\frac{\sqrt[]{33}}{7} \\ \tan (A)=\frac{-4}{-\sqrt[]{33}}=\frac{4\cdot\sqrt[]{33}}{33} \end{gathered}[/tex]Therefore, the exact values for cos(A) and tan(A) are:
[tex]\begin{gathered} \cos (A)=-\frac{\sqrt[]{33}}{7} \\ \tan (A)=\frac{4\cdot\sqrt[]{33}}{33} \end{gathered}[/tex]