Answer:
1/2 and 3/2Step-by-step explanation:
Given the expression u= av + bw where u = 2i + j, v = i + j, and w = i - j, to find the scalars a and b, we will first substitute the vectors into the expression given as shown;
u= av + bw
2i + j = a( i + j)+b(i-j)
open the parenthesis
2i + j = ai+aj+bi-bj
collect like terms at the right hand side of the equation
2i + j = ai+bi+aj-bj
2i + j = (a+b)i+(a-b)j
compare the coefficient of complex value i and j on both sides
a+b = 2 ............... 1
a-b = 1................. 2
solve the resulting equation simultaneously using elimination method
Subtract both equation from each other.
(a-a) +(b-(-b))= 2-1
0+2b = 1
b = 1/2
Substitue b = 1/2 into equation 2 to get a
From equation 2, a = 1+b
a = 1 + 1/2
b = 3/2
Hence the scalars a and b are 1/2 and 3/2 respectively
[tex]a=\frac{3}{2},b=\frac{1}{2}[/tex]
A complex number is of the form [tex]\boldsymbol{a+ib}[/tex] where [tex]a,b[/tex] are real numbers.
[tex]u=2i+j[/tex]
[tex]v=i+j[/tex]
[tex]w=i-j[/tex]
[tex]u=av+bw[/tex]
So,
[tex]2i+j=a(i+j)+b(i-j)[/tex]
[tex]2i+j=(a+b)i+(a-b)j[/tex]
Therefore,
[tex]2=a+b\\1=a-b[/tex]
Add these equations.
[tex]3=2a\\\frac{3}{2}=a[/tex]
So,
[tex]1=\frac{3}{2}-b\\\\b=\frac{3}{2}-1\\[/tex]
[tex]=\frac{1}{2}[/tex]
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Find the following integral.
Integrate x + 5/x^2 +10x +26 dx
Answer:
[tex]\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx = \frac{ln|x^2 + 10x + 26|}{2} + C[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Integration
IntegralsDefinite/Indefinite IntegralsIntegration Constant CIntegration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
U-Substitution
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx[/tex]
Step 2: Integrate Pt. 1
Set variables for u-substitution.
Set u: [tex]\displaystyle u = x^2 + 10x + 26[/tex][u] Differentiate [Addition/Subtraction, Basic Power Rule]: [tex]\displaystyle du = 2x + 10 \ dx[/tex]Step 3: Integrate Pt. 2
[Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx = \frac{1}{2}\int {\frac{2(x + 5)}{x^2 + 10x + 26}} \, dx[/tex][Integrand] Expand: [tex]\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx = \frac{1}{2}\int {\frac{2x + 10}{x^2 + 10x + 26}} \, dx[/tex][Integral] U-Substitution: [tex]\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx = \frac{1}{2}\int {\frac{1}{u}} \, du[/tex][Integral] Logarithmic Integration: [tex]\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx = \frac{1}{2}ln|u| + C[/tex]Back-Substitute: [tex]\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx = \frac{1}{2}ln|x^2 + 10x + 26| + C[/tex]Simplify: [tex]\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx = \frac{ln|x^2 + 10x + 26|}{2} + C[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
use the figure to determine the intersection of planes EAB and EFG
FG
EA
EF
no intersection
Answer:
Option (3). EF
Step-by-step explanation:
From the figure attached,
Plane defined by EAB can be represented by the face EABF of the square prism also.
Similarly, plane EFG can be represented by the face EFGH of the prism.
Now these sides EABF and EFGH are joining each other at the edge EF of the cuboid.
Therefore, intersection of the given planes is EF.
Option (3) will be the answer.
What is the domain for the green segment in the piecewise function
below? *
y
6
Х
1-8
1-4/
o
4
8
6
- 12
-18
1-24
O (-10,10)
O (2,10)
O 13,10]
O (1,9)
Answer:
(2,10)
Step-by-step explanation:
The x value for the green segment starts at 2 and ends at 10
y=x^2-6x-16 in vertex form
Answer:
[tex]y=(x-3)^{2} -25[/tex]
Step-by-step explanation:
The standard form of a quadratic equation is [tex]y=ax^{2} +bx+c[/tex]
The vertex form of a quadratic equation is [tex]y=a(x-h)^{2} +k[/tex]
The vertex of a quadratic is (h,k) which is the maximum or minimum of a quadratic equation. To find the vertex of a quadratic, you can either graph the function and find the vertex, or you can find it algebraically.
To find the h-value of the vertex, you use the following equation:
[tex]h=\frac{-b}{2a}[/tex]
In this case, our quadratic equation is [tex]y=x^{2} -6x-16[/tex]. Our a-value is 1, our b-value is -6, and our c-value is -16. We will only be using the a and b values. To find the h-value, we will plug in these values into the equation shown below.
[tex]h=\frac{-b}{2a}[/tex] ⇒ [tex]h=\frac{-(-6)}{2(1)}=\frac{6}{2} =3[/tex]
Now, that we found our h-value, we need to find our k-value. To find the k-value, you plug in the h-value we found into the given quadratic equation which in this case is [tex]y=x^{2} -6x-16[/tex]
[tex]y=x^{2} -6x-16[/tex] ⇒ [tex]y=(3)^{2} -6(3)-16[/tex] ⇒ [tex]y=9-18-16[/tex] ⇒ [tex]y=-25[/tex]
This y-value that we just found is our k-value.
Next, we are going to set up our equation in vertex form. As a reminder, vertex form is: [tex]y=a(x-h)^{2} +k[/tex]
a: 1
h: 3
k: -25
[tex]y=(x-3)^{2} -25[/tex]
Hope this helps!
Any places a dot on a coordinate plane (-3,6) she wants to place another dot across the y axis and it must be 8 points away . Where will any place the other dot
Answer:
(5,6)
Step-by-step explanation:
move the dot 8 spaces to the right because that is the across the y axis.
a) Let U = {1,2, ., 9} be the universal set, and let
A = {1,2,3,4,5),
C= {5, 6, 7, 8, 9), E= {2,4,6,8),
B = {4, 5, 6, 7),
D= {1, 3, 5, 7, 9), F = {1,5,9}.
Find: i) AUC
Step-by-step explanation:
(i) AUC={1,2,3,4,5,6,7,8,9}
A local library claims it has 2.73x103 books. If there are 42 people in the building, how many books per capita are there?
Answer:
[tex]Per\ Capita = 64[/tex]
Step-by-step explanation:
Given
[tex]Books = 2.7 * 10^3[/tex]
[tex]People = 42[/tex]
Required
Determine the per capita
The per capita is calculated by dividing number of books by people as follows;
[tex]Per\ Capita = \frac{Books}{People}[/tex]
Substitute 2.7 * 10^3 for books and 42 for people
[tex]Per\ Capita = \frac{2.7 * 10^3}{42}[/tex]
[tex]Per\ Capita = 64[/tex] (Approximated)
Hence, the per capita is 64
The radius of a circle measures 8 yd. What is the circumference of the circle? Use 3.14 for pi, and do NOT round your answer. Be sure to include the correct unit in your answer.
Answer:
The answer is
50.24 ydStep-by-step explanation:
Circumference of a circle = 2πr
where
r is the radius
From the question
r = 8 yd
π = 3.14
Substitute the values into the above formula and solve
Circumference = 2(8)(3.14)
We have the final answer as
50.24 ydHope this helps you
Write 20% as a decimal.
Answer:
0.2
Steps:
To convert to a decimal, divide the percent number by 100.
So, 20 divided by 100 = 0.2
8x-9x = 150/10 helpp
Answer:
x =-15
Step-by-step explanation:
[tex]8x-9x = \frac{150}{10} \\\\\mathrm{Add\:similar\:elements:}\:8x-9x=-x\\-x=\frac{150}{10}\\\\\mathrm{Divide\:the\:numbers:}\:\frac{150}{10}=15\\-x=15\\\\\mathrm{Divide\:both\:sides\:by\:}-1\\\frac{-x}{-1}=\frac{15}{-1}\\\\Simplify\\x=-15[/tex]
3/4+5/36+7/144+.......+17/5184+19/8100
(a)0.95 (b)1 (c) 0.99 (d)0.98
Answer:
B) 1
Step-by-step explanation:
All I did was simplify the equation
1509/1600
(Decimal: 0.943125)
HOPE THIS HELPS!!!! :)
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A rectangle's perimeter is 92 ft. Its length is 5 ft shorter than two times of its width. Use an equation
to find the rectangle's length and width.
Field data have indicated that Unit A has a failure rate of 0.0004 failure per hour. Calculate the reliability of the unit for a 150-hour mission.
Answer:
Reliability= 149.94
Step-by-step explanation:
Field data have indicated that Unit A has a failure rate of 0.0004 failure per hour.
Reliability per hour=1- failure rate
Reliability per hour= 1-0.0004
Reliability per hour= 0.9996
Reliability= rate if reliability * number of hours
Rate of reliability= 0.9996 per hour
Number of hours= 150 hours
Reliability= 0.9996*150
Reliability= 149.94
Which of the following expressions is equal to 5^6/5^2? A) 5 • 5 • 5 • 5 B) 5 • 5 • 5 C) 1/5•5•5•5 D) 3
Step-by-step explanation: To simplify, we will apply the Quotient Rule.
The 5's in this problem are bases so as you apply the quotient rule,
subtract the exponents but leave the base alone to get 5⁴.
We can also write 5⁴ as 5 · 5 · 5 · 5.
Answer:
A) 5*5*5*5
Step-by-step explanation:
How many ways can you choose seven people from a group of twenty?
Answer:
77520 according to google
Summarize the data by filling in the frequency (to the nearest whole number), the relative frequency (3 decimals), and the percent frequency (1 decimal) values below. Class Frequency Relative Frequency Percent Frequency 12-14 ( ) ( ) ( ) 15-7 ( ) ( ) ( ) 18-20 ( ) ( ) ( ) 21-23 ( ) ( ) ( ) 24-26 ( ) ( ) ( ) Total ( ) ( ) ( )
Answer:
The answer is below
Step-by-step explanation:
The relative frequency is the proportion of a class or interval within a category. It is represented by the formula:
Relative frequency = frequency / total frequency
The percent frequency is the relative frequency expressed in percentage, it is given by the formula:
Percent frequency = Relative frequency × 100%
Using the formulas above, the table is filled:
The total frequency = 39
Class Frequency Relative frequency Percent frequency
12 - 14 2 2/39 = 0.0513 0.0513 × 100% = 5.13%
15 - 17 8 8/39 = 0.2051 0.2051 × 100% = 20.51%
18 - 20 11 11/39 = 0.2821 0.2821 × 100% = 28.21%
21 - 23 9 9/39 = 0.2308 0.2308 × 100% =23.08%
24 - 26 9 9/39 = 0.2308 0.2308 × 100% =23.08%
Class Frequency Relative frequency Percent frequency
12 - 14 2 0.0513 5.13%
15 - 17 8 0.2051 20.51%
18 - 20 11 0.2821 28.21%
21 - 23 9 0.2308 23.08%
24 - 26 9 0.2308 23.08%
Ashley Traveled 10 miles north, then 8 miles east, then 9 miles south on her boat
Which of the following Expressions has a coefficient of 10 and a constant of 5?
Answer:
10r + 5
Step-by-step explanation: This is the correct answer also i just got my answer wrong lol for reals this is the correct answer
Answer:
its 10x + 5
Step-by-step explanation:
Please answer I need this asap
Answer:
84
Step-by-step explanation:
The equation that models this situation is 1110 = 15x + 150 where x = cameras sold (note that this doesn't include the first 20 cameras, we'll add those on at the end). Let's solve for x!
1110 = 15x + 150
960 = 15x
x = 64 so the answer is 64 + 20 = 84 cameras.
3.7m∧5÷3.7m∧2 How do I complete this expression?
The average ocean depth is 3.7 × 103 m, and the area of the oceans is 3.6 × 1014 m2. What is the total volume of the ocean in liters? (One cubic meter contains 1000 liters. Round your answer to one decimal place.)
a/x + b/x =c, c ne0
i’ll give more points
Answer:
x = (a +b)/c
Step-by-step explanation:
Perhaps you want to solve for x.
Multiply by x:
a + b = cx
x = (a +b)/c . . . . . divide by c
Determine whether each point lies on the graph of the equation.(a) 2x - y - 3 = 0 (a) (1, 2)Yes, the point is on the graph.No, the point is not on the graph.(b) (1, -1)Yes, the point is on the graph.No, the point is not on the graph.
Answer:
a) (1, 2) not on the graph.
b) (1, -1) is on the graph.
Step-by-step explanation:
Given the equation of the line as:
[tex]2x - y - 3 = 0[/tex]
The points given are
a) (1, 2)
To determine whether it is on the graph of the line or not.
To do so, we can do 2 things:
1. Draw the graph and plot the point on the graph to check whether it is on the graph or not.
2. To put the point in the given equation of the line, whether the equation is satisfied or not.
For method 1: Kindly refer to the attached image of the line and point plotted.
Method 2:
Let us put [tex]x=1, y=2[/tex] in the Left Hand Side (LHS) of equation.
[tex]2\times 1 - 2 - 3\\\Rightarrow -3 \neq 0 , RHS[/tex]
(1, 2) Not on the graph.
(b) (1, -1)
For method 1: Kindly refer to the attached image of the line and point plotted.
Method 2:
Let us put [tex]x=1, y=-1[/tex] in the Left Hand Side (LHS) of equation.
[tex]2\times 1 - (-1) - 3\\\Rightarrow 3-3 = 0 = RHS[/tex]
(1, -1) is on the graph.
Need help with this question. Will crown brainliest
what number is the opposite of -24
Answer:
+24 is the opposite of -24 I think
Step-by-step explanation:
Hope it will help ^_^
it 3x - 2 = 7, then 3x = 9
is an example of the ?
Answer:
Addition Property of Equality
Step-by-step explanation:
So we had:
[tex]3x-2=7[/tex]
And then we got:
[tex]3x=9[/tex]
We did this by adding 2 to both sides of the equation. When we added 2, the left side cancelled and the right side equated to 9:
[tex](3x-2)+2=(7)+2\\3x=9[/tex]
Since we added 2 to both sides, this is an example of the addition property of equality.
If costheta=4/5 then what is sec^2theta
Answer:
[tex]\sec^2(\theta)=\frac{25}{16}[/tex]
Step-by-step explanation:
Recall what the relationship between cosine and secant:
[tex]\sec(\theta)=\frac{1}{\cos(\theta)}[/tex]
In other words, secant is the reciprocal of cosine.
So, if we know cosine, we only need to find its reciprocal to find secant.
We are given that cosine is:
[tex]\cos(\theta)=\frac{4}{5}[/tex]
Then secant must be:
[tex]\sec(\theta)=\frac{5}{4}[/tex]
So:
[tex]\sec^2(\theta)=(\frac{5}{4})^2[/tex]
Square:
[tex]\sec^2(\theta)=\frac{25}{16}[/tex]
And we're done!
What is the volume of a sphere that has
a radius of 11.5 units?
Answer:
6370.63 is the volume
Step-by-step explanation:
V=4
3πr3=4
3·π·11.53≈6370.6263
Please help me I’m rly confused
Joe’s dad works 45 hours per week. He receives a weekly salary of $850, plus a 3.5% commission on sales over $6,500. Given the functions f(x) = 0.035x and g(x) = x - 6,500 and the fact that Joe’s dad sold enough this week to earn commission, which composition function represents Joe’s dad’s commissions ? (Ques 26)
Answer:
f(x) * g(x)
Step-by-step explanation:
Hours worked per week = 45 hours
Weekly salary = $850 plus ;
3.5% commission on sales above $6,500
Given the functions :
f(x) = 0.035x and
g(x) = x - 6,500
and the fact that Joe’s dad sold enough this week to earn commission, which composition function represents Joe’s dad’s commissions ?
Commission = 3.5 / 100 = 0.035
Let his sales for the week = x ; since his sales exceeds 6,500
Hence, sales on which commission is payable is x - 6,500
Hence, commission for the week :
0.035 × (x - 6,500)
f(x) = 0.035 ; g(x) = (x - 6,500)
Composition function:
f(x) * g(x)
HELP ASAPPPPP Becca is using a sheet of scrapbook paper with a width of x inches, and a length of 2x inches. She cut off one inch of both the length and the width. What is the area ( in square inches) of the newly cut paper, in terms of x?
Answer:
52 square in
Step-by-step explanation:
x2 is 25 by 50
The area of the new rectangle in terms of x is 2x² - 3x + 1 sq inches.
What are the area and perimeter of a rectangle?The area of a rectangle is the product of its length and width.
The perimeter of a rectangle is the sum of the lengths of all the sides.
We know the area of a rectangle is (length×width).
Given, A paper with a width of x inches, and a length of 2x inches.
So, The area is (x×2x) sq inches.
= 2x² sq inches.
Now, he cut off one inch of both the length and the width which is,
= (x - 1)(2x - 1) sq inches.
= 2x² - x - 2x + 1 sq inches.
= 2x² - 3x + 1 sq inches.
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