The vector in component form is given by:
V = i - j.
How to find a vector?A vector is given by the terminal point subtracted by the initial point, hence:
(3,-4) - (2, -3) = (3 - 2, -4 - (-3)) = (1, -1)
How a vector is written in component form?A vector (a,b) in component form is:
V = a i + bj.
Hence, for vector (1,-1), we have that:
V = i - j.
More can be learned about vectors at https://brainly.com/question/24606590
#SPJ1
consider the function y=-2-cos(x-pi). What effect does pi have on the basic graph?
Using translation concepts, it is found that pi is the phase shift of the graph, and since it is negative, the graph is shifted right pi units.
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.
In this problem, the change is given as follows:
x -> x - pi
It means that the change is in the domain, in which pi is the phase shift of the graph, and since it is negative, the graph is shifted right pi units.
More can be learned about translation concepts at https://brainly.com/question/4521517
#SPJ1
s−3(s+6)= please help
Answer:
-2s-18 is the simplified answer
Step-by-step explanation:
s-3(s+6)
distributive property
s-3s-18
-2s-18
please help!!!
maths functions
Answer:
Point A: (2, 10)
Point B: (-3, 0)
Point C: (-5, -4)
Point D: (-5, -32)
Step-by-step explanation:
Part (a)Points A and B are the points of intersection between the two graphs.
Therefore, to find the x-values of the points of intersection, substitute one equation into the other and solve for x:
[tex]\implies 2x+6=-2x^2+18[/tex]
[tex]\implies 2x^2+2x-12=0[/tex]
[tex]\implies 2(x^2+x-6)=0[/tex]
[tex]\implies x^2+x-6=0[/tex]
[tex]\implies x^2+3x-2x-6=0[/tex]
[tex]\implies x(x+3)-2(x+3)=0[/tex]
[tex]\implies (x-2)(x+3)=0[/tex]
[tex]\implies x=2, -3[/tex]
From inspection of the graph:
The x-value of point A is positive ⇒ x = 2The x-value of point B is negative ⇒ x = -3To find the y-values, substitute the found x-values into either of the equations:
[tex]\begin{aligned} \textsf{Point A}: \quad 2x+6 & =y\\2(2)+6 & =10\\ \implies & (2, 10)\end{aligned}[/tex]
[tex]\begin{aligned} \textsf{Point B}: \quad -2x^2+18 & =y\\-2(-3)^2+18 & =0\\ \implies & (-3,0)\end{aligned}[/tex]
Therefore, point A is (2, 10) and point B is (-3, 0).
Part (b)If the distance between points C and D is 28 units, the y-value of point D will be 28 less than the y-value of point C. The x-values of the two points are the same.
Therefore:
[tex]\textsf{Equation 1}: \quad y=2x+6[/tex]
[tex]\textsf{Equation 2}: \quad y-28=-2x^2+18[/tex]
As the x-values are the same, substitute the first equation into the second equation and solve for x to find the x-value of points C and D:
[tex]\implies 2x+6-28=-2x^2+18[/tex]
[tex]\implies 2x^2+2x-40=0[/tex]
[tex]\implies 2(x^2+x-20)=0[/tex]
[tex]\implies x^2+x-20=0[/tex]
[tex]\implies x^2+5x-4x-20=0[/tex]
[tex]\implies x(x+5)-4(x+5)=0[/tex]
[tex]\implies (x-4)(x+5)=0[/tex]
[tex]\implies x=4,-5[/tex]
From inspection of the given graph, the x-value of points C and D is negative, therefore x = -5.
To find the y-value of points C and D, substitute the found value of x into the two original equations of the lines:
[tex]\begin{aligned} \textsf{Point C}: \quad 2x+6 & =y\\2(-5)+6 & =-4\\ \implies & (-5,-4)\end{aligned}[/tex]
[tex]\begin{aligned} \textsf{Point D}: \quad -2x^2+18 & = y \\ -2(-5)^2+18 & =-32\\ \implies & (-5, -32)\end{aligned}[/tex]
Therefore, point C is (-5, -4) and point D is (-5, -32).
Answer:
a) A = (2, 10) and B = (-3, 0)
b) C = (-5, -4) and D = (-5, -32)
Explanation:
a) To determine the coordinates of A and B, find the intersection points of the line "y = 2x + 6" and curve "y = -2x² + 18".
Solve the equation's simultaneously:
y = y
⇒ 2x + 6 = -2x² + 18
⇒ 2x² + 2x + 6 - 18 = 0
⇒ 2x² + 2x - 12= 0
⇒ 2x² + 6x - 4x - 12 = 0
⇒ 2x(x + 3) - 4(x + 3) = 0
⇒ (2x - 4)(x + 3) = 0
⇒ 2x - 4 = 0, x + 3 = 0
⇒ x = 2, x = -3
Then find value of y at this x points,
at x = 2, y = 2(2) + 6 = 10
at x = -3, y = 2(-3) + 6 = 0
Intersection points: A(2, 10) and B(-3, 0)
b) Given that CD = 28 units. Also stated parallel to y axis so x coordinates for both will be same but differ in y coordinate.
[tex]2x + 6 = -2x^2 + 18 + 28[/tex]
[tex]-2x^2 + 18 + 28-2x - 6 = 0[/tex]
[tex]-2x^2-2x+40=0[/tex]
[tex]-2x^2-10x+8x+40=0[/tex]
[tex]-2(x+5)+8(x+5)=0[/tex]
[tex](-2x+8)(x+5)=0[/tex]
[tex]x = -5, 4[/tex]
[tex]\leftrightarrow \sf C(-5, y_2), \ D(-5, y_2)[/tex]
Find y value for Point C : 2x + 6 = 2(-5) + 6 = -4
Find y value for Point D : -2x² + 18 = -2(-5)² + 18 = -32
[tex]\sf \rightarrow Point \ C = (-5, -4)\\ \\\rightarrow Point \ D = (-5, -32)[/tex]
a dust mite is 400 pm long under a microscope, it looks 100,000 pm long. what magnification scale was used
Answer: 250:1
Step-by-step explanation:
Divide the measurement of appearance by the actual measurement.
[tex]100,000/400=250[/tex]
The magnification scale is 250:1. The scale is used by a 250X magnifying lense.
Help me asappp w this question
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
The straight angles are angles that form a straight line, and their measure = 180°
In the given figure, the Straight angles is :
GEB[tex] \qquad \large \sf {Conclusion} : [/tex]
Correct choice is D15. Find the young modulus of a brass rod of diameter 25mm and of length 250mm which is
subjected to a tensile load of 50KN when the extension of the rod is equal to 0.3mm.
The Young Modulus of the brass rod ≈ 8.5 × 10¹⁰ Pa or N/m².
How to find the young modulus of a brass rod?
Given: Physical Properties of the Brass rod exist
Length, L = 250 mm
Radius, R = Diameter / 2 = 25/2 mm
Load of 50 kN hanging on the brass rod which results in elongation of the rod to 0.3 mm.
To estimate the Young Modulus of the brass rod we will require Stress on the rod, Ratio of elongation to the total length.
To find the stress on the rod,
Stress(Pressure) = Force / Area
⇒ Stress = 50,000 / (Area of cross-section)
The rod exists shaped like a cylinder then the area of its cross-section will be πR² ( R = radius)
⇒ Stress = 50000 / (π (25/2 mm)² )
The force exists given in newton, then we must convert the unit of the area into m² to obtain the solution in pascal.
1 mm = 0.001 m
⇒ Stress = 50,000 / 22/7 ( 25/2 × 10⁻³ m)²
take, π = 22/7
⇒ Stress = 50000 / ( 22/7 × 625 / 4 × 10⁻⁶ )
⇒ Stress = 50000 / ( 11/14 × 625 × 10⁻6 )
⇒ Stress = 50000 × 14 × 10⁶ / 625×11
⇒ Stress = 80 × 14 × 10⁶ / 11
⇒ Stress = 1120/11 × 10⁶ Pa
Now, that we maintain Stressed, to estimate Strain (ratio of elongation to original length),
⇒ Strain = (Elongation) / (Original length)
⇒ Strain = 0.3 / 250
⇒ Strain = 3 / 2500
Therefore, Young Modulus, Y = Stress / Strain
⇒ Y = 1120/11 × 10⁶ / 3/2500
⇒ Y = 1120/11 × 2500/3 × 10⁶
⇒ Y = 101.8 × 833.3 × 10⁶
⇒ Y = 84829.94 × 10⁶
⇒ Y ≈ 8.5 × 10¹⁰ Pa
Therefore, the Young Modulus of the brass rod ≈ 8.5 × 10¹⁰ Pa or N/m².
To learn more about Young Modulus refer to:
https://brainly.com/question/13257353
#SPJ9
Suppose you know that the distribution of sample proportions of fifth grade students in a large school district who read below grade level in samples of 100 students is normal with a mean of 0.30 and a standard deviation of 0.12. You select a sample of 100 fifth grade students from this district and find that the proportion who read below grade level in the sample is 0.54. This sample proportion lies 2.0 standard deviations above the mean of the sampling distribution. What is the probability that a second sample would be selected with a proportion greater than 0.54 ?
Based on the mean of the sample and the proportion who read below grade level, the probability that a second sample would have a proportion greater than 0.54 is 0.9772.
What is the probability of the second sample being greater than 0.54?The probability that the second sample would be selected with a proportion greater than 0.54 can be found as:
P (x > 0.54) = P ( z > (0.54 - 0.30) / 0.12))
Solving gives:
P (x > 0.54) = P (z > 2)
P (x > 0.54) = 0.9772
In conclusion, the probability that the second sample would be selected with a proportion greater than 0.54 is 0.9772.
Find out more on probability at https://brainly.com/question/25638875
#SPJ1
one of the acute angles of a right triangle is 36° find the other acute angles
Step-by-step explanation:
An acute angle is an angle less or equal to 90 degrees.
let the other angle be x.
x + 36 = 90
grouping like terms
X = 90 - 36
X = 54 degrees
Answer:
54
Step-by-step explanation:
= ∠A + ∠B + ∠C = 180o … [∵sum of the angles of a triangle is 180]
or 180 = 36 + 90 + ∠C
C=54
Jake goes to the grocery store and buys 3 apples, 2 cans of soup, and 1 box of cereal. The apples cost $0.89 each; the soup costs $2.98 per can; and the box of cereal costs $4.99. Write an equation that represents the total cost c of Jake’s purchases.
The equation that represents the total cost, c, of the purchases made by Jake is: c = 3(0.89) + 2(2.98) + 4.99.
How to Write the Equation of Total Cost?To write the equation that represents the total cost of a given scenario like the one given, you can use variables to represent each component that makes up the total cost in the given situation.
Thus, let:
a = apple = 3a
s = can of soup = 2s
b = box of cereal = b
c = total cost
We know that unit price of each items are:
Apple = $0.89
Can of soup = $2.98
Box of cereal = $4.99
Equation would be:
c = 3a + 2s + b
Plug in the values
c = 3(0.89) + 2(2.98) + 4.99
Therefore, the equation that represents the total cost, c, of the purchases made by Jake is: c = 3(0.89) + 2(2.98) + 4.99.
Learn more about equations on:
https://brainly.com/question/13818690
#SPJ1
Evaluate the limits
[tex]x > \ln(x)[/tex] for all [tex]x[/tex], so
[tex]\displaystyle \lim_{x\to\infty} (\ln(x) - x) = - \lim_{x\to\infty} x = \boxed{-\infty}[/tex]
Similarly, [tex]\displaystyle \lim_{x\to\infty} (x-e^x) = - \lim_{x\to\infty} e^x = \boxed{-\infty}[/tex]
We can of course see the limits are identical by replacing [tex]x\mapsto e^x[/tex], so that
[tex]\displaystyle \lim_{x\to\infty} (\ln(x) - x) = \lim_{x\to\infty} (\ln(e^x) - e^x) = \lim_{x\to\infty} (x - e^x)[/tex]
You can also rewrite the limands to accommodate the application of l'Hôpital's rule. For instance,
[tex]\displaystyle \lim_{x\to\infty} (x - e^x) = \ln\left(\exp\left(\lim_{x\to\infty} (x - e^x)\right)\right) = \ln\left(\lim_{x\to\infty} e^{x-e^x}\right) = \ln\left(\lim_{x\to\infty} \frac{e^x}{e^{e^x}}\right)[/tex]
Using the rule, the limit here is
[tex]\displaystyle \lim_{x\to\infty} \frac{(e^x)'}{\left(e^{e^x}\right)'} = \lim_{x\to\infty} \frac{e^x}{e^x e^{e^x}} = \lim_{x\to\infty} \frac1{e^{e^x}} = 0[/tex]
so the overall limit is
[tex]\displaystyle \lim_{x\to\infty} (x - e^x) = \ln\left(\lim_{x\to\infty} \frac{e^x}{e^{e^x}}\right) = \ln(0) = \lim_{x\to0^+} \ln(x) = -\infty[/tex]
What is the domain of the relation graphed below?
The domain of the relation shown in the graph is -4 <= x <= 4
How to determine the domain of the relation shown in the graph?The relation on the graph is an ellipse function.
The domain is the set of x values the function can take
From the graph, we have the following x values
This means that the domain of x in the graph is -4 <= x <= 4
Hence, the domain of the relation shown in the graph is -4 <= x <= 4
Read more about domain at:
https://brainly.com/question/1770447
#SPJ1
-y(-6y-3) I need to combine like terms and simplify. I got 6y^2 + 3y by distributing. It's wrong and in the explanation it says use the distributive property to remove the parentheses and it gave -6y -3 -y as the first step answer. How the heck did they remove the parentheses or use the distributive property correctly?
if any number or variable is outside a parenthesis it will go to all the number rin the parenthesis.
in this case -y is outside the parenthesis so it will got to both the sides :
-y(-6y-3) = -6y x y - 3 x y
= -6y² -3y//
Two discs are randomly taken without replacement from a bag containing 3 red discs and 2 blue discs. What is the probability of taking 2 red discs ?
Answer:
3/10
Step-by-step explanation:
There are totally 5 disks. On the first pick, there are 3 red discs. So
P(red disc on first pick ) = 3/5
Assuming that a red disc has been picked the first time, there will be 2 red discs and 2 blue discs so
(P red disc on second pick given red disc on first pick) = 2/4
P(red disc on both picks) = 3/5 x 2/4 = 3/10
the figure at the right shows the dimensions of the garden in Marissa's back yard. What is the area of the garden?
The area of the garden is 74 square feet
How to determine the area of the garden?The complete question is added as an attachment
From the attached figure, we have the following shapes and dimensions:
Rectangle: 10 by 5 feetTrapezoid: Bases = 10 and 6; Height = 3The rectangular area is
A1 = 10 * 5
Evaluate
A1 = 50
The area of the trapezoid is
A2 = 0.5 * Sum of parallel bases * height
This gives
A2 = 0.5 * (10 + 6) * 3
Evaluate
A2 = 24
The total area is
Total = A1 + A2
This gives
Total = 50 + 24
Evaluate
Total = 74
Hence, the area of the garden is 74 square feet
Read more about areas at:
https://brainly.com/question/24487155
#SPJ1
Instructions: Find the missing side of the triangle.
X= Answer
Answer: 15
Step-by-step explanation: Using the pythagorean theorem A^2 + B^2 = C^2 where C is the hypotenuse of the triangle and any A and B are any sides of the triangle that are not the hypotenuse; we can rewrite this as A^2 = C^2 - B^2 to find out the missing side. I picked A to be the unknown side but you could also choose B. We know the hypotenuse is 39, and side B is 36. 39^2 - 36^2, or 1521 - 1296, is 225.
Now we have A^2 = 225. To get rid of the square, we have to take the square root of both sides. The A^2 cancels to be A, and sqrt(225) is 15. Thus, side A, or the missing side is 15.
Hope this helped!
PLEASE HELP I will give 50 points! PLEASE ANSWER CORRECTLY
In a school, 10% of the students have green eyes. Find the experimental probability that in a group of 4 students, at least one of them has green eyes. The problem has been simulated by generating random numbers. The digits 0-9 were used. Let the number "9" represent the 10% of students with green eyes. A sample of 20 random numbers is shown. 7918 7910 2546 1390 6075 2386 0793 7359 3048 1230 2816 6147 5978 5621 9732 9436 3806 5971 6173 1430 Experimental Probability = [?]% =
Answer: 45%
Step-by-step explanation:
Given:
A sample of 20 random numbers.The number "9" represents the 10% of students having green eyes.Find:
The experimental probability that in a group of 4 students, at least one of them has green eyes.The number of groups which contains 9 is 9 and the total number of groups are 20.
So,
P = 9 / 20 = 0.45 = 45%
Therefore, the experimental probability is 45%
Which function represents g(x), a reflection of f(x) = 1/2(3)^x across the y-axis?
The equation for the reflected function is:
g(x) = (1/2)*(3)^(-x)
How to get the reflected function?
Here we know that function g(x) is a reflection of f(x) across the y-axis.
Remember that the reflection is written as:
g(x) = f(-x)
Here we know that:
f(x) = (1/2)*(3)^x
Then, replacing that in the equation for g(x), we get:
g(x) = f(-x) = (1/2)*(3)^(-x)
g(x) = (1/2)*(3)^(-x)
If you want to learn more about reflections:
https://brainly.com/question/4289712
#SPJ1
(4x³-12x +11) + (2x - 2)
Identify the most reasonable unit to measure the time spent at school on an average school day. Seconds, minutes, or hours?
The most reasonable unit to measure the time spent at school on an average school day is in hours.
How to find the most reasonable unit for a measure?
In anything we are going to measure, we have to pay special attention to the unit. When doing this, we want to keep the absolute value of the measure, that is, the number small, hence the do this conversion of units may be used.
For example, it is easier and more practical to say one day instead of 87,840 seconds. The same logic is applied to this problem, in which the most reasonable unit to measure the time spent at school on an average school day is in hours, as it keeps the numerical measure smaller than it would be in minutes or seconds.
More can be learned about the most reasonable unit of a measure at https://brainly.com/question/21896360
#SPJ1
The volume of a rectangular prism is given by the following function:
V(x) = 2x³ + x² - 16 - 15
The length of the rectangular prism is (x − 3) and the width is (2x + 5). What is the height?
[tex]v(x) = wlh \\ 2x {}^{3} + {x}^{2} - 16 x- 15 =(2x + 5)(x - 3)h \\ h = \frac{2x {}^{3} + x {}^{2} - 16x - 15 }{2x {}^{2} - x - 15 } \\ using \: long \: divison \\ h = x + 1[/tex]
HOW MANY DIFFERENT ARRANGEMENTS CAN BE MADE WITH THE NUMBERS
28535852
Using the arrangements formula, it is found that 1680 arrangements can be made with these numbers.
What is the arrangements formula?The number of possible arrangements of n elements is given by the factorial of n, that is:
[tex]A_n = n![/tex]
When there are repeated elements, repeating [tex]n_1, n_2, \cdots, n_n[/tex] times, the number of arrangements is given by:
[tex]A_n^{n_1, n_2, \cdots, n_n} = \frac{n!}{n_1!n_2! \cdots n_n!}[/tex]
For the number 28535852, we have that:
There are 8 numbers.5 repeats 3 times.2 repeats two times.8 repeats two times.Hence the number of arrangements is:
[tex]A_8^{3,2,2} = \frac{8!}{3!2!2!} = 1680[/tex]
More can be learned about the arrangements formula at https://brainly.com/question/24648661
#SPJ1
Find a potential function for the vector field
(a) We want to find a scalar function [tex]f(x,y,z)[/tex] such that [tex]\mathbf F = \nabla f[/tex]. This means
[tex]\dfrac{\partial f}{\partial x} = 2xy + 24[/tex]
[tex]\dfrac{\partial f}{\partial y} = x^2 + 16[/tex]
Looking at the first equation, integrating both sides with respect to [tex]x[/tex] gives
[tex]f(x,y) = x^2y + 24x + g(y)[/tex]
Differentiating both sides of this with respect to [tex]y[/tex] gives
[tex]\dfrac{\partial f}{\partial y} = x^2 + 16 = x^2 + \dfrac{dg}{dy} \implies \dfrac{dg}{dy} = 16 \implies g(y) = 16y + C[/tex]
Then the potential function is
[tex]f(x,y) = \boxed{x^2y + 24x + 16y + C}[/tex]
(b) By the FTCoLI, we have
[tex]\displaystyle \int_{(1,1)}^{(-1,2)} \mathbf F \cdot d\mathbf r = f(-1,2) - f(1,1) = 10-41 = \boxed{-31}[/tex]
[tex]\displaystyle \int_{(-1,2)}^{(0,4)} \mathbf F \cdot d\mathbf r = f(0,4) - f(-1,2) = 64 - 41 = \boxed{23}[/tex]
[tex]\displaystyle \int_{(0,4)}^{(2,3)} \mathbf F \cdot d\mathbf r = f(2,3) - f(0,4) = 108 - 64 = \boxed{44}[/tex]
5. Write the following inequality in slope-intercept form. −8x + 4y ≥ −52 y ≤ 2x − 13 y ≥ 2x − 13 y ≤ 2x + 13 y ≥ 2x + 13
Answer:35443
Step-by-step explanation:
Does this set of ordered pairs represent a function? {(–2, 3), (–1, 3), (0, 2), (1, 4), (5, 5)} A. The relation is a function. Each input value is paired with more than one output value. B. The relation is a function. Each input value is paired with one output value. C. The relation is not a function. Each input value is paired with only one output value. D. The relation is not a function. Each input value is paired with more than one output value.
The correct option regarding whether the relation is a function is:
B. The relation is a function. Each input value is paired with one output value.
When does a relation represent a function?A relation represent a function if each value of the input is paired with one value of the output.
In this problem, when the input - output mappings are given by:
{(–2, 3), (–1, 3), (0, 2), (1, 4), (5, 5)}.
Which means that yes, each input value is paired with one output value, hence the relation is a function and option B is correct.
A similar problem, in which we verify if a relation is a function, is given at https://brainly.com/question/12463448
#SPJ1
find the fourth roots of i
Answer:
1∠22.5°, 1∠112.5°, 1∠202.5°, 1∠292.5°
Step-by-step explanation:
A root of a complex number can be found using Euler's identity.
ApplicationFor some z = a·e^(ix), the n-th root is ...
z = (a^(1/n))·e^(i(x/n))
Here, we have z = i, so a = 1 and z = π/2 +2kπ.
Using r∠θ notation, this is ...
i = 1∠(90° +k·360°)
and
i^(1/4) = (1^(1/4))∠((90° +k·360°)/4)
i^(1/4) = 1∠(22.5° +k·90°)
For k = 0 to 3, we have ...
for k = 0, first root = 1∠22.5°
for k = 1, second root = 1∠112.5°
for k = 2, third root = 1∠202.5°
for k = 3, fourth root = 1∠292.5°
A company's sales increased 20% this year, to $6740. What were their sales last year?
The sales last year of the company is 5617
How to determine the sales last year?The given parameters are:
Company sales = 6740
Percentage increase = 20%
The sales of the company last year (x) to date is calculated as:
Company sales = (1 + Proportion) * Last year sales
Substitute the known values in the above equation
6740 = (1 + 20%) * x
Evaluate the sum
6740 = 1.2 * x
Divide both sides by 1.2
x = 5617
Hence, the sales last year is 5617
Read more about percentages at:
https://brainly.com/question/843074
#SPJ1
Consider the ordinary differential equation (answer questions in picture)
a. Given the 2nd order ODE
[tex]y''(x) = 4y(x) + 4[/tex]
if we substitute [tex]z(x)=y'(x)+2y(x)[/tex] and its derivative, [tex]z'(x)=y''(x)+2y'(x)[/tex], we can eliminate [tex]y(x)[/tex] and [tex]y''(x)[/tex] to end up with the ODE
[tex]z'(x) - 2y'(x) = 4\left(\dfrac{z(x)-y'(x)}2\right) + 4[/tex]
[tex]z'(x) - 2y'(x) = 2z(x) - 2y'(x) + 4[/tex]
[tex]\boxed{z'(x) = 2z(x) + 4}[/tex]
and since [tex]y(0)=y'(0)=1[/tex], it follows that [tex]z(0)=y'(0)+2y(0)=3[/tex].
b. I'll solve with an integrating factor.
[tex]z'(x) = 2z(x) + 4[/tex]
[tex]z'(x) - 2z(x) = 4[/tex]
[tex]e^{-2x} z'(x) - 2 e^{-2x} z(x) = 4e^{-2x}[/tex]
[tex]\left(e^{-2x} z(x)\right)' = 4e^{-2x}[/tex]
[tex]e^{-2x} z(x) = -2e^{-2x} + C[/tex]
[tex]z(x) = -2 + Ce^{2x}[/tex]
Since [tex]z(0)=3[/tex], we find
[tex]3 = -2 + Ce^0 \implies C=5[/tex]
so the particular solution for [tex]z(x)[/tex] is
[tex]\boxed{z(x) = 5e^{-2x} - 2}[/tex]
c. Knowing [tex]z(x)[/tex], we recover a 1st order ODE for [tex]y(x)[/tex],
[tex]z(x) = y'(x) + 2y(x) \implies y'(x) + 2y(x) = 5e^{-2x} - 2[/tex]
Using an integrating factor again, we have
[tex]e^{2x} y'(x) + 2e^{2x} y(x) = 5 - 2e^{2x}[/tex]
[tex]\left(e^{2x} y(x)\right)' = 5 - 2e^{2x}[/tex]
[tex]e^{2x} y(x) = 5x - e^{2x} + C[/tex]
[tex]y(x) = 5xe^{-2x} - 1 + Ce^{-2x}[/tex]
Since [tex]y(0)=1[/tex], we find
[tex]1 = 0 - 1 + Ce^0 \implies C=2[/tex]
so that
[tex]\boxed{y(x) = (5x+2)e^{-2x} - 1}[/tex]
6.a) The differential equation for z(x) is z'(x) = 2z(x) + 4, z(0) = 3.
6.b) The value of z(x) is [tex]z(x) = 5e^{2x} - 2[/tex].
6.c) The value of y(x) is [tex]y(x) = \frac{5e^{2x}}{4} - \frac{1}{4e^{2x}} -1[/tex].
The given ordinary differential equation is y''(x) = 4y(x) + 4, y(0) = y'(0) = 1 ... (d).
We are also given a substitution function, z(x) = y'(x) + 2y(x) ... (z).
Putting x = 0, we get:
z(0) = y'(0) + 2y(0),
or, z(0) = 1 + 2*1 = 3.
Rearranging (z), we can write it as:
z(x) = y'(x) + 2y(x),
or, y'(x) = z(x) - 2y(x) ... (i).
Differentiating (z) with respect to x, we get:
z'(x) = y''(x) + 2y'(x),
or, y''(x) = z'(x) - 2y'(x) ... (ii).
Substituting the value of y''(x) from (ii) in (d) we get:
y''(x) = 4y(x) + 4,
or, z'(x) - 2y'(x) = 4y(x) + 4.
Substituting the value of y'(x) from (i) we get:
z'(x) - 2y'(x) = 4y(x) + 4,
or, z'(x) - 2(z(x) - 2y(x)) = 4y(x) + 4,
or, z'(x) - 2z(x) + 4y(x) = 4y(x) + 4,
or, z'(x) = 2z(x) + 4y(x) - 4y(x) + 4,
or, z'(x) = 2z(x) + 4.
The initial value of z(0) was calculated to be 3.
6.a) The differential equation for z(x) is z'(x) = 2z(x) + 4, z(0) = 3.
Transforming z(x) = dz/dx, and z = z(x), we get:
dz/dx = 2z + 4,
or, dz/(2z + 4) = dx.
Integrating both sides, we get:
∫dz/(2z + 4) = ∫dx,
or, {ln (z + 2)}/2 = x + C,
or, [tex]\sqrt{z+2} = e^{x + C}[/tex],
or, [tex]z =Ce^{2x}-2[/tex] ... (iii).
Substituting z = 3, and x = 0, we get:
[tex]3 = Ce^{2*0} - 2\\\Rightarrow C - 2 = 3\\\Rightarrow C = 5.[/tex]
Substituting C = 5, in (iii), we get:
[tex]z = 5e^{2x} - 2[/tex].
6.b) The value of z(x) is [tex]z(x) = 5e^{2x} - 2[/tex].
Substituting the value of z(x) in (z), we get:
z(x) = y'(x) + 2y(x),
or, 5e²ˣ - 2 = y'(x) + 2y(x),
which gives us:
[tex]y(x) = \frac{5e^{2x}}{4} - \frac{1}{4e^{2x}} -1[/tex] for the initial condition y(x) = 0.
6.c) The value of y(x) is [tex]y(x) = \frac{5e^{2x}}{4} - \frac{1}{4e^{2x}} -1[/tex].
Learn more about differential equations at
https://brainly.com/question/18760518
#SPJ1
The width of a rectangle measures
(7h+3) centimeters, and its length measures
(8h−4) centimeters. Which expression represents the perimeter, in centimeters, of the rectangle?
The expression that represents the perimeter of the rectangle is 30h - 2
How to determine the perimeter expression?The dimensions of the rectangle are given as:
Length = 7h + 3
Width = 8h - 4
The perimeter of the rectangle is given as:
P = 2 * (Length + Width)
Substitute the known values in the above equation
P =2 * (7h + 3 + 8h - 4)
Evaluate the like terms
P = 2 * (15h - 1)
Evaluate the product
P = 30h - 2
Hence, the expression that represents the perimeter of the rectangle is 30h - 2
Read more about perimeter at:
https://brainly.com/question/24571594
#SPJ1
TIME REMAINING
44:36
Which point is an x-intercept of the quadratic function f(x) = (x + 6)(x – 3)?
(0,6)
(0,–6)
(6,0)
(–6,0)
Answer:
d. (-6, 0)
Step-by-step explanation:
no explanation bc u need this quick
Solve the following system of equations. 4x + 3y = -5
- 3x + 7y=13
Answer: 13
Step-by-step explanation:
4x+3y=-5 solve x :
4x = -3y + -5 | -3y
1x = -0.75y + -1.25 | : 4
-3x + 7y = 13 solve x :
-3x + 7y = 13 | -7y
-3x = -7y = 13 | : (-3)
Equalization Method Solution: -0.75y+-1.25=2.333y+-4.333
-0, 75y - 1,25 = 2,333y - 4, 333 solve y:
-0, 75y - 1,25 = 2,333y -4, 333 | -2,333y
-3, 083y - 1,25 = -4,333 | + 1, 25
-3. 083y = -3,088 | : (-3, 083)
y = 1
Plug y = 1 into the equation 4x + 3y = -5 :
4x + 3 · 1 | Multiply 3 with 1
4x + 3 = -5 | -3
4x = -8 | : 4
x = -2
So the solution is:
y = 1, x = -2
ヘ( ^o^)ノ\(^_^ )If you want to learn more about mathematics, I share this link to complement your learning:
https://brainly.com/question/20333461