The statement that the function is decreasing in the given interval is incorrect.
We have,
To justify why the function with f' = 2cos(x)sin(x) is decreasing from π/2 to 3π/2, we need to analyze the sign of the derivative in that interval.
The derivative f' represents the instantaneous rate of change of the function f(x) at any given point.
When f' is positive, it indicates that the function is increasing, and when f' is negative, it indicates that the function is decreasing.
In this case,
f' = 2cos(x)sin(x).
To determine the sign of f' in the interval [π/2, 3π/2], we can examine the behavior of cos(x) and sin(x) in that range.
In the interval [π/2, 3π/2], cos(x) is negative, and sin(x) is negative as well. When both factors in the expression 2cos(x)sin(x) are negative, their product is positive.
Since f' = 2cos(x)sin(x) is positive in the interval [π/2, 3π/2], it indicates that the function f(x) is increasing in that interval.
Therefore, the function is not decreasing from π/2 to 3π/2.
Thus,
The statement that the function is decreasing in the given interval is incorrect.
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Use the diagram as a reference to answer the question below.
If arctan 3/3 = B find the measures of angles a and b and side lengths h, k, and r.
The measures of angles α and β and the side lengths h, k, and r, found using trigonometric ratios and the Pythagorean Theorem are;
α = 60°, β = 30°, h = √3, k = 3, r = 2·√3
Pythagorean Theorem states that the square of the length of the hypotenuse side is equivalent to the sum of the squares of the lengths of the legs of a right triangle.
The triangle formed by the sides h, k, and r, is a right triangle
The hypotenuse side of the right triangle = r
The legs of the right triangle are h and k
The measure of arctan(√(3)/3) = β...(1)
The tangent of angle β = h/k
Therefore; arctan(h/k) = β...(2)
Comparison the equations (1) and (2), we get;
h/k = (√3)/3
When h = √3, k = 3
Arctan(√3)/3 = 30°
Therefore;
β = 30°Angle α and angle β are complementary angles, therefore;
α + β = 90° (definition of complementary angles)
α = 90° - β
α = 90° - 30° = 60°
α = 60°Pythagorean Theorem indicates that we get;
r² = h² + k²
Therefore; r² = (√3)² + 3² = 3 + 9 = 12
r = √(12) = 2·√3
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The graph shows the cube root parent function
Which statement best describes the function?
(image attached)
Answer:
Step-by-step explanation:
As you move from left to right along the graph, each y-value gets larger and larger. Therefore the y value is always increasing. The correct answer is D.
An object of mass m=2 kg with momentum <2,2,6> kg m/s, is struck with a stick. Right after the collision it now has a momentum of <2,-2,1> kg m/s.
1) Calculate the change in momentum expierence by the object
2) Let the system be just the mass m, it should be obvious this system is not isolated. Give a reason why we know this.
3) IF the collision only took 5 millisecond, what was the net force expierenced by the object?
4)What was the acceleration expierenced by the object due to the collison
The change in momentum experienced by the object is <0,-4,-5> kg m/s.
What was the acceleration expierenced by the object due to the collison?We know the system is not isolated because the momentum of the object changed, meaning that something outside of the system must have acted on the object to cause the momentum change.The net force experienced by the object due to the collision was F = Δp/Δt = <0,-4,-5> kg m/s/0.005 s = <0,-800,-10000> N.The acceleration experienced by the object due to the collision was a = F/m = <0,-800,-10000> N/2 kg = <0,-400,-5000> m/s2.The change in momentum experienced by the object is <0,-4,-5> kg m/s. This can be found by subtracting the initial momentum of <2,2,6> kg m/s from the final momentum of <2,-2,1> kg m/s.We know that this system is not isolated because the momentum of the object has changed, which implies that a net force has been applied to the object.The net force experienced by the object in the collision can be calculated by dividing the change in momentum by the time taken to experience this change. Therefore, the net force experienced by the object is <0,-800,-1000> N.The acceleration experienced by the object due to the collision can be calculated by dividing the net force experienced by the object by the mass of the object. Therefore, the acceleration experienced by the object is <0,-400,-500> m/s2.To learn more about impulse acceleration refer to:
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Exercise 1.1.105: Solve y' = y", y(0) = 1, where n is a positive integ consider different cases.
The solution for different cases of positive integers by using the initial condition y(0) = 1.
A differential equation is an equation that relates an unknown function y to its derivatives, represented by symbols such as y' or y". On the other hand, a positive integer is a whole number greater than zero. For example, 1, 2, 3, and so on are positive integers.
Here we need to to solve a differential equation where y' is equal to y". In this exercise, we will consider different cases where n is a positive integer.
In the given equation,
=> y' = y", y(0) = 1,
we are trying to find the value of y at any given time a. To solve this equation, we need to use the initial condition y(0) = 1.
By using this condition, we can find the solution for different cases of positive integers.
For the first case, if n = 1, then y(t) = eᵃ.
For the second case, if n = 2, then y(t) = (1 + a²)/2.
Then the solution for different cases of positive integers by using the initial condition y(0) = 1.
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suppose you deposit $1,000 into an account which pays 4 nnual interest, compounded quarterly. approximately how long will it take for the amount of money in the account to double?
The number of years will be 5 years, will it take for the amount of money in the account to double.
Here, we have,
The number of years will be computed using the excel formula as:
=N per(rate,pmt,pv,fv,type)
where
N per is number of years
rate is 4%
But is compounded quarterly,
So,
rate = 4% × 4
rate = 16%
pmt is $0
pv is Present value which is -$1,000
fv is future value which is $2,000 (as it will get doubled)
Putting the values above:
=N per(4%× 4,0,-1000,2000,0)
= 4.67 years or 5 years
Therefore, the amount will get double in 5 years will it take for the amount of money in the account to double.
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Give a number you can cube to give an answer bigger than 400 but smaller than 600
Answer:
8
Step-by-step explanation:
7 cubed would be 343 (smaller than 400)
8 cubed is 512 (fits in the range!)
9 cubed would be 729 (bigger than 600)
Hope this helps :)
(T<3, –2> ◦ D5)(ΔTUV) for T(–1, –1), U(–1, 2), V(2, 1)?
Points T''(x, y) = (10, - 15), U''(x, y) = (10, 0) and V''(x, y) = (25, - 5) are the images of vertices T(x, y) = (- 1, - 1), U(x, y) = (- 1, 2) and V(x, y) = (2, 1).
How to apply rigid transformations
In this problem we need to make use of rigid transformations on a triangle, rigid transformations are transformations applied on geometric locus such that Euclidean distance is conserved in every site of the locus. We need to make use of the following two operations:
Translation
P'(x, y) = P(x, y) + T(x, y)
Dilation
P'(x, y) = k · P(x, y)
Where:
P(x, y) - Original pointP'(x, y) - Resulting pointT(x, y) - Translation vectork - Dilation factorFirst, write the three vertices of the triangle:
T(x, y) = (- 1, - 1), U(x, y) = (- 1, 2), V(x, y) = (2, 1)
Second, apply translation on every vertex:
T'(x, y) = (- 1, -1) + (3, - 2)
T'(x, y) = (2, - 3)
U'(x, y) = (- 1, 2) + (3, - 2)
U'(x, y) = (2, 0)
V'(x, y) = (2, 1) + (3, - 2)
V'(x, y) = (5, - 1)
Third, apply dilation on every image:
T''(x, y) = 5 · (2, - 3)
T''(x, y) = (10, - 15)
U''(x, y) = 5 · (2, 0)
U''(x, y) = (10, 0)
V''(x, y) = 5 · (5, - 1)
V''(x, y) = (25, - 5)
Vertices T(x, y) = (- 1, - 1), U(x, y) = (- 1, 2) and V(x, y) = (2, 1) have the following images T''(x, y) = (10, - 15), U''(x, y) = (10, 0) and V''(x, y) = (25, - 5), respectively.
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If the median of the given data is 19, find the value of P. Ansc 6-12 12-18 18-24 24-30 30-36 36-42 P 4 3 3 Age in year Number of students 4 10 n
Answer:
Step-by-step explanation:
To find the value of P, we need to use the information that the median of the data is 19. The median is the middle value in a set of data when it is arranged in numerical order. The first step is to arrange the data in numerical order which is 6-12 12-18 18-24 24-30 30-36 36-42.
We know that n = 4+3+3+4+10+P = 24+P, where P is the number of students in the age group 36-42.
Since the median is 19, half of the total number of students (n) will have ages less than 19. Therefore, we need to find the number of students whose ages are less than 19. Since 4+3+3 = 10, we know that 10 students are less than 19 years old.
To find P, we need to subtract the number of students whose ages are less than 19 from n, which is 24+P.
n = 24+P
10 = 24+P
P = -14
The value of P cannot be negative, so the given data is not correct. There is missing information that is needed to solve this problem.
Find
10. Triangle ABC and Triangle DEC are similar. Solve for the missing variable.
A
11
X D
B
E
15
The value of ED = 150 m if Triangle ABC and Triangle DEC are similar.
What is triangle ?
Triangle can be defined in which it consists three sides, three angles and sum of three angles is always 180 degrees.
Given ,
Triangle ABC and Triangle DEC are similar.
Let AB = 1.25 m
BC = 1.5 m
BD = 181.5 m
Then,
CD = BD - BC
= 181.5 - 1.5
CD = 180 m
So,
we know that ,
AB / ED = BC / CD
1.25 / ED = 1.5 / 180
1.5 ED = 180 * 1.25
ED = 180 * 1.25 / 1.5
ED = 150 m .
Therefore, The value of ED = 150 m if Triangle ABC and Triangle DEC are similar.
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If u put 3500 in a bank account, with 2% interet for 5 year, how much money will have in the bank account
In 5 years after depositing 3500, the bank account will have 3850
What is simple interest?It is the operation in which we calculate the profit produced by a capital loaned at a given percentage.
The formula for simple interest and procedure we will use to solve this exercise is:
S.I.= (P*R*T)/100
Where:
P = principalR = rate of interest in % per annumT = timeInformation about the problem:
P = 3500R = 2%T = 5 yearsTotal amount = ?Applying the simple interest formula, we get:
S.I.= (P*R*T)/100
S.I.= (3500*2*5)/100
S.I.= 350
Calculating the total amount that would be in my savings account, we get:
Total amount = P + S.I.
Total amount = 3500 + 350
Total amount = 3850
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Given the first four terms of a linear sequence : 3,x,y,30 a) Calculate the value of x and y
Answer:
Step-by-step explanation:
It should be noted that based on the information given, the values of x and y in the linear pattern will be 12 and 21.
How to illustrate the information?
It should be noted that since this is a linear pattern, it should be noted that it illustrates that the values in between them are the same.
It should be noted that the numbers that fits into the scenario are 3, 12, 21 and 30. There's a difference of 9 between them.
Therefore, it should be noted that based on the information given, the values of x and y in the linear pattern will be 12 and 21.
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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.
John and his cousin Abby are picking apples in their grandparents' orchard. John has filled 13 baskets with apples and is filling them at a rate of 3 baskets per hour. Abby has 7 full baskets and will continue picking at 6 baskets per hour. Once the cousins get to the point where they have filled the same number of baskets, they will carry them to the barn and then eat lunch. How long will that take? How much fruit will they have picked by then?
Each will have picked 19, and 19 baskets.
What is the system of equations?One or many equations having the same number of unknowns that can be solved simultaneously called as simultaneous equation. And simultaneous equation is the system of equation.
Given:
John and his cousin Abby are picking apples in their grandparents' orchard.
John has filled 13 baskets with apples and is filling them at a rate of 3 baskets per hour.
Abby has 7 full baskets and will continue picking at 6 baskets per hour.
John is at 13+3x, starting now at t=0
Abby is at 7+6x.
Once the cousins get to the point where they have filled the same number of baskets, they will carry them to the barn and then eat lunch.
Set them equal
13+3x = 7+6x
x=2 hours from now.
Each will have picked 13+3(2)=19, and 7+6(2)= 19 baskets.
Therefore, all the required values are given above.
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Write a system of linear equations for the graph below.
Y=
Y=
A system of linear equations for the graph below is y = -3x + 3 Y=1/3x-7
What is slope of a line?Generally, the slope of a line gives the measure of its steepness and direction. The slope of a straight line between two points says (x1,y1) and (x2,y2) can be easily determined by finding the difference between the coordinates of the points. The slope is usually represented by the letter 'm'.
Whenever the equation of a line is written in the form y = mx + b, it is called the slope-intercept form of the equation. The m is the slope of the line. And b is the b in the point that is the y-intercept (0, b). For example,
for the equation
y = 3x – 7,
the slope is 3,
and
the y-intercept is (0, −7).
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293 added to the product of 312 and q is the same as 306
The numerical form of the statement is 293 + 312q = 306 and the value of q is 1/24.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
Given, A statement 293 is added to the product of 312 and q is the same
as 306.
Now, The numerical form of the statement is,
293 + 312q = 306.
312q = 306 - 293.
312q = 13.
q = 13/312.
q = 1/24.
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the null and alternative hypotheses for a hypothesis test of the difference in two population means are: null hypothesis:mu 1 equals mu 2 alternative hypothesis:mu 1 does not equal mu 2 notice that the alternative hypothesis is a two-tailed test. suppose ttest ind method from scipy module is used to perform the test and the output is (-1.99, 0.0512). what is the p-value for this hypothesis test? select one.
As the alternate hypothesis is a two-tailed test the p-value for this hypothesis test is 0.0512.
Here the null hypothesis, H₀, is μ₁ =μ₂
Alternate hypothesis, H₁ , is μ₁ ≠ μ₂
The statistical test is said to be a two-tailed test. So the critical area of of distributions are on both sides of the range.
Here t test-ind method for spicy module is used.
The output is given as (-1.99 , 0.0512)
In such an output the first value is the test statistic value and second value is the p-value.
So the test statistic value = -1.99
p-value = 0.0512.
So the p-value is 0.0512.
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Which of the following is a geometric sequence?
Answer:
(A). 5,-25,125,-625
Are the triangles similar?
The two given triangles are not similar. The correct option is D.
What is the similarity?If two objects are having the same shape then they will be termed as similar. So in mathematics, if two figures have the same shapes, lines or angles then they are called similar.
Given that there are two triangles one triangle has sides 20 and 24. The second triangle has sides 36 and 42.
If the ratio of the two lines should be equal to the other two lines then the triangle will be similar.
( 20 / 36) = ( 24 / 42 )
4 / 9 = 6 / 7
Hence, the two triangles will not be similar.
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A gas company's delivery truck has a cylindrical tank that is 14 feet in diameter and 40 feet long.
Use the space in your supplement to draw the tank, if that would help. Use 3.14 as an approximation of
π
. Round to the nearest tenth if needed.
How much gas can fit in the tank?
The required volume of the gas that can fit in the cylindrical tank is given as 6154.4 cubic feet.
What is volume?Volume is defined as the mass of the object per unit density while for geometry it is calculated as profile area multiplied by the length at which that profile is extruded.
Here,
Radius = 14/2 = 7 feet
The volume of the cylinder is given as,
= πr²h
= 3.14 × 7 ² × 40
= 6154.4 cubic feet
Thus, the required volume of the gas that can fit in the cylindrical tank is given as 6154.4 cubic feet.
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Prove that
secθ+1 / tanθ = tanθ / secθ−1
To prove that secθ + 1 / tanθ = tanθ / secθ−1, we can use trigonometric identities.
First, we will define secθ as 1/cosθ and tanθ as sinθ/cosθ.
Now, we can substitute these definitions into the equation to get:
secθ + 1 / tanθ = (1/cosθ) + 1 / (sinθ/cosθ)
And, tanθ / secθ−1 = (sinθ/cosθ) / (1/cosθ) - 1
Expanding both sides using basic algebraic manipulations, we get:
(1/cosθ) + 1 / (sinθ/cosθ) = (1/cosθ) * ((sinθ/cosθ) + cosθ/cosθ)
= (sinθ/cosθ) / (1/cosθ) - 1
= (sinθ/cosθ) * (cosθ/1) - 1
= (sinθ) / 1 - 1
= sinθ.
Therefore, secθ + 1 / tanθ = tanθ / secθ−1 has been proven.
2.A container holds marble . A marble will randomly selected from the bag.
-!7 red marbles
-8 blue
-16 green
-9 black
What is the probability in decimal form that the selected marble will be green or black.
Answer:
To find the probability of selecting a green or black marble, we need to add the number of green marbles (16) to the number of black marbles (9) and divide by the total number of marbles (7 + 8 + 16 + 9 = 40). So the probability is:
(16 + 9) / 40 = 0.625 or 62.5% in decimal form.
At the county fair, Amy played a game to win a big stuffed bear. The game she
played used 4 cubes in a box: 1 red cube, 1 blue cube, 1 yellow cube, and 1 green
cube. Without peeking, Amy reached into the box and drew 1 cube. She put the
cube back and drew again. If the cubes on both draws were the same color, Amy
won a bear! How many different combinations could Amy draw? How many of the
would be winning combinations?
In mathematics, a combination is a decision taken from a set of independent elements when the order of the choices is unimportant.
What is meant by combination?A combination is a choice made from a set of separate elements in mathematics when the order of the choices is irrelevant.
Combinations are mathematical operations that count the variety of configurations that can be made from a set of objects, where the order of the selection is irrelevant. You can choose any combination of the things in any order.
Combination and permutation are two alternative strategies to divide up a collection of elements into subsets in mathematics. The components of the subset may be arranged in any order when combined. The subset's components are listed in a permutation in a certain order.
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Total number of combinations formed equals 10, She will have a total of 4 winning combinations.
What is meant by combination?In mathematics, a combination is a decision taken from a set of independent elements when the order of the choices is unimportant.
Combinations are mathematical procedures that count the number of possible configurations from a set of things, where the order of the selection of the elements is not important. You can select any set of items in any arrangement.
In mathematics, there are two additional methods for dividing a set of components into subsets: combination and permutation. When merged, the subset's elements can be put in any order. The elements of the subset are listed in a permutation in a certain sequence.
Given,
There are total 4 number of cubes in the box,
She will win if she chooses cube of same color,
Total number of combinations = [tex]\frac{n!}{r!(n-r)!}[/tex]
where n = total number of items [tex]\frac{4!}{2!(2!)}[/tex],
r = 2
Putting the values in the above formula,
⇒ combinations = [tex]\frac{4 \times 3 \times 2 \times 1}{2 \times 2}[/tex]
⇒ 6
With repetition,
she will have 4 more combinations of same color,
Total combinations = 6 + 4
Total winning combinations will be = Red Red, Blue Blue, Yellow Yellow, Green Green = 4
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A bag factory is to produce 2 000 bags in four different colors-red, white, yellow,
and black in the ratio 3:4:5:8. How many bags of each color should be made?
To find out how many bags of each color should be made, we need to first find the total ratio of bags, which is 3+4+5+8=20.
Next, we find the number of bags in each color by dividing the color's ratio by the total ratio and multiplying by the total number of bags to be made (2000):
Red bags: (3/20) * 2000 = 300 bags
White bags: (4/20) * 2000 = 400 bags
Yellow bags: (5/20) * 2000 = 500 bags
Black bags: (8/20) * 2000 = 800 bags
So the factory should produce 300 red bags, 400 white bags, 500 yellow bags, and 800 black bags.
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The table below shows the ratio of materials in a composting container.
Brown Material (gallons) 12.75 18.6 24 E
Green Material (gallons) 4.25 B D 11.5
Total (gallons) A C 32 46
If Jeremy needs 52 gallons of compost to plant his garden in the spring, how much brown material will he need? How much green material will he need?
Jeremy will need 40 gallons of brown material and 12 gallons of green material.
Jeremy will need 13 gallons of brown material and 39 gallons of green material.
Jeremy will need 39 gallons of brown material and 13 gallons of green material.
Jeremy will need 37 gallons of brown material and 15 gallons of green material.
The amounts needed of brown and green material for 52 gallons of compost are
Brown material: 39 gallons.
Green material: 13 gallons.
What is a proportion?In general, the term "proportion" refers to a part, share, or amount that is compared to a total.
According to the concept of proportion, two ratios are in proportion when they are equal.
Two ratios or fractions are equal when an equation or a declaration to that effect is utilised.
According to proportion, two sets of provided numbers are said to be directly proportional to one another if they increase or decrease in the same ratio. "::" or "=" are symbols used to indicate proportions.
Given:
On the first column of the table, the total amount is:
12.75 + 4.25 = 17 gallons.
The fractions of each material are given as follows:
Brown material: 12.75/17 = 0.75
Green material: 4.25/17 = 0.22.
Thus, the amount of brown material needed with 52 gallons is:
0.75 x 52 = 39 gallons.
and, the amount of green material needed with 52 gallons is:
0.25 x 52 = 13 gallons.
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Answer:
Brown material: 39 gallons
Green material: 13 gallons.
Step-by-step explanation:
1. the f/a 18 aircraft lands at approximately 160 knots. determine the landing speed in a. mph (miles per hour) & ft/sec (feet per second) b. km/h (kilometer per hour) & meter/sec (meter per second
The landing speed of the F/A 18 aircraft in mph and ft/sec is 184 mph and 270.2488 ft/sec, respectively. The landing speed in km/h and m/sec is 298.72 km/h and 81.82752 m/sec, respectively.
What is speed ?
The speed of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time.
1 knot = 1.15078 miles per hour (mph)
1 knot = 1.68781 feet per second (ft/sec)
1 knot = 1.852 km per hour (km/h)
1 knot = 0.51444 meter per second (m/sec)
Given that the F/A 18 aircraft lands at approximately 160 knots, we can convert its landing speed to different units:
a. mph and ft/sec:
160 knots x 1.15078 mph/knot = 184 mph
160 knots x 1.68781 ft/sec/knot = 270.2488 ft/sec
b. km/h and m/sec:
160 knots x 1.852 km/h/knot = 298.72 km/h
160 knots x 0.51444 m/sec/knot = 81.82752 m/sec
Therefore, the landing speed of the F/A 18 aircraft in mph and ft/sec is 184 mph and 270.2488 ft/sec, respectively. The landing speed in km/h and m/sec is 298.72 km/h and 81.82752 m/sec, respectively.
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Find the length of side x to the nearest tenth.
Heat loss, h, in calories per hour through a glass window varies jointly as the difference, d,
between the inside and outside temperatures and as the area, A, of the window and
inversely as the thickness, t, of the pane of glass. If the temperature difference is 30
degrees, there is a heat loss of 9000 calories per hour through a window with an area of
1500 square centimeters and thickness of 0.25 centimeter. Find the heat loss through a
window with the same area and a thickness of 0.2 centimeter when the temperature
difference is 15 degrees.
The heat lost when the temperature difference is 15 degrees Celsius and thickness 0.2 cm is 5625 Calories.
What is the heat lost through the window?
The heat lost through the window is calculated by determining the value of constant in the joint variation.
h = ( kdA / t )
where;
k is the constantd is the difference between inside and outside temperatureA is the area of the windowt is the thickness of the windowk = ( ht ) / ( dA )
k = ( 9000 x 0.25 ) / ( 30 x 1500 )
k = 0.05
The heat lost when the temperature difference is 15 degrees Celsius and thickness 0.2 cm is calculated as;
h = ( kdA / t )
h = ( 0.05 x 15 x 1500 ) / ( 0.2 )
h = 5,625 calories
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Help me please, I'm running out of points.
10. In the inequality x + y ≥ 6, which of the following is NOT a solution?
A. (2,4)
B. (2,3)
C. (1,7)
D. (3,4)
11. What are the possible values of y if x = 4 in the inequality x + 2y ≥ 8?
A. y≥4
B. y≥2
C. y≤2
D. y≤4
Answer:
please see below
Step-by-step explanation:
{10} B
{11} B
2x-7 divided by 4 = x+3 divided by 3
Answer:
x = 33/2
Step-by-step explanation:
2x-7/4 = x+3/3
By cross multiplication we get
3(2x-7) = 4(x+3)
6x - 21 = 4x + 12
6x - 4x = 12 + 21
2x = 33
x = 33/2
Answer:
[tex]x=16.5[/tex] [tex]or[/tex] [tex]\frac{33}{2}[/tex]
Step-by-step explanation:
[tex]\frac{2x-7}{4}=\frac{x+3}{3}[/tex]
In this situation, you can cross-multiply:
[tex](2x-7)3[/tex] [tex]=[/tex] [tex](x+3)4[/tex]
[tex]6x-21 = 4x+12[/tex]
[tex]+21[/tex] [tex]+21[/tex]
[tex]6x=4x+33[/tex]
[tex]-4x[/tex] [tex]-4x[/tex]
[tex]2x=33[/tex]
[tex]\frac{2x}{2}=\frac{33}{2}[/tex]
[tex]x=16.5[/tex] [tex]or[/tex] [tex]\frac{33}{2}[/tex]
Solve the following initial value problems for y(t) by antidifferentiation or by
aseparation of variables:
e-21 dt =y(1-e); y(0) = 0
On Solving the initial value problem y(t) , e⁻²¹ dy/dt =y⁻¹(1-e⁻²t); y(0) = 0 , we get the solution is y² = e^(2t) - 2t - 1 .
The function is : e⁻²¹ dy/dt =y⁻¹(1-e⁻²t) ;
⇒ e^(-2t)dy/dt = (1/y)(1-e⁻²t) ..because y⁻¹ = 1/y ;
⇒ ydy = e^(2t)[1-e^(-2t)) dt ;
⇒ ydy = (e^(2t) - 1)dt ;
now integrating both sides of the problem ;
⇒ y²/2 = [e^(2t)/2 - t]+c ;
Now we know that initial value : y(0) = 0 ;
Substituting y(0)= 0 , we get ;
⇒ 0²/2 = [e⁰/2 - 0]+c ;
⇒ c = -1/2 ;
So , the solution is y²/2 = [e^(2t)/2 - t] - 1/2 .
it can be written as : y² = e^(2t) - 2t - 1 .
Therefore , the solution of the initial value problem is y² = e^(2t) - 2t - 1 .
Solve the following initial value problems for y(t) by antidifferentiation or by aseparation of variables : e⁻²¹ dy/dt =y⁻¹(1-e⁻²t); y(0) = 0 .
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Match the sentences with the missing words.
No, the point (2, 3) is not a solution to the linear inequality y < 2x-3, since 3 is not less than 2*2-3, which is 1.
Determine if the point is a solution of the linear inequality?the point (2,3) is not a solution of the linear inequality y < 2x-3.This is because when the given point is substituted into the linear inequality, the result is 3 < 2(2)-3, which is false. The linear inequality y < 2x-3 is an expression of the boundary between points that are in the solution area and points that are not.The boundary is a solid line described by the equation y = 2x-3.The solution area is a shaded area below the line, which consists of all the points that make the inequality true.The point (2,3) does not make the inequality true, and therefore is not a solution of the linear inequality.To learn more about The linear inequality refer to:
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