Laplace transform using a table of Laplace transforms, we get v(t) = (mg/b)(1 - e^(-bt/m)) + v(0)e^(-bt/m)
a) To solve the differential equation using Laplace transforms, we first take the Laplace transform of both sides:
L[mv' + bv] = L[mg]
Using the linearity of the Laplace transform and the fact that L[v'] = sV(s) - v(0), we can simplify the left side:
m(sV(s) - v(0)) + bV(s) = mg/(s)
Solving for V(s), we get:
V(s) = (mg/m)/(s + b/m) + v(0)/(s + b/m)
Taking the inverse Laplace transform using a table of Laplace transforms, we get:
v(t) = (mg/b)(1 - e^(-bt/m)) + v(0)e^(-bt/m)
b) Yes, the object reaches a terminal velocity. As t approaches infinity, the exponential term e^(-bt/m) approaches zero, and the velocity approaches:
v(t) = mg/b
This is the terminal velocity, which is constant and independent of the initial conditions.
c) When b = 0, the differential equation reduces to:
mv' = mg
which can be easily solved by integrating both sides:
v(t) = (mg/m)t + v(0)
This gives a linear increase in velocity with time, in contrast to the exponential increase when b is nonzero. The solution with b = 0 corresponds to free fall under gravity without air resistance.
Here are rough sketches of the solutions for both cases:
Velocity vs. time for b > 0 (blue) and b = 0 (red):
The blue curve shows an exponential increase in velocity that approaches the terminal velocity (shown as a horizontal line) as t approaches infinity. The red curve shows a linear increase in velocity that continues indefinitely without approaching a terminal velocity.
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How do I solve this problem 34=p+19
Hello,
[tex]34 = p + 19[/tex]
[tex]34 - 19 = p + 19 - 19[/tex]
[tex]15 = p[/tex]
[tex]p = 15[/tex]
#1) a normal distribution has a mean of 101 and a standard deviation of 4.2. what is the z-score of 105?
Step-by-step explanation:
z = (specific score - mean) / standard deviation
in our case
z = (105 - 101)/4.2 = 4/4.2 = 0.952380952... ≈ 0.95
as z-tables usually round the z-scores to hundredths.
The z-score of the normal distribution is 0.95.
What is normal distribution?A probability distribution which is symmetric about the mean is the normal distribution, sometimes referred to as the Gaussian distribution.
It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean. The normal distribution shows as a "bell curve" on a graph.
Z-score: A Z-score is just a metric that quantifies how closely a value relates to a mean of a set of values. Standard deviations from of the mean are used to measure Z-score.
A Z-score of zero means the data point is actually score is the same as the mean score.
The formula for z-score is-
[tex]Z=\frac{x-\mu}{\sigma}[/tex]
Z = standard score
x is observed value = 105
[tex]\mu[/tex] is mean of the sample 101.
[tex]\sigma[/tex] is standard deviation of the sample 4.2.
Substitute all the values in the z-score formula;
z = ( 105 - 101)/4.2
z = 0.9523..
z = 0.95
Therefore, the z-score the given normal distribution is 0.95.
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How many different 4 digit numbers can be formed using only digits 1,2,3 and 4 given that they contain each of these digits as many times as you want?
The number of different 4 digit numbers that can be formed using only digits 1,2,3 and 4 if they are repeated is; 256 possible numbers
How to find the probability combination?Since the numbers can be used as many times as we want, then it means they can be repeated.
Thus, if we are looking at the number of 4 digit numbers we can create using the numbers 1, 2, 3, and 4, we can calculate that the following way:
For each digit (thousands, hundreds, tens, ones), we have 4 possible choices of numbers. And so we can create;
4 × 4 × 4 × 4 = 256 numbers
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Multiply the following numbers. Reduce the answer to lowest terms.
Answer:
B 4 1/3
Step-by-step explanation:
Triangle X Y Z is shown. Angle X Z Y is 90 degrees and angle X Y Z is 41 degrees. The length of Z X is 22.
Which equation could be used to solve for the length of XY?
XY = (22)sin(41°)
XY = (22)cos(41°)
XY = StartFraction 22 Over cosine (41 degrees) EndFraction
XY = StartFraction 22 Over sine (41 degrees) EndFraction
The equation that could be used to solve for the length of XY is XY = 22/sin(41)
What is an equation?An equation is an expression that shows the relationship between two or more variables and numbers.
Sine rule is used to show the relationship between the sides and angles of a trinagles.
Using sine rule on triangle XYZ:
XY / sin(Z) = XZ / sin(Y)
XY / sin(90) = 22/sin(41)
XY = 22/sin(41)
The equation that could be used to solve for the length of XY is XY = 22/sin(41)
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Answer:
D
Step-by-step explanation:
Which inequality can be used to explain why these three segments cannot be used to construct a triangle? ac ab > cb ac cb < ab ac cb > ab ac ab < cb
The response option that demonstrates why the three segments cannot be combined to form a triangle is AC + CB< AB.
What is triangle inequalities?According to the triangle inequality, the length of any two sides of a triangle added together must be longer than the length of the third side. Since a straight line is the shortest distance between two places, this conclusion follows. If the triangle isn't degenerate, the inequality is strict (meaning it has a non-zero area).
Which inequality explains why a triangle cannot be made from the three segments?Since the length of any two sides of a triangle added together is longer than the length of the third side, it follows from the triangle inequalities theorem that this is the case.
As a result, the necessary inequality is AC + CB AB because the sum of sides AC + CB is less than AB.
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I understand that question you are looking for is:
Which inequality can be used to explain why these three segments cannot be used to construct a triangle?
A. AC + AB > CB
B. AC + CB < AB
C. AC + CB > AB
D. AC + AB < CB
mAFD=90 mAFB=31 Find mCFE
The value of angle CFE in the diagram attached is 51 degrees.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
From the diagram:
∠AFB = ∠CFD; ∠BFC = ∠DFE
Hence:
∠AFC = ∠CFE
∠AFD = ∠AFC + ∠CFD = ∠AFC + ∠AFB
90 = ∠AFC + 31
∠AFC = 51° = ∠CFE
The value of angle CFE in the diagram attached is 51 degrees.
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Please help me i don’t understand
The graph of the inverse of the function is shown in the image attached below.
How to graph the inverse of a function
In this question we must apply the following transformation rule to the three points to determine the points of the inverse function:
(x, y) → (y, x)
Now we proceed to find the points:
(- 7, 3) → (3, - 7)
(-3, 0) → (0, -3)
(-2, -2) → (-2, -2)
Lastly, we proceed to construct the graph of the inverse of the function.
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given that f.x 3x-2 over x+1 g[x] x +5 evaluate f[-4] and gf [-2]
The value of f[ -4 ] and g°f[-2] are [tex]\frac{14}{3}[/tex] and 13 respectively.
What is the value of f[-4] and g°f[-2]?Given the function;
[tex]f(x) = \frac{3x-2}{x+1}[/tex][tex]g(x)=x+5[/tex]f[ -4 ] = ?g°f[ -2 ] = ?For f[ -4 ], we substitute -4 for every variable x in the function.
[tex]f(x) = \frac{3x-2}{x+1}\\\\f(-4) = \frac{3(-4)-2}{(-4)+1}\\\\f(-4) = \frac{-12-2}{-4+1}\\\\f(-4) = \frac{-14}{-3}\\\\f(-4) = \frac{14}{3}[/tex]
For g°f[-2]
g°f[-2] is expressed as g(f(-2))
[tex]g(\frac{3x-2}{x+1}) = (\frac{3x-2}{x+1}) + 5\\\\g(\frac{3x-2}{x+1}) = \frac{3x-2}{x+1} + \frac{5(x+1)}{x+1}\\\\g(\frac{3x-2}{x+1}) = \frac{3x-2+5(x+1)}{x+1}\\\\g(\frac{3x-2}{x+1}) = \frac{8x+3}{x+1}\\\\We\ substitute \ in \ [-2] \\\\g(\frac{3x-2}{x+1}) = \frac{8(-2)+3}{(-2)+1}\\\\g(\frac{3x-2}{x+1}) = \frac{-16+3}{-2+1}\\\\g(\frac{3x-2}{x+1}) = \frac{-13}{-1}\\\\g(\frac{3x-2}{x+1}) = 13[/tex]
Therefore, the value of f[ -4 ] and g°f[-2] are [tex]\frac{14}{3}[/tex] and 13 respectively.
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It takes you two and a half hours to drive 175 miles. how fast were you driving? (mph)
If one travels 175 miles in 2hour 30 mins then his speed is 70 mph .
Calculation of the speedDistance travels= 175 miles
time is taken to cover 175 miles = 2.5 hours
We have to find out the speed in mph.
We know that,
Distance = speed x time
⇒speed = distance / time
Putting the value of distance and time, we get:
speed = 175 / 2.5 = 70 mph
Hence the final answer is 70 mph.
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Explain or show how to determine if the following coordinates are a solution to the following system of linear inequalities. y<4x+8 y>x-6
The inequalities y < 4x + 8 and y > x - 6, the solution to the inequalities is the darker region shown in the graph.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Given the inequalities y < 4x + 8 and y > x - 6, the solution to the inequalities is the darker region shown in the graph.
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Can someone help? ASAP
Answer:
11. [tex]\frac{\sqrt{2} }{2}[/tex]
12. [tex]\sqrt{3}[/tex]
13. [tex]\frac{\sqrt{3} }{2}[/tex]
14. [tex]\frac{\sqrt{3} }{3}[/tex]
15. [tex]\sqrt{2}[/tex]
16. [tex]\frac{2\sqrt{3} }{3}[/tex]
Step-by-step explanation:
Hope this helps
Find the fifth term of the sequence.
a) Substituting in n = 2,
[tex]\frac{3(2)^{2}}{2}=6[/tex]
b) Substituting in n = 5,
[tex]\frac{3(5)^{2}}{2}=\frac{75}{2}[/tex]
c) Setting the nth term to be greater than 50,
[tex]\frac{3n^{2}}{2}>50 \\ \\ n^{2} >\frac{100}{3} \\ \\ n>\frac{10}{\sqrt{3}} \\ \\ \lceil n \rceil = 6[/tex]
Brainliest for answer
A quadratic equation is graphed to the left. Which of the following equations could be paired with the graphed equation to create a system of equations whose solution set is comprised of the points (2,-4)and (-4,2)?
The linear function that goes through the points (2,-4) and (-4,2) is:
y = -x + 2.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.A linear function going through the points (2,-4) and (-4,2) would intersect the parabola at these points, hence these points would be the solution for the system of equations.
The slope of the line is:
m = (-4 - 2)/(2 - (-4)) = -1.
The line goes through point (2,-4), that is, when x = 2, y = -4, which we use to find the y-intercept b.
y = -x + b
-4 = -2 + b
b = -2.
Hence the equation is:
y = -x + 2.
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Mr. lee walks into the courtyard after school and wants to know the height of the tree. he measures an angle of 47* between a line of sight to the top of the tree and the ground and he is 22 feet away from the tree. how tall is the tree? *
The tree is 23.54 feet tall.
What is an angle?An angle is a figure in plane geometry that is created by two rays or lines that have a shared endpoint. The Latin word "angulus," which meaning "corner," is the source of the English term "angle." The shared terminus of two rays is known as the vertex, and the two rays are referred to as sides of an angle. It is not necessary for the angle to be in Euclidean space if it lies in the plane. Angles are referred to as dihedral angles if they are created by the intersection of two planes in a space other than Euclidean.
According to the question,
Height of tree= 22*tan [tex]47^{o}[/tex]
= 22* 1.07
= 23.54 ft.
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Which of the following parabolas has its vertex the furthest away from the x-axis
Oy
x² - 1
y = 3x² + 4
Oy
=
Y =
1/2x²
Oy = x² - 10
Question 12 (1 point)
Answer:
y=x^2-10
Step-by-step explanation:
In a sample of 16 employees randomly selected at a company, the average age is 25 years and the standard deviation is 2 years. We want to test if the average age of all employees is greater than 24. If ages are normally distributed in the population, using a significance level of 0.05, we can claim that the population mean is: Group of answer choices
If ages are normally distributed then using a significance level of 0.05 , we can claim that the population mean is greater than 24.
Given sample mean of 25 years , sample size of 16 ,standard deviation of 2 years.
We have to find whether the population average is greater than 24.
We have to use t test because n is less than 30. It is right tailed.
We have to first form Hypothesis.
[tex]H_{0}[/tex]:μ>24,
[tex]H_{1}[/tex]:μ<24.
t=X-μ/S/[tex]\sqrt{n}[/tex]
t critical at 5% significance level and degree of freedom=15 (16-1)=1.7531
t=24-25/2/[tex]\sqrt{16}[/tex]
=-1/0.5
=-2
Because 1.7531 is greater than -2 so we will accept the null hypothesis.
Hence we can say that the average of population is greater than 24.
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A right-triangle has the hypotenuse length of 5 units long. One of the other sides of the triangle is 3 units. What is the length of the remaining side
Answer:
4
Step-by-step explanation:
[tex] {3}^{2} + {x}^{2} = {5}^{2} \\ \\ 9 + {x}^{2} = 25 \\ \\ {x}^{2} = 16 \\ \\ x = 4[/tex]
An entomologist can model the number of mosquitos in a forest as a function of
rainfall. In a similar way, they can also model the number of bats in the forest.
The function f(x) = 5x
22
gives the number of mosquitos in the forest (in
millions), and the function g(x) = 3a 0.5a² gives the number of bats (in millions
-
In both equations z represents rainfall (in centimeters). Here are the graphs of f and
G
The interpretation of the point of intersection of the curves is (c) they give a solution to the equation 5x - x^2 = 3x - 0.5x^2
How to interpret the points of intersection?The graph of the two functions alongside their equations are the given parameters of this question
The functions are given as:
f(x) = 5x - x^2
g(x) = 3x - 0.5x^2
When the graphs of two curves or lines intersect on a coordinate plane, it means that the point of intersection represents where the graphs have equal value
In this case, it represents
f(x) = g(x)
Substitute the known values in the above equation
5x - x^2 = 3x - 0.5x^2
So, the interpretation of the point of intersection of the curves is (c) they give a solution to the equation 5x - x^2 = 3x - 0.5x^2
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Juan bought a sweater for $15.95 and two shirts for $9.00 each. how much did juan spend on clothes?
Answer:
Step-by-step explanation:
Well first you have:
$15.95 for one sweater
$9.00 each for two shirts
You would double the amount $9.00 which would be:
9 + 9 = 18 or Multiply 9 x 2 = 18
So this would total to 15 + 18 = $33
At birth, one puppy weighs 15/16 pounds. A second puppy weighs 1 and 1/8 pounds. How much more did the second puppy weigh?
Answer:
3/16 pounds more.
Step-by-step explanation:
It is 1 1/8 - 15/16
= 9/8 - 15/16
= 18/16 - 15/16
= 3/16.
How many quarters are there in 7 1/2?
Answer: 30.
Step-by-step explanation:
[tex]7\frac{1}{2} =\frac{7*2+1}{2} \\\frac{14+1}{2}=\frac{15}{2}\\ \frac{15*2}{2*2}=\frac{30}{4} \\ \frac{30}{4}=30*\frac{1}{4} .[/tex]
There are total 30 quarters in [tex]7\frac{1}{2}[/tex].
What is one quarter?A quarter is one out of four equal parts. One quarter is the same as the fraction ¼.
What is mixed fraction?A fraction represented with its quotient and remainder is a mixed fraction.
How to convert a mixed fraction into an improper fraction?To change a mixed fraction to an improper fraction, we multiply the whole number by the denominator and then add this product with the numerator.
According to the given question.
We have a mixed fraction [tex]7\frac{1}{2}[/tex].
So, the improper fraction of the given mixed fraction is given by
[tex]7 \frac{1}{2} = \frac{15}{2}[/tex]
To find the number of quarters in the given mixed fraction, we will divide [tex]7\frac{1}{2}[/tex] by [tex]\frac{1}{4}[/tex].
Therefore,
[tex]\frac{7\frac{1}{2} }{\frac{1}{4} }[/tex]
= [tex]\frac{\frac{15}{2} }{\frac{1}{4} }[/tex]
[tex]= \frac{15\times 4}{2}[/tex]
[tex]= 15\times 2[/tex]
[tex]= 30[/tex]
Hence, there are total 30 quarters in [tex]7\frac{1}{2}[/tex].
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For the graph x = 1 find the slope of a line that is perpendicular to it and the slope of a line parallel to it. Explain your answer with two or more sentences.
The slope of the parallel line is undefined and the slope of the perpendicular line is 0
How to determine the slope?The equation is given as:
x = 1
The above do not have any y value.
This means that the slope of the line is undefined
i.e.
m = undefined
The slope of the parallel line is:
Slope = m
This gives
Slope = undefined
The slope of the perpendicular line is:
Slope =-1/m
This gives
Slope = -1/undefined
Slope = 0
Hence, the slope of the parallel line is undefined and the slope of the perpendicular line is 0
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In London, the charge of a tour bus consists of a fixed rental price plus a per kilometre rate, that depends on the distance of the trip.
The per kilometre rate is $12/km. You decide to go on a 5 km trip, and it costs you $70.
a) What is the fixed rental cost? Show your work.
b) Determine the equation relating the cost, C, in dollars, and the distance, d, in km.
d) What is the cost of an 18 km trip?
e) What is the distance travelled if the total cost is $250?
PLEASEEEEEEEE
Using a linear function, we have that:
a) The fixed cost is of $10.
b) The equation is: C = 10 + 12d.
c) The cost of an 18 km trip is $226.
d) The distance traveled is of 20 km.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.For this problem, the slope is of m = 12. When d = 5, C(d) = 70, hence we find the fixed cost as follows:
C(d) = 12d + b
70 = 60 + b
b = 10.
Hence the equation is:
C(d) = 12d + 10.
For a trip of 18 km, d = 18, hence the cost is:
C(12) = 12 x 18 + 10 = $226.
When the cost is of $250, the distance is found as follows:
250 = 12d + 10
12d = 240
d = 240/12
d = 20 km.
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Find the indicated term of the geometric sequence.
10th term of 0.6, 0.06, 0.006,...
Answer:
0.0000000006
Step-by-step explanation:
0.6 * (0.1)^9 = 0.0000000006
The functions f(x) and g(x) are shown on the graph.
f(x) = log(x)
g(x)
6
Using fix), what is the equation that represents gix)?
A) g(x) = log5(x) - 3
B) g(x) = log5(x) + 3
C) g(x) = log5(x-3)
D) g(x) = log5(x+3)
Using translation concepts, the equation of g(x) is:
D) [tex]\log_5{x + 3}[/tex]
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
Researching this problem on the internet, g(x) is a shift left of 3 units of [tex]f(x) = \log_5{x}[/tex], hence:
[tex]g(x) = f(x + 3) = \log_5{x + 3}[/tex], which means that option D is correct.
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according to the graph, what is the value of the constant in the equation below?
Answer: B
Step-by-step explanation:
The product of height and width is constant.
If we take the point (0.5,1), we get the product to be 0.5.
Raj measures the height of 70 plants calculate the estimate of the mean height of the plants
The mean height of plants is 44.07cm.
Given that a raj measures the height of 70 plants which is shown in the table in the image attached below.
The mean is defined as the mean value equal to the ratio of the total number of a given set of values to the total number of values contained in that set.
Firstly, we will find the midpoint of the height by using the formula
(upper limit +lower limit)/2
When h is 10<h<20 then the midpoint is
x₁=(10+20)/2
x₁=15
When h is 20<h<0 then the midpoint is
x₂=(20+40)/2
x₂=30
When h is 40<h<50 then the midpoint is
x₃(40+50)/2
x₃=45
When h is 50<h<60 then the midpoint is
x₄=(50+60)/2
x₄=55
When h is 60<h<90 then the midpoint is
x₅=(60+90)/2
x₅=75
Now, we will find the mean height by using the formula
x=(f₁x₁+f₂x₂+f₃x₃+f₄x₄+f₅x₅)/(Σfi)
Here, f₁=7,f₂=15,f₃=27,f₄=13,f₅=8
and Σfi=7+15+27+12+8=70
Substitute these values in the formula, we get
x=(7×15+15×30+27×45+12×55+8×75)/70
x=(105+450+1215+715+600)/70
x=(3085)/70
x=44.07
Hence, the estimate of the mean height of the plants when raj measures the height of 70 plants is 44.07cm.
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URGENT, PLEASE ANSWER THESE QUESTIONS
Can send helicopter 3 but not helicopter 1 and 2.
What is distance formula?
Consider two points (x1, y1) and (x2, y2) in the two-dimension, then the distance between these two points is given by,
[tex]d=\sqrt{(x2-x1)^2+(y2-y1)^2}[/tex]
We will solve given problem as shown below:
Let fire have coordinate F (-3,-5)
Let Helicopter 1 have coordinate H1(1,4)
Let Helicopter 2 have coordinate H2(-2,3)
Let Helicopter 1 have coordinate H3(4, -2)
A. Distance between helicopter 1 and fire
[tex]=\sqrt{(1-(-3))^2+(4-(-5))^2}[/tex]
[tex]=\sqrt{16+81}[/tex]
[tex]=\sqrt{97}=9.8[/tex]
No, cannot send helicopter 1
B. Distance between helicopter 2 and fire
[tex]=\sqrt{(-3-(-2))^2+(-5-3)^2}[/tex]
[tex]=\sqrt{1+64}[/tex]
[tex]=\sqrt{65}=8.06[/tex]
No, cannot send helicopter 2
Distance between helicopter 3 and fire
[tex]=\sqrt{(4-(-3))^2+(-2-(-5))^2}[/tex]
[tex]=\sqrt{49+9}[/tex]
[tex]=\sqrt{58}=7.6[/tex]
Yes, can send helicopter 3
Hence, can send helicopter 3 but not helicopter 1 and 2.
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Which function could be a stretch of the exponential decay function shown on the graph?
f(x) = 2(6)x
f(x) = One-half(6)x
f(x) = 2(one-sixth) Superscript x
f(x) = One-half (one-sixth) Superscript x
The function that could be a stretch of the exponential decay function shown on the graph is B. f(x) = One-half(6)x.
How to illustrate the function?From the information given, it can be seen that function A is a monotonically increasing function.
Also, it can be seen that C and D don't pass through (0, 1).
Therefore, the function could be a stretch of the exponential decay function shown on the graph is B.
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Answer:
b
Step-by-step explanation: