y' + 6y = f(t)
where
[tex]f(t)=\begin{cases}0&\text{if }0\le t\le2\\12&\text{if }2<t\le6\\0&\text{if }6<t<\infty\end{cases}[/tex]
You can write f(t) in terms of the unit step (i.e. Heaviside theta) function u(t), which is defined as
[tex]u(t)=\begin{cases}0&\text{if }t<0\\1&\text{if }t\ge0\end{cases}[/tex]
Then the DE is written as
y' + 6y = 12 u (t - 2) - 12 u (t - 6)
(a) Take the Laplace transform of both sides:
LT[y' + 6y] = LT[12 u (t - 2) - 12 u (t - 6)]
s Y - y (0) + 6Y = 12 (exp(-2s) - exp(-6s))/s
(b) Solve for Y :
(s + 6) Y = 12 (exp(-2s) - exp(-6s))/s + y (0)
Y = 12 (exp(-2s) - exp(-6s))/(s (s + 6)) + y (0)/(s + 6)
(c) Take the inverse transform:
LT⁻¹ [Y] = LT⁻¹[12 (exp(-2s) - exp(-6s))/(s (s + 6)) + y (0)/(s + 6)]
y = 12 LT⁻¹ [(exp(-2s) - exp(-6s))/(s (s + 6))] + y (0) LT⁻¹ [1/(s + 6)]
y = 12 u (t - 2) LT⁻¹ [1/(s (s + 6))] - 12 u (t - 6) LT⁻¹ [1/(s (s + 6))] + y (0) exp(-6t )
For the remaining inverse transform, break up into partial fractions:
1/(s (s + 6)) = a/s + b/(s + 6)
1 = a (s + 6) + bs
1 = (a + b) s + 6a
==> 6a = 1, a + b = 0 ==> a = 1/6, b = -1/6
y = 2 u (t - 2) LT⁻¹ [1/s - 1/(s + 6)] - 2 u (t - 6) LT⁻¹ [1/s - 1/(s + 6)] + y (0) exp(-6t )
y = 2 u (t - 2) (1 - exp(-6t )) - 2 u (t - 6) (1 - exp(-6t )) + y (0) exp(-6t )
lim(x-0) (sinx-1/x-1)
lim ( sinx-1)/(x-1)
x=>0
apply x=0
(sin(0)-1)/(0-1)
(0-1)/(-1)
=1
I) Find the volume in terms of pie
ii) curved surface area in terms of pie
iii) capacity in litres (correct to nearest litre)
Answer:
i) pi×4500 cm³
ii) pi×600 cm²
iii) 14 liters
Step-by-step explanation:
in general : the diameter is 30 cm, the radius is half of that (15 cm)
i)
the volume of a cylinder is base area times height.
Vc = pi×r²×h = pi×15²×20 = pi×225×20 = pi×4500 cm³
ii)
similar to volume, the side "mantle" area of the cylinder is the circumference of the base area times height.
surface area of the cylinder mantle is
Scm = 2×pi×r×h = 2×pi×15×20 = pi×30×20 = pi×600 cm²
iii)
for this we need now to do the multiplication with pi and then convert the cm³ to liters.
1 liter = a cube of 10 cm side length = 10×10×10 = 1000 cm³
pi×4500 = 14137.17 cm³ = 14.13717 liters or rounded 14 liters
Please help on 25 it’s confusing me I need the correct answer
Answer:
X * 0.8 = $64
x = $80
Step-by-step explanation:
Answer:
$80 (D)
Step-by-step explanation:
If Richard is getting a discount and his final price is $64 that means the answer must be above 64. That eliminates A, B, and C.
Use the formula: 20% of x = $64 and substitute the other two options. 20 percent of 84 is 16.8. 84-16.8=67.2 (not the correct answer). 20 percent of 80 is 16. 80-16=64(The Correct Answer).The answer must be $80 (D)
The length of a rectangle is four more than three times the width. If the perimeter of this rectangle is at least 70 square centimeters. Write an inequality that can be solved to find the width of the rectangle
Answer:
Step-by-step explanation:
Let L represent the length of the triangle.
Let W represent the width of the triangle.
The length of a rectangle is four more than three times the width. This means that
L = 3W + 4
The formula for determining the perimeter of a rectangle is expressed as
Perimeter = 2(L + W)
If the perimeter of this rectangle is at least 70 square centimeters, an inequality that can be solved to find the width of the rectangle is
2(L + W) ≥ 70
L + W ≥ 70/2
L + W ≥ 35
Answer:
6w +8 ≥70
Step-by-step explanation:
Let w be the width
The length is then 3w+4 ("the length is 4 more than 3 times the width")
Since a rectangle has opposite sides equal, the perimeter would be 2(l+w) or 2(w+3w+4) which would be 6w +8. If the perimeter is at least 70, that is, 70 or more, the inequality would be
6w + 8 ≥ 70.
The units, however, would not be SQUARE centimeters, just centimeters. If the question were asking for area, the units would be square units, but since perimeter is a linear measurement, the units would have to be linear.
help asap pleaseeee asap
Please help!!!!! Nowwww
Answer:
It has 1 term and a degree of 4.
Step-by-step explanation:
3j⁴k-2jk³+jk³-2j⁴k+jk³
= 3j⁴k-2j⁴k-2jk³+jk³+jk³
= j⁴k
So, in this expression, there is 1 term, and it has a degree of 4.
if f(x)=3x²-7 and f(x+n)=3x²+24x+41, what is the value of n?
Answer:
n=4
Step-by-step explanation:
f(x+n)=3(x+n)^2-7=3x^2+24x+41
3x^2+3n^2+6xn-7=3x^2+24x+41
Comparing and we will get, n=4
Matematykakdbebox
Jaggbn
Answer:
theres no question....
Step-by-step explanation:
???
A variety of trigonometric functions are shown in the answer choices below.
Which trigonometric function has an inverse over the domain x2≤x≤3x2
A-f(x)=cos(x−1/2)+3/2
B-f(x)=cos(x+π/2)
C-f(x)=sin(x−1/2)+3/2
D-f(x)=sin(x+π/2)
Car drove 2hours at a speed of 100km per hour & 3 hour at a speed of 50 km per hour . What was the average speed of the car during the trip?
Answer:
200 kilometers and 150 kilometers
3/8n+5(n-6)=1 7/8n-2
Answer:
n = 112/13 = 8.615
Step-by-step explanation:
(3/8) n + 5n - 30 = (17/8)n - 2
(3/8)n +5n - (17/8)n = 30-2
(13/4)n = 28
n = 28 * 4/13
n = 112/13
n = 8.615
Find the value of x. Round to the nearest tenth.
Answer:
1.6 ft
Step-by-step explanation:
If you use the Pythagorean Theorem to solve for x, you get:
[tex]x=\sqrt{2.1^2-1.4^2}[/tex]
[tex]x=\sqrt{2.45} = 1.56524758425[/tex]
Rounded to the nearest tenth, the answer is 1.6
7x to the power of 2 is a what is it
a) monomial
b) binomial
c) Trinomial
Two trains are 500 miles apart when they first start moving towards each other. If in two hours the distance between them is 300 miles and one train goes 20 miles faster than another, find the speed of the faster train. (Note: there are two possible solutions. Could you please find both?)
Answer:
Step-by-step explanation:
They travel 500 - 300 = 200 miles in 2 hours so their combined speed is
100 mph.
If their respective speed are x and y mph then we have the system
x + y = 100
x - y = 20
Adding the 2 equations
2x = 120
x = 60
and y = 40.
The other solution is that y = 60 mph and x = 40 mph.
Every high school senior takes the SAT at a school in St. Louis. The high school guidance director at this school collects data on each graduating senior’s GPA and their corresponding SAT test score. The guidance director is conducting a _________ in this experimental design.
A. sample survey
B. census
C. sample poll
D. random sample
The guidance director is conducting a sample poll in this experimental design.
What is sample?Sample is a part of population. It does not comprises whole population. It is representatitive of whole population.
How to fill blank?We are required to fill the blank with appropriate term among the options.
The correct option is sample poll because the guidance director collects data in his school only.
Census collects the whole population of the country.
Sample poll means collecting data from small population.
Random sample means collecting data from a part of popultion without identifying any variable.
Hence we found that he was doing sample poll.
Learn more about sample at https://brainly.com/question/24466382
#SPJ2
Let a submarine be at a constant depth of 5 km. It is headed in the direction of a lighthouse. If the distance between the submarine and the base of the lighthouse is decreasing at a rate of 24 km/h when the sub is 13 km away from the base, then what is the speed of the submarine
Answer:
24 km/h
Step-by-step explanation:
Given:
Constant speed of submarine = 24 km/h
Depth under sea = 5 km
Distance of submarine from lighthouse = 13 km
Find:
Speed of the submarine
Computation:
At steady speed, the distance between both the submarine and the lighthouse base decreases at a rate of 24 km/hr.
So, when it is 13 kilometres from its starting point, the speed remains constant at 24 kilometres per hour.
Is it true that every whole number is a solution of x > 0? Use complete sentences to explain your reasoning.
Whole numbers are natural numbers, and natural numbers do not include negatives, decimals, fractions, or roots, so all whole numbers can indeed satisfy the inequality.
Two mechanics worked on a car. The first mechanic worked for 10 hours, and the second mechanic worked for 5 hours. Together they charged a total of
$750. What was the rate charged per hour by each mechanic if the sum of the two rates was $105 per hour?
Step-by-step explanation:
let's convert the statement into equation..
let the charge of 1st mechanic be x and second be y..
by the question..
10x+5y=750...(i)
x+y=105..(ii)
from eqn(ii)..
x+y=105
or, x=105-y...(iii)
substituting the value of x in eqn (i)..
10x+5y=750
or, 10(105-y)+5y=750
or, 1050-10y+5y=750
or, 1050-750=5y
or, y=300/5
•°• y=60
substituting the value of y in eqn(iii).
x=105-y
or, x=105-60
•°• x= 45..
the rate charged by two mechanics per hour was 60$ and 45$
At a large Midwestern university, a simple random sample of 100 entering freshmen in 1993 found that 20 of the sampled freshmen finished in the bottom third of their high school class. Admission standards at the university were tightened in 1995. In 1997, a simple random sample of 100 entering freshmen found that only 10 finished in the bottom third of their high school class. Let p 1 and p 2 be the proportions of all entering freshmen in 1993 and 1997, respectively, who graduated in the bottom third of their high school class. What is a 90% plus four confidence interval for p 1 – p 2?
Answer:
The 90% confidence interval for the difference of proportions is (0.01775,0.18225).
Step-by-step explanation:
Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
p1 -> 1993
20 out of 100, so:
[tex]p_1 = \frac{20}{100} = 0.2[/tex]
[tex]s_1 = \sqrt{\frac{0.2*0.8}{100}} = 0.04[/tex]
p2 -> 1997
10 out of 100, so:
[tex]p_2 = \frac{10}{100} = 0.1[/tex]
[tex]s_2 = \sqrt{\frac{0.1*0.9}{100}} = 0.03[/tex]
Distribution of p1 – p2:
[tex]p = p_1 - p_2 = 0.2 - 0.1 = 0.1[/tex]
[tex]s = \sqrt{s_1^2+s_2^2} = \sqrt{0.04^2 + 0.03^2} = 0.05[/tex]
Confidence interval:
[tex]p \pm zs[/tex]
In which
z is the z-score that has a p-value of [tex]1 - \frac{\alpha}{2}[/tex].
90% confidence level
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The lower bound of the interval is:
[tex]p - zs = 0.1 - 1.645*0.05 = 0.01775 [/tex]
The upper bound of the interval is:
[tex]p + zs = 0.1 + 1.645*0.05 = 0.18225 [/tex]
The 90% confidence interval for the difference of proportions is (0.01775,0.18225).
Elena drank 3 liters of water yesterday. Jada drank 3/4 times as much water as elena. Link drank twice as much as Jada. Did jada drink more or less then elena? Explain how you know
Answer:
Step-by-step explanation:
3\4 bc on a nuberline it would be 3 3\4
I--I----(etc)
so yeah hope i helped
Solve the formula for the given variable.
-2x - 6 = 4x
Please helppp
Answer:
s snsnnssjsjjsnsnsjs
es 17 es
State two similarities and one difference between the graphs of f(x)= 3^x and g (x)= (1/3) ^x
Find the lengths of AD, EF, and BC in the trapezoid below.
We know that,
[tex]EF=\dfrac{AD+BC}{2}[/tex]
which is
[tex]x=\dfrac{x-5+2x-4}{2}[/tex]
Now solve for x,
[tex]x=\dfrac{3x-9}{2}[/tex]
[tex]2x=3x-9[/tex]
[tex]x=9[/tex]
Since x is 9, the lengths are,
[tex]AD=x-5=9-5=\boxed{4}[/tex]
[tex]EF=x=\boxed{9}[/tex]
[tex]BC=2x-4=18-4=\boxed{14}[/tex]
Hope this helps :)
Please help on my hw
Answer:
Base is 3 centimeter and hight is 12 cent.
What are the coordinates of point K?
A (-2,4)
B (-2,-4)
C (2,-4)
D (2, 4)
Answer:
A
Step-by-step explanation:
I guess that is the answer
Find the value of x.
X 9 9 7 x = [?]
Tom and Jerry had a race. Tom started at 0200 running at 4.5km per hour. Jerry started
later at 6:30am running at 6.5km per hour. After five hours, who was ahead, and by how
much (answer in kilometres)
Answer:
Tom, 10.25
Step-by-step explanation:
Distance covered by Tom=4.5*(4 1/2+5)=81/4=42.75km
Distance covered by Jerry=6.5*(5)=32.5km
Tom is ahead from Jerry by 10.25km
write the equation of the graph, y=? SOMEBODY PLEASE HELP
Answer:
[tex]y=\left(6\right)^{x}\ -3[/tex]
Step-by-step explanation:
What number x gives maximum value for c(12,x)?
Answer:
i will say methods try yourself:
How to Determine Maximum Value
How to Determine Maximum ValueIf your equation is in the form ax2 + bx + c, you can find the maximum by using the equation:
How to Determine Maximum ValueIf your equation is in the form ax2 + bx + c, you can find the maximum by using the equation:max = c - (b2 / 4a).
How to Determine Maximum ValueIf your equation is in the form ax2 + bx + c, you can find the maximum by using the equation:max = c - (b2 / 4a).The first step is to determine whether your equation gives a maximum or minimum. ...
How to Determine Maximum ValueIf your equation is in the form ax2 + bx + c, you can find the maximum by using the equation:max = c - (b2 / 4a).The first step is to determine whether your equation gives a maximum or minimum. ...-x2 + 4x - 2.
How to Determine Maximum ValueIf your equation is in the form ax2 + bx + c, you can find the maximum by using the equation:max = c - (b2 / 4a).The first step is to determine whether your equation gives a maximum or minimum. ...-x2 + 4x - 2.Since the term with the x2 is negative, you know there will be a maximum point.
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plzz follow and make brainlist
what is the simple definition of realnumbers
Answer:
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion).
