Suppose that the height (in centimeters) of a candle is a linear function of the amount of time (in hours) it has been burning. After 9 hours of burning, a candle has a height of 25.4 centimeters. After 23 hours of burning, its height is 19.8 centimeters. What is the height of the candle after 22 hours?
Answers
Answer:
Suppose that the height (in centimeters) of a candle is a linear function of
the amount of time (in hours) it has been burning.
After 11 hours of burning, a candle has a height of 23.4 centimeters.
After 30 hours of burning, its height is 12 centimeters.
What is the height of the candle after 13 hours?
:
Assign the given values as follows:
x1 = 11; y1 = 23.4
x2 = 30; y2 = 12
:
Find the slope using: m = %28y2-y1%29%2F%28x2-x1%29
m = %2812-23.4%29%2F%2830-11%29 = %28-11.4%29%2F19
:
Find the equation using the point/slope formula: y - y1 = m(x - x1)
y - 23.4 = -11.4%2F19(x - 11)
y - 23.4 = -11.4%2F19x + 125.4%2F19
y = -11.4%2F19x + 125.4%2F19 + 23.4
y = -11.4%2F19x + 125.4%2F19 + 23.4
y = -11.4%2F19x + 125.4%2F19 + 444.6%2F19
y = -11.4%2F19x + 570%2F19
y = -11.4%2F19x + 30, is the equation
:
What is the height of the candle after 13 hours?
x = 13
y = -11.4%2F19(13) + 30
y = -148.2%2F19 + 30
y = -7.8 + 30
y = 22.2 cm after 13 hrs
Related Questions
Given that log 2 = 0.3010 and log 7 = 0.8451, find the following (a) log 49 (b) log 560
Answers
Answer:
ok babababr
Step-by-step explanation:
skwjshehhehdhhshwhhdhwhbeujsbsgehbedheb
Answer:
Log 49= 1.6902
Log 560= 1.7481
Step-by-step explanation:
log49= log7×7
From the rules of logarithms, we have that
log a×b= log a + log b
So log49= log7×7= log7 + log7
= 0.8451 + 0.8451 = 1.6902
Log 560 = log7×8 = log7 + log8
log8= log2^3 = 3log2
log8= 3×0.3010 =0.930
log 560= 0.8451 + 0.930 = 1.7481
1. Quadratics: The path of the longest shot put by the Women’s track team at Sun Devil U is modeledby h(x) = -0.015x2 + 1.08x + 5.8, where x represents the horizontal distance from the start and h(x) isthe height of the shot put above the ground. (Both x and h(x) are measured in feet.)a. [3 pts] Determine h(24). Round your answer to 2 decimal places.
Answers
Answer:
23.08 feetStep-by-step explanation:
If the path of the longest shot put by the Women’s track team at Sun Devil U is modeled by h(x) = -0.015x² + 1.08x + 5.8 where x represents the horizontal distance from the start and h(x) is the height of the shot put above the ground, to determine h(24), we will have to substitute x = 24 into the modeled equation as shown;
[tex]h(x) = -0.015x^2 + 1.08x + 5.8\\\\if \ x = 24;\\\\h(24) = -0.015(24)^2 + 1.08(24) + 5.8\\\\h(24) = -0.015(576)+25.92+5.8\\\\h(24) = -8.64+31.72\\\\h(24) = 23.08\\[/tex]
Hence the value of the height at the horizontal distance of 24 feet is 23.08 feet to 2 decimal place.
If a person invested half of her money at 9% and half at 7% and received $160 interest, find the total amount of money invested.
Answers
Answer:
$2000
Step-by-step explanation:
let x be the money she invested
lets assume this was for 1 year
0.09(x/2) + 0.07(x/2) = 160
multiply each side by 2 to cancel the denominators:
0.09x + 0.07x = 320
0.16x = 320
x = 2000
Answer: $2000
Let the amount of money she invested be x
Lets assume the time of investment as 1 year
ATQ
0.09(x/2) + 0.07(x/2) = 160
0.09x + 0.07x = 320
0.16x = 320
x = 2000
Must click thanks and mark brainliest
Makayla wants to make 200 mL of a 18% saline solution but only has access to 8% and 24% saline mixtures.
Which of the following system of equations correctly describes this situation if x represents the amount of the 8% solution used, and y represents the amount of the 24% solution used?
Answers
Answer:
x + y = 200
0.08x + 0.24y = 0.18(200)
Step-by-step explanation:
x + y = 200
0.08x + 0.24y = 0.18(200)
The equations which describes the amount of 8% solution used and the amount of 24% solution used are: x+ y=200 and x+3y=450.
What is equation?An equation is a relationship between two or more variables. They are mostly present in equal to form and are equated to find the value of variables present in them.
How to form equation?let the amount of 8% solution used be x and the amount of 24% solution used be y.
According to question the amount of total solution will be 200ml, So, the equation will be:
x +y=200
Now we have been said that the solution will be 18% saline and 8% saline mixture and 24% saline mixtures are used, So the next equation will be:
0.08x+ 0.24y=0.18*200
8x/100+24/100=18/100*200
8x+24y=3600
8(x+3y)=3600
x+3y=450
Hence the equations which shows the amount of solutions will be x+y=200 and x+3y=450.
Learn more about equations at https://brainly.com/question/2972832
#SPJ2
Patios can be made by mixing cubic meters of ash, stone, and wood chips in the ratio 5:7:3. How much stone is needed to make 45 cubic meters of patio?
Answers
Answer:
21 m^3
Step-by-step explanation:
5 + 7 + 3 = 15
The ratio of stone to the total is
7:15
If the total needed is 45 m^3, then we multiply both parts of the ratio by 3.
7 * 3 : 15 * 3
21:45
Answer: 21 m^3
write the equation of a horizontal ellipse with a major axis of 18, and minor axis of 10, and a center at (-4, 5).
Answers
See the attached picture
[tex]\bold{\text{Answer:}\quad \dfrac{(x+4)^2}{81}+\dfrac{(y-5)^2}{25}=1}[/tex]
Step-by-step explanation:
A "horizontal" ellipse means that the x-radius is bigger than the y-radius. Thus, x is the major axis and y is the minor axis.
The equation of an ellipse is: [tex]\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1[/tex] where
(h, k) is the center of the ellipsea is the radius on the x-axisb is the radius on the y-axisIt is given that the center is at (-4, 5) --> h = -4, k = 5
It is given that the major axis has a length of 18 --> x-radius = 9
It is given that the minor axis has a length of 10 --> y-radius = 5
Input those values into the equation of an ellipse to get:
[tex]\dfrac{(x-(-4))^2}{9^2}+\dfrac{(y-5)^2}{5^2}=1[/tex]
Simplify to get:
[tex]\dfrac{(x+4)^2}{81}+\dfrac{(y-5)^2}{25}=1[/tex]
find the slope of the line (-3,-2) (1,6)
Answers
Answer:
slope = 2
Step-by-step explanation:
Calculate the slope m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (- 3, - 2) and (x₂, y₂ ) = (1, 6)
m = [tex]\frac{6-(-2)}{1-(-3)}[/tex] = [tex]\frac{6+2}{1+3}[/tex] = [tex]\frac{8}{4}[/tex] = 2
Kevin made a business trip of 312.5 miles. He averages 60 mph for the first part of the trip and 55 mph for the second part. If the trip took a 5.5 hours how long did he travel at each rate
Answers
Answer:
Kevin travels for 2 hours at 60 mph and 3.5 hours at 55 mph
Step-by-step explanation:
Let's say that Kevin spends x hours going 60 mph and y hours going 55 mph. We can say that the sum of the two parts is 5.5, so x+y = 5.5 . Next, he goes 60 miles per hour for the first part of the trip, so for each hour he goes 60 mph, he travels 60 miles. We can then denote 60 * x as the distance traveled during the first part of his trip as he goes 60 mph for x hours. Similarly, 55 * y denotes the distance Kevin travels during the second part of his trip. His total distance is thus 60 * x + 55 * y = 312.5 miles
We have
x + y = 5.5
60 * x + 55 * y = 312.5
One way we can solve this is to solve for y in the first equation and plug that into the second. Subtracting x from both sides in the first equation, we get
y = 5.5 - x
Plugging that into the second equation, we get
60 * x + 55 * (5.5-x) = 312.5
60 * x + 55 * 5.5 - 55x = 312.5
5x +302.5 = 312.5
subtract 302.5 from both sides to isolate the x and its coefficient
5x = 10
divide both sides by 5 to solve for x
x = 2
y = 5.5 - x = 5.5 - 2 = 3.5
Therefore, Kevin travels for 2 hours at 60 mph and 3.5 hours at 55 mph
Point A is at (2, -8) and point C is at (-4, 7).
Find the coordinates of point B on AC such that the ratio of AB to BC is 2:1.
Answers
Answer:
(-2, 2)
Step-by-step explanation:
Given:
Point A is at (2, -8) and point C is at (-4, 7)Difference of coordinates:
Δx = 2 - (-4) = 6Δy = - 8 - 7 = - 15The ratio of AB to AC is 2:1. So:
AB = 2*AC/3 and BC = AC/3Then coordinates of point B should be 2/3 from the point A:
x = 2- 6*2/3 = 2 - 4 = -2y = - 8 - (-15)*2/3 = -8 + 10 = 2So point B has coordinates of (-2, 2)
An Image point after a 180° rotation Is Z(3, 7). What were the coordinates of the pre-Image point?
Z-3-7)
2173)
27-3)
21-7-3)
Answers
Answer:
Does the answer help you?
I NEED ALGEBRA HELP! Can you solve a system of equations using the substitution by solving one equation for x or y and then using the substitution method? x + 6y = 6 and 7x - 5y = -5
Answers
Answer:
let x be y
NOW,
X+6Y=6
Y+6Y=6
7Y=6
Y=0.87
An urn contains two blue balls (denoted B1 and B2) and three white balls (denoted W1, W2, and W3). One ball is drawn, its color is recorded, and it is replaced in the urn. Then another ball is drawn and its color is recorded.
a. Let B1 W2 denote the outcome that the first ball drawn is B1 and the second ball drawn is W2. Because the first ball is replaced before the second ball is drawn, the outcomes of the experiment are equally likely. List all 25 possible outcomes of the experiment.
b. Consider the event that the first ball that is drawn is blue. List all outcomes in the event. What is the probability of the event?
c. Consider the event that only white balls are drawn. List all outcomes in the event. What is the probability of the event?
Answers
Answer:
(a) Shown below.
(b) The probability that the first ball drawn is blue is 0.40.
(c) The probability that only white balls are drawn is 0.36.
Step-by-step explanation:
The balls in the urn are as follows:
Blue balls: B₁ and B₂
White balls: W₁, W₂ and W₃
It is provided that two balls are drawn from the urn, with replacement, and their color is recorded.
(a)
The possible outcomes of selecting two balls are as follows:
B₁B₁ B₂B₁ W₁B₁ W₂B₁ W₃B₁
B₁B₂ B₂B₂ W₁B₂ W₂B₂ W₃B₂
B₁W₁ B₂W₁ W₁W₁ W₂W₁ W₃W₁
B₁W₂ B₂W₂ W₁W₂ W₂W₂ W₃W₂
B₁W₃ B₂W₃ W₁W₃ W₂W₃ W₃W₃
There are a total of N = 25 possible outcomes.
(b)
The sample space for selecting a blue ball first is:
S = {B₁B₁, B₁B₂, B₁W₁, B₁W₂, B₁W₃, B₂B₁, B₂B₂, B₂W₁, B₂W₂, B₂W₃}
n (S) = 10
Compute the probability that the first ball drawn is blue as follows:
[tex]P(\text{First ball is Blue})=\frac{n(S)}{N}=\frac{10}{25}=0.40[/tex]
Thus, the probability that the first ball drawn is blue is 0.40.
(c)
The sample space for selecting only white balls is:
X = {W₁W₁, W₂W₁, W₃W₁, W₁W₂, W₂W₂, W₃W₂, W₁W₃, W₂W₃, W₃W₃}
n (X) = 9
Compute the probability that only white balls are drawn as follows:
[tex]P(\text{Only White balls})=\frac{n(X)}{N}=\frac{9}{25}=0.36[/tex]
Thus, the probability that only white balls are drawn is 0.36.
Urgent please I neeed some help !!!!!!!!!!!!!!!!!!!!!! URGENT 20 point bonus
Answers
Answer:
113.1
Step-by-step explanation:
use the formula to solve for volume
2. A dolphin leaps 5 feet in the air at the same time an orca whale dives 12 feet below the water. How far apart are the whale and the dolphin?
Answers
Hope this helps
Hi Mathies, Help with this vectors excercise pls. Givan that OA (segment) = 13x+7y , OB = 5x+12y and CO = -15+12y write down each of the following vectors in its simplest form a) BA = 8x+ 5y (l got it, i ve done it) b) AC= ?? i cant find vector AC thanks in advance
Answers
Answer:
AC = 2x-19yStep-by-step explanation:
Given vector the following vector equations OA = 13x+7y , OB = 5x+12y and CO = -15x+12y, the following expression is true about vector OA, OB and OC;
OA+OB = CO (CO is the resultant since its is moving in the opposite direction compare to OA and OB)
Also BO+OA = BA and AO+OC = AC
If OB = 5x+12y, then BO = -(5x+12y)
BO = -5x-12y (BO = -OB)
Since BO+OA = BA
BA = -5x-12y + 13x+7y
BA = -5x+13x-12y+7y
BA = 8x-5y
Similarly AO+OC = AC
Since AO = -OA and OC = -CO
-OA-CO = AC
AC = -(13x+7y)-(-15x+12y)
AC = -13x-7y+15x-12y
AC = -13x+15x-7y-12y
AC = 2x-19y
Find the work done by the force field F(x,y,z)=6xi+6yj+6k on a particle that moves along the helix r(t)=3 cos(t)i+3sin(t)j+2 tk,0≤t≤2π.
Answers
Answer:
the work done by the force field = 24 π
Step-by-step explanation:
From the information given:
r(t) = 3 cos (t)i + 3 sin (t) j + 2 tk
= xi + yj + zk
∴
x = 3 cos (t)
y = 3 sin (t)
z = 2t
dr = (-3 sin (t)i + 3 cos (t) j + 2 k ) dt
Also F(x,y,z) = 6xi + 6yj + 6k
∴ F(t) = 18 cos (t) i + 18 sin (t) j +6 k
Workdone = 0 to 2π ∫ F(t) dr
[tex]\mathbf{= \int \limits ^{2 \pi} _{0} (18 cos (t) i + 18 sin (t) j +6k)(-3 sin (t)i+3cos (t) j +2k)\ dt}[/tex]
[tex]\mathbf{= \int \limits ^{2 \pi} _{0} (-54 \ cos (t).sin(t) + 54 \ sin (t).cos (t) + 12 ) \ dt}[/tex]
[tex]\mathbf{= \int \limits ^{2 \pi} _{0} 12 \ dt}[/tex]
[tex]\mathbf{= 12 \times 2 \pi}[/tex]
= 24 π
Find the surface area of the regular pyramid shown in the accompanying diagram. If necessary, express your answer in simplest radical form.
Answers
Answer:
The area of the pyramid is 360 unit²
Step-by-step explanation:
Given
Base Edge, a = 10
Height, h = 12
Required
Determine the surface area
The surface area of a regular pyramid is calculated as thus;
[tex]A = a^2 + 2a\sqrt{\frac{a^2}{4} + h^2}[/tex]
Substitute values for a and h
[tex]A = 10^2 + 2 * 10 * \sqrt{\frac{10^2}{4} + 12^2}[/tex]
Evaluate all squares
[tex]A = 100 + 2 * 10 * \sqrt{\frac{100}{4} + 144}[/tex]
[tex]A = 100 + 2 * 10 * \sqrt{25 + 144}[/tex]
[tex]A = 100 + 2 * 10 * \sqrt{169}[/tex]
Take positive square root of 169
[tex]A = 100 + 2 * 10 * 13[/tex]
[tex]A = 100 + 260[/tex]
[tex]A = 360[/tex]
Hence, the area of the pyramid is 360 unit²
Answer:
B.) 360 units2
Step-by-step explanation:
I got it correct on founders education
Match each function name with its equation.
Answers
Answer:
a. Quadratic--[tex]y=x^{2}[/tex]
b. Absolute Value--[tex]y=|x|[/tex]
c. Linear--[tex]y=x[/tex]
d. Reciprocal Squared--[tex]y=\frac{1}{x^{2} }[/tex]
e. Cubic--[tex]y=x^{3}[/tex]
f. Square Root--[tex]y=\sqrt{x}[/tex]
g. Reciprocal--[tex]y=\frac{1}{x}[/tex]
h. Cube root--[tex]y=\sqrt[3]{x}[/tex]
Answer:
Step-by-step explanation:
y=[tex]x^{2}[/tex] is quadratic
y=x is an absolute value
y= |x| os linear
y= [tex]\frac{1}{x}[/tex] is reciprocal
y= [tex]x^{3}[/tex] is cubic
y= [tex]\sqrt{x}[/tex] is square root
y= [tex]\frac{1}{x^{2} }[/tex] is reciprocal squared
y= [tex]\sqrt[3]{x}[/tex] is cube root
A study of 200 computer service firms revealed these incomes after taxes: Income After Taxes Number of Firms Under $1 million 102 $1 million up to $20 million 61 $20 million or more 37 What is the probability that a particular firm selected has $1 million or more in income after taxes
Answers
Answer:
The probability that a particular firm selected has $1 million or more in income after taxes is 49%.
Step-by-step explanation:
We are given a study of 200 computer service firms revealed these incomes after taxes below;
Income After Taxes Number of Firms
Under $1 million 102
$1 million up to $20 million 61
$20 million or more 37
Total 200
Now, the probability that a particular firm selected has $1 million or more in income after taxes is given by;
Total number of firms = 102 + 61 + 37 = 200
Number of firms having $1 million or more in income after taxes = 61 + 37 = 98 {here under $1 million data is not include}
So, the required probability = [tex]\frac{\text{Firms with \$1 million or more in income after taxes}}{\text{Total number of firms}}[/tex]
= [tex]\frac{98}{200}[/tex]
= 0.49 or 49%
The probability that a particular firm selected has $1 million or more in income after taxes is 0.49 or 49%.
What is probability?Probability means possibility. It deals with the occurrence of a random event. The value of probability can only be from 0 to 1. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.
A study of 200 computer service firms revealed these incomes after taxes:
Income After Taxes Number of Firms Under
$1 million 102
$1 million up to $20 million 61
$20 million or more 37.
Then the total event will be
Total event = 102 + 37 +61 = 200
The probability that a particular firm selected has $1 million or more in income after taxes will be
Favorable event = 37 + 61 = 98
Then the probability will be
[tex]\rm P = \dfrac{98}{200} \\\\P = 0.49 \ or \ 49 \%[/tex]
More about the probability link is given below.
https://brainly.com/question/795909
The price of a technology stock was $ 9.56 yesterday. Today, the price rose to $ 9.69 . Find the percentage increase. Round your answer to the nearest tenth of a percent.
Answers
Answer and Step-by-Step explanation:
% increase = 100 x [(new price) - (original price)] / (original price)] = 100 (9.67 - 9.56) / 9.56
% increase ≅ 1.2% (to the nearest tenth)
You know only the given information about
the measures of the angles of a triangle. Find the probability that the triangle is equiangular.
39. Each is a multiple of 12.
Answers
Since they are multiples if 12
The possibilities are
12, 12, 156
12,24,144
12,36,132
12,48,120
12,60,108
12,72,96
12,84,84
24,24,132
24,36,120
24,48,108
24,60,96
24,72,84
36,36,108
36,48,96
36,60,84
36,72,72
48,48,84
48,60,72
60,60,60
Hence the probability is 1/19 or 0.0526
If the sides of a square measure 9.3 units the.find the length of the diagonal
Answers
Answer:
Approximately 13.1521 units.
Step-by-step explanation:
To find the diagonal, we can use the Pythagorean Theorem.
Since the figure is a square, all four sides are equivalent. A square also has four right angles. Therefore, we can use the Pythagorean Theorem to find the diagonal d. Therefore:
[tex]a^2+b^2=c^2[/tex]
Substitute 9.3 for a and b, and let c equal d:
[tex](9.3)^2+(9.3)^2=d^2[/tex]
Instead of squaring, add the like-terms:
[tex]2(9.3)^2=d^2[/tex]
Take the square root of both sides:
[tex]d=\sqrt{2(9.3)^2}[/tex]
Expand:
[tex]d=\sqrt{2}\cdot\sqrt{(9.3)^2}[/tex]
The right cancels:
[tex]d=\sqrt2\cdot(9.3)\\d=9.3\sqrt2\\d\approx13.1521\text{ units}[/tex]
4. Find the area of the polygon.
Answers
Answer:
20
Step-by-step explanation:
Emile is a long-distance trucker. In one week he drives miles from his home in Fort Lauderdale, FL, to Benson, NC. He then drives miles to Barstow, CA, and continues driving miles to Bakersfield, CA. From there, Emile drives miles to Seattle, WA. Estimate the total distance Emile travels by first rounding each distance to the nearest hundred. Do not put units in your answer.
Answers
Answer:
Estimated total distance is 1,900 miles.
Step-by-step explanation:
We begin by adding each distance traveled by Emile:
1. Fort Lauderdale, FL, to Benson, NC = 748 miles
2. Barstow, CA, to Bakersfield, CA = 130 miles
3. Bakersfield, CA. to Seattle, WA = 1030 miles
Total miles = 1,908.
Therefore, in one week Emile's total distance to the nearest hundred is 1,900.
Note: the distances where gotten via Google Map.
Jack is going to the fair. The fair charges $10 to enter and $0.25 per ticket. How much will be spent by Jack? t = tickets 0.25 + 10 10 + 0.250. pless help
Answers
Answer:
10 + .25t
Step-by-step explanation:
The total amount spent is equal to the amount to get in plus the cost of the tickets times the number of tickets
cost = 10 + .25t
is perpendicular to . How many 90° angles are formed by the intersection?
Answers
Answer:
if a is perpendicular to b then four 90 degree angles are formed
Step-by-step explanation:
if a line is perpendicular to another that means that it forms a 90 degree angle on all of the angles
Answer:
Four
That is the right answer for Edmentum and Plato users
Like and Rate!
How many fluid ounces are there in 4pints?
Answers
Answer: 64 fluid ounces
Step-by-step explanation:
1 pint=16 fl oz
16*4=64
help me besties pls and have a good Bestie
Answers
Answer:
6
Step-by-step explanation:
Area = length x width
Input the numbers:
Area = 78
length = 13
78 = 13 x width
width = 78 / 13
width = 6
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
Answer:
Width=6m
Step-by-step explanation:
Area=78m^2
Length=13m
width=?(let width be x)
[AREA OF RECTANGLE=length× width]
78=13×x
78=13x
×=6
how many years will it take for a sum of money to double at 10% compounded annually
Answers
Answer:
t=7.27 years
Step-by-step explanation:
Let the money be p and t will be the number of years that will be needed for the money to get double.
ATQ, 2p=p*(1+0.1)^t
2=(1.1)^t
log(2)/log(1.1)=t, t=7.27
Consider the differential equation:
2y'' + ty' − 2y = 14, y(0) = y'(0) = 0.
In some instances, the Laplace transform can be used to solve linear differential equations with variable monomial coefficients.
If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . ,then
ℒ{tnf(t)} = (-1)^n d^n/ds^n F(s)
to reduce the given differential equation to a linear first-order DE in the transformed function Y(s) = ℒ{y(t)}.
Requried:
a. Sovle the first order DE for Y(s).
b. Find find y(t)= ℒ^-1 {Y(s)}
Answers
(a) Take the Laplace transform of both sides:
[tex]2y''(t)+ty'(t)-2y(t)=14[/tex]
[tex]\implies 2(s^2Y(s)-sy(0)-y'(0))-(Y(s)+sY'(s))-2Y(s)=\dfrac{14}s[/tex]
where the transform of [tex]ty'(t)[/tex] comes from
[tex]L[ty'(t)]=-(L[y'(t)])'=-(sY(s)-y(0))'=-Y(s)-sY'(s)[/tex]
This yields the linear ODE,
[tex]-sY'(s)+(2s^2-3)Y(s)=\dfrac{14}s[/tex]
Divides both sides by [tex]-s[/tex]:
[tex]Y'(s)+\dfrac{3-2s^2}sY(s)=-\dfrac{14}{s^2}[/tex]
Find the integrating factor:
[tex]\displaystyle\int\frac{3-2s^2}s\,\mathrm ds=3\ln|s|-s^2+C[/tex]
Multiply both sides of the ODE by [tex]e^{3\ln|s|-s^2}=s^3e^{-s^2}[/tex]:
[tex]s^3e^{-s^2}Y'(s)+(3s^2-2s^4)e^{-s^2}Y(s)=-14se^{-s^2}[/tex]
The left side condenses into the derivative of a product:
[tex]\left(s^3e^{-s^2}Y(s)\right)'=-14se^{-s^2}[/tex]
Integrate both sides and solve for [tex]Y(s)[/tex]:
[tex]s^3e^{-s^2}Y(s)=7e^{-s^2}+C[/tex]
[tex]Y(s)=\dfrac{7+Ce^{s^2}}{s^3}[/tex]
(b) Taking the inverse transform of both sides gives
[tex]y(t)=\dfrac{7t^2}2+C\,L^{-1}\left[\dfrac{e^{s^2}}{s^3}\right][/tex]
I don't know whether the remaining inverse transform can be resolved, but using the principle of superposition, we know that [tex]\frac{7t^2}2[/tex] is one solution to the original ODE.
[tex]y(t)=\dfrac{7t^2}2\implies y'(t)=7t\implies y''(t)=7[/tex]
Substitute these into the ODE to see everything checks out:
[tex]2\cdot7+t\cdot7t-2\cdot\dfrac{7t^2}2=14[/tex]