Particular solution of the differential equation that satisfies the initial condition f'(s) = 10s - 4s³ , f(3) = 5 is f(s) = 5s² - s⁴ + 41.
We determine the particular solution to the differential equation. We do this by integrating the given differential equation and then applying the given initial condition. The power rule should be used for the given expression.
[tex]\int\ {x^{a} } \, dx[/tex] = [tex]\frac{x^{a+1} }{a+1} }[/tex]
we have given that,
f'(s) = 10s - 4s³
f(s) = ∫ f'(s)ds
= ∫(10s - 4s³)ds
= [tex]\frac{10s^{2} }{2} - \frac{4s^{2} }{4}+ C[/tex]
f(s) = 5s² - s⁴ + C
Now we will apply the initial condition,
f(3) = 5
⇒5(3)² - (3)⁴ + C =5
-36 + C = 5
C =41
Therefore the solution is expressed as,
f(s) = 5s² - s⁴ + 41
This is the particular solution.
Given question is incomplete. Complete question is given as:
find the particular solution of the differential equation that satisfies the initial condition(s). f '(s) = 10s - 4s³, f(3) = 5.
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(10)
Write the equation
of the line in
Slope-Intercept
Form.
Answer: y= -3x -1
Step-by-step explanation: y=mx+b
The Y-intercept is at (0,-1), so b is -1
The slope is rise/run, and the graph is down 3, right 1, so the slope is -3/1
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Question
A square based pyramid has a height of 6 feet and the edge length of the base is 2 feet. A second square based pyramid is 2 feet taller and the edge length of the base is 0.5 feet greater than the first pyramid. How much greater is the volume of the second pyramid than the volume of the first pyramid? Round your answer to the nearest hundredth.
please help
A pyramid is a solid shape with a specified shaped base. The required answer is 8.67 cubic feet.
A pyramid is a solid shape with a specified shaped base. While a square based pyramid is a type of pyramid which has its base to be a square.
Volume of a square based pyramid = [tex]\frac{1}{3}[/tex] x base area x height
Also, are of a square = length x length
So that,
i. For the first pyramid;
volume = [tex]\frac{1}{3}[/tex] x (2 x 2) x 6
= 8 cubic feet
The volume of the first pyramid is 8 cubic feet.
ii. For the second pyramid, we have;
height = 8 feet
base side = 2.5 feet
So that.
Volume = [tex]\frac{1}{3}[/tex] x (2.5 x 2.5) x 8
= 16.6667
Volume = 16.67 cubic meters
The volume of the second pyramid is 16.67 cubic meters.
Therefore, the volume of the second pyramid is greater than the volume of the first pyramid by;
volume of the second pyramid - volume of the first pyramid = 16.67 - 8.0
= 8.67 cubic feet
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For the conditional write the converse , inverse , and contrapositive: Walking in front of a moving car is dangerous to your health .
The converse of the conditional statement is “If dangerous to your health then walking in front of a moving car”.
The inverse of the conditional statement is “If not dangerous to your health then not walking in front of a moving car”.
The contrapositive of the conditional statement is “if not walking in front of a moving car then not dangerous to your health."
What is the conditional statement?The conditional statement is defined as the two fundamental components that make up a conditional statement Hypothesis (if) and Conclusion (then).
The given conditional statement is "walking in front of a moving car is dangerous to your health."
The converse of the conditional statement is “If dangerous to your health then walking in front of a moving car”.
The inverse of the conditional statement is “If not dangerous to your health then not walking in front of a moving car”.
The contrapositive of the conditional statement is “if not walking in front of a moving car then not dangerous to your health."
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Find the area of the triangle below. Be sure to include the correct unit in your answer. 8 cm 9 cm 18 cm
Answer:
648
Step-by-step explanation:
An easy way to do this is to multiply all sides, then divide by 2.
8 × 9 × 18 = 1296
Now divide that by 2:
1296 ÷ 2 = 648
The area of the triangle is 648.
For the high school basketball game, it costs $12 for every 3 tickets.
Complete the table below showing the cost and the number of tickets.
Based on the unit rate of the table, the missing values that shows the cost and the number of tickets are:
$32 and 9 tickets.
How to Complete the Table Showing Cost and Number of Tickets?To calculate the missing values in the table shown in the attachment below, we have to find the unit rate, which is the cost of 1 ticket.
Given that 3 tickets will cost $12, therefore:
Unit rate = 12 / 3 = $4 per ticket.
Using this rate, find the missing values as explained below.
For 8 tickets, we will have the following cost:
8 × $4 = $32
For $36, we will have the following tickets:
$36/$4 = 9 tickets
Thus, the missing values that will complete the table will be:
$32 and 9 tickets.
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What is the location of point G, which partitions the directed line segment from D to F into a 5:4 ratio? –1 0 2 3
The location of G is (3, 0).
Given question in that we have a number line.
A number line goes from negative 5 to positive 10.
Point D is at negative 2 and point F is at positive 7.
A line is drawn from point D to point F.
So we can write coordinate of point D as (-2,0) and point F as (7,0).
line segment from D to F divides as point G as in ratio of 5:4
Consider two points P(x1, y1) and Q(x2, y2). We have to find the coordinates of the point R which divides PQ in the ratio m : n
x = [tex]m*x2+n*x1/(m + n)[/tex]
[tex]y = m*y2+n*y1/(m + n)[/tex]
so here in our question m =5 n=4
x1=-2 x2=7 y1=0 y2=0
x = 5*7+4*(-2)/9
x = 3
y = 5*0+4*0/9 = 0
So the location of G is (3, 0)
Given Question is incomplete complete question here.
What is the location of point G, which partitions the directed line segment from D to F into a 5:4 ratio? A number line goes from negative 5 to positive 10. Point D is at negative 2 and point F is at positive 7. A line is drawn from point D to point F.
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Find the scale factor
Please help me!
Answer:
[tex]\frac{7}{5}[/tex] or 7:5
Step-by-step explanation:
You are going from the smaller triangle to the larger.
Let x = the scale factor. Find two corresponding sides that you know both lengths.
50x = 70 Divide both sides by 50
x = [tex]\frac{70}{50}[/tex] Divide both the numerator and denominator by 10
[tex]\frac{7}{5}[/tex]
Find the slope: m=
Find the y-intercept: b=
Equation: y=
Answer:
y = - [tex]\frac{5}{4}[/tex] x + 5
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + b ( m is the slope and b the y- intercept )
calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (0, 5) and (x₂, y₂ ) = (4, 0) ← 2 points on the line
m = [tex]\frac{0-5}{4-0}[/tex] = [tex]\frac{-5}{4}[/tex] = - [tex]\frac{5}{4}[/tex]
the line crosses the y- axis at (0, 5 ) ⇒ b = 5
y = - [tex]\frac{5}{4}[/tex] x + 5 ← equation of line
A marching band director needs to divide 44 members into two groups. He wants the ratio of band members in group 1 to band members in group 2 to be 1 to 3. How many band members will be in group 2?.
If the ratio of band members in group 1 to band members in group 2 is 1 to 3, then for every 3 band members in group 2, there will be 1 band member in group 1.Therefore, group 2 will have 11 band members.
What is ratio and proportion?Comparing two numbers or quantities results in a ratio. Usually, it is expressed in the form "a:b" or "a/b," with a and b standing for the two quantities under comparison. Contrarily, a percentage is an equation that states that two ratios are equal. Alternatively said, a proportion is a claim that one ratio is equal to another.
For example, if the ratio of apples to oranges in a basket is 2:3, this implies that there are three oranges per each two apples. We may alternatively state that the ratio of apples to total number of fruits in the basket is 2:5, because 2 + 3 = 5. The two ratios (2:3 and 2:5) are proportionate in this instance. Also, ratio of band members in group 1 to band members in group 2 to be 1 to 3
How to solve?
If the director wants to divide 44 band members into two groups, with this ratio, then the number of band members in group 2 will be 44 / (1 + 3) = 44 / 4 = 11 band members.
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a paddle wheel on a boat is 4 feet in diameter. the fins along the outer edge travel at a speed of 1.3 feet per second. how long does it take the paddle wheel to complete 100 full revolutions ? round to the nearest second.
_ minutes and _ seconds.
Answer: To find out how long it takes the paddle wheel to complete 100 full revolutions, we need to calculate the circumference of the paddle wheel and divide that by the speed at which the fins along the outer edge are traveling.
The formula for the circumference of a circle is:
C = 2πr
where C is the circumference and r is the radius of the circle.
The radius of the paddle wheel is 2 feet (the diameter of 4 feet divided by 2), so the circumference is:
C = 2π * 2 = 8π feet
To find out how long it takes the paddle wheel to complete 100 full revolutions, we divide the number of feet traveled in one revolution (the circumference of 8π feet) by the speed at which the fins are traveling (1.3 feet per second):
100 * 8π feet / 1.3 feet per second = 615.38 seconds
Rounded to the nearest second, it takes approximately 615 seconds for the paddle wheel to complete 100 full revolutions.
In the following diagram, AD || BC and AB || DC.
Prove angle A is congruent to angle C
1. Line L contains points (4, -1) and (4,9). Point P has coordinates (1,6)
The distance between the point and the line is 3 units.
Find the equation for line L
Begin by finding the slope of the line through the given points, (4,-1) and (4,9) ; its intercept:
m = (y₂ - y₁) / (x₂ - x₁)
= (9 - (-1) ) / 4 - 4
= 10 / 0
= Undefined
Because the slope is undefined, then line L is a vertical line which has an x-intercept of (4,0):
x=4
Because, line L is vertical, then the line w perpendicular to it is a horizontal line that passes through (1,6)(1,6) so its equation is y=6. Hence, L and w intersect at (4,6) which we will denote as Q.
The perpendicular distance is the distance from point P(1,6) to point Q(4,6) which is simply the absolute value of the difference of the x-coordinates:
d = | 1 - 4 |
= |-3|
= 3
Hence, The distance between the point and the line is 3 units.
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In the coordinate plane, the point X(-4,-2) is translated to the point X'(-6,-5) . Under the same translation, the points Y(0,0) and Z(-2,2) are translated to Y' and Z' , respectively. What are the coordinates of Y' and Z' ?
Under the same translation as X and X' the coordinates of Y' = (-2, -4) and Z' = (-4, -1)
What is translation of a point in coordinate system?
A translation is a transformation that occurs when a figure is moved from one location to another location without changing its size, shape or orientation. In the coordinate plane we can draw the translation if we know the direction and how far the figure should be moved.
According to the given question:
It is given about the points X (-4, -2) and X' (-6, -5).
We see that the translation is moving -2 units horizontally to the left and -3 units vertically down.
Therefore, the point Y(0,0) is translated to Y'(0-2,-1-3) = Y'(-2,-4)
and the point Z(-2,2) is translated to Z'(-2-2,2-3) = Y'(-4,-1).
Hence, under the same translation Y' = (-2, -4) and Z' = (-4, -1)
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-2x2 + 8x = 10 using the Quadratic Formula. Simplify if possible.
The required solutions are x = 2-i and x = 2+i to the given quadratic function.
What is a quadratic function?
The quadratic function is defined as a function containing the highest power of a variable is two.
We have been given that quadratic function as
-2x² + 8x - 10 = 0
Compare the given function to the standard quadratic function f(x) = ax² + b x + c.
We get a = -2, b = 8 and c = -10
Using the quadratic formula to solve the above quadratic function
[tex]x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
[tex]x=\dfrac{-8\pm \sqrt{8^2-4\left(-2\right)\left(-10\right)}}{2\left(-2\right)}[/tex]
[tex]x=\dfrac{-8\pm \:4i}{2\left(-2\right)}[/tex]
[tex]x=\dfrac{-8+4i}{2\left(-2\right)},\:x=\dfrac{-8-4i}{2\left(-2\right)}[/tex]
x = 2-i, x = 2+i
Thus, the required solutions are x = 2-i and x = 2+i to the given quadratic function.
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Nora placed 12 cans in each box.How many boxes will she need if she has 240 cans?
Answer:
20 boxes
Step-by-step explanation:
12 cans = 1 box
240 = x
apply rule of three
(240/12) x 1 = x
20 = x
Answer:
She will need 20 boxes
Step-by-step explanation:
Take the total number of cans and divide by the number of cans per box
240/12
20
She will need 20 boxes.
50 POINTS PLEASE HELP QUICK
Which expression is equivalent to 2 and three tenths raised to the fourth power divided by nine tenths raised to the fifth power, all raised to the fourth power?
A: 2 and three tenths raised to the eighth power divided by nine tenths raised to the ninth power
B: 2 and three tenths raised to the sixteenth power divided by nine tenths raised to the twentieth power
C: 1.4 raised to the forth power
D: 1.4 raised to the ninth power
Answer:
The answer is B
Step-by-step explanation:
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A semicircle is divided into two circular sectors of which one is 4/5 of the other. The radius of the circle measures 18cm. Calculate the length of the two arcs and the area of the two circular sectors
The length of the two arcs and the area of the two circular sectors can be calculated as follows:
Since the semicircle is divided into two circular sectors, each sector has an angle of 180/2 = 90 degrees. Therefore, the smaller circular sector has an angle of 90 * (4/5) = 72 degrees, and the larger circular sector has an angle of 90 - 72 = 18 degrees.
The length of the arc of a circle is equal to the circumference of the circle multiplied by the ratio of the central angle of the arc to 360 degrees. Therefore, the length of the smaller arc is 18 * pi * (72/360) = 8pi cm, and the length of the larger arc is 18 * pi * (18/360) = 2pi cm.
The area of a circular sector is equal to the area of the circle multiplied by the ratio of the central angle of the sector to 360 degrees. Therefore, the area of the smaller circular sector is 18^2 * pi * (72/360) = 288pi cm^2, and the area of the larger circular sector is 18^2 * pi * (18/360) = 72pi cm^2.
Therefore, the length of the two arcs is 8pi cm and 2pi cm, and the area of the two circular sectors is 288pi cm^2 and 72pi cm^2.
You currently have $40 saved up. You get $5 every
time you shovel the driveway. Your friend has no
money saved up and gets $10 every time she
shovels her driveway (hers is twice as big as
yours!) After how many driveways will you have
the same amount of money as your friend?
2) Solve: 115 − 10 = −12 − 5(2 − 6)
Answer:
False
Step-by-step explanation:
Simplify 115 − 10 : 105Subtract the numbers 115 − 10 = 105
2.Simplify 12 − 5(2 − 6) : 8
105=8
this shows that the sides are not equal so it is false
Find the three consecutive integers such that the smallest and three times the largest is 330. What is the smallest integer?
The smallest integer out of the three integers is 109
What is a number system?A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set in a consistent manner using digits or other symbols. In different numeral systems, the same sequence of symbols can represent different numbers.
A zero, a positive natural number, or a negative integer with a minus sign is an integer. The negative numbers are the additive inverses of their positive counterparts.
Three consecutive integers:
x, x+1, x+2
The sum of the integers,
x+(x+1)+(x+2) = 330
3x + 3 = 330
3x= 327
x= 327/3
x= 109
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What is the slope of the line that passes through the points (10, 3)(10,3) and (7, 0)(7,0)? write your answer in simplest form.
The slope of the line that passes through the points (10, 3)(10,3) and (7, 0)(7,0)= 1.
What is slope?A line's steepness and direction are measured by the line's slope.
Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all.
Any two different points on a line can be used to calculate the slope of any line.
A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively.
Two locations located on a straight line can be used to determine a line's slope. We can use the slope of line formula if we know the two spots' coordinates. These two points' coordinates should be-
P1 = (x1, y1) and P2 = (x2, y2)
According to our question-
The slope m using the slope formula
m = y2-y1/x2-x1
with (x₁, y₁ ) = (10, 3 ) and (x₂, y₂ ) = (7, 0 )
-3/-3 = 1
Hence, the slope is = 1.
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solve for b in the trapezoid.
180 - 57 = 123 ( ALLIED ANGLE )
You have rented a moving truck to move into your first apartment. ABC truck has interior dimensions of 168 inches times 96 inches times 77 inches. The dimensions of your storage boxes are 24 inches times 13 inches times 28 inches. Determine how many boxes can fit into the truck.
what is this in Algebra tiles (x-1) (x+3)
please help me.
The area of a rectangle with the dimensions (x-1) and (x+3) is x²+2x-3 square units.
Given that, the dimensions of a rectangle (x-1) and (x+3).
What is the area of a rectangle?The area occupied by a rectangle within its boundary is called the area of the rectangle. The formula to find the area of a rectangle is Area = Length × Breadth.
Here, the area of a rectangle
= (x+3)×(x-1)
= x²+3x-x-3
= x²+2x-3
Hence, the area of a rectangle with the dimensions (x-1) and (x+3) is x²+2x-3 square units.
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"Your question is incomplete, probably the complete question/missing part is:"
Find the area of a rectangle with the dimensions (x-1) and (x+3).
Alex has $600 in her checking account. She wants to spend a part of this money on a computer.
She wants to have at least $250 left in her checking account after buying the computer. Write an
inequality that can be used to find t, the amount of money in dollars that Alex can spend on the
computer, and solve fort.
Amount of computer can be 350 or more than 350.
What do you mean by inequality?
Inequalities specify the relationship between two values that are not equal. Equal does not imply inequality. Typically, we use the "not equal sign ()" to indicate that two values are not equal. But several inequalities are utilised to compare the numbers, whether it is less than or higher than.
Inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions.
It is given that Alex has $600 in her checking account and She wants to have at least $250 left in her checking account after buying the computer.
Let the amount of computer be t
According to the question , inequality become:
600 - t ≥ 250
600 - 250 ≥ t
350 ≥ t
Therefore, amount of computer can be 350 or more than 350.
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Paige can sort her toys in 6 fewer hours that Madison can. When they work together, it
takes them only 4 hours to sort the toys. How long would it take for each of them to sort
the toys alone?
Let X be the number of hours it takes Paige to sort the toys alone.
It takes Madison X + 6 hours to sort the toys alone.
Together, it takes them X + (X + 6) = 4 hours to sort the toys.
Combining like terms, we get 2X = 4
Dividing both sides by 2, we get X = 2
Therefore, it takes Paige 2 hours to sort the toys alone.
It takes Madison 2 + 6 = 8 hours to sort the toys alone.
solve for 2x² = 80?
Answer:
x=2√10,−2√10
Step-by-step explanation:
Joseph and Raquel are building a scale model of the Alamo Mission for their Texas history project using a scale in which 2 inches represents 30 feet. The enclosed area of the Alamo Mission was 480 feet long and 160 wide.
What should the length of the enclosed area of the scale model be in inches?
Th length of the enclosed area of the scale model in inches is 32 inches.
How to find the length of the enclosed area of the scale model in inches?Joseph and Raquel are building a scale model of the Alamo Mission for their Texas history project using a scale in which 2 inches represents 30 feet.
The enclosed area of the Alamo Mission was 480 feet long and 160 feet wide.
The length of the enclosed area of the scale model in inches can be calculated as follows:
30 feet = 2 inches
480 feet = ?
cross multiply
length of the enclosed area = 480 × 2 / 30
length of the enclosed area = 960 / 30
length of the enclosed area = 32 inches
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Write an equation in slope-intercept form of the line that is perpendicular to the one given and passes through the given point. (4,8); y=-2x - 4
answer:
step by step explanation:
(4,8); y = 2x — 4 = ?
TO FIND THE ANSWER TO YOUR QUESTION:help meeeeeeeeeeeeeeeeeeeeeeeeeee pleaseeeeeeeeeeeeeeeeeeeeee!!!!!
Answer:
a) 23.1 million
b) year 2013
Step-by-step explanation:
Given a population in millions is modeled by A = 23.1e^(0.0152t) for t years after 2000, you want to know the population in year 2000, and the year in which the population is 28.3 million.
a) Initial populationWhen t=0, the exponential factor is 1. The population is given by the coefficient of that factor: 23.1
The population in 2000 was 23.1 millions.
b) Year of 28.3MWe can solve for t to find the year the population is 28.3 million:
28.3 = 23.1e^(0.0152t)
28.3/23.1 = e^(0.0152t) . . . . .divide by 23.1
ln(28.3/23.1) = 0.0152t . . . . . take natural logs
t ≈ 13.36
The population will reach 28.3 million in the year 2013.
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