The Laplace transform method is applied to solve the differential equation y' + 6y = e^4t with the initial condition y(0)=2. After transforming, simplifying, and solving for Y(s), we use inverse Laplace transform to find the solution y(t) in the time domain.
Explanation:Laplace transform is a powerful tool in the field of mathematics used for solving differential equations. To solve the given initial value problem y' + 6y = e^4t ; y(0)=2, we can start by taking the Laplace transform of both sides of the equation.
The Laplace transform of y' is sY(s) - y(0) and the Laplace transform of y is Y(s). Therefore, the Laplace transform of y' + 6y gives sY(s) - y(0) + 6Y(s). Given that y(0)=2, this simplifies to sY(s) + 6Y(s) - 2.
On the right-hand side, the Laplace transform of e^4t is 1/(s-4). Thus, we have the equation sY(s) + 6Y(s) - 2 = 1/(s-4).
By solving for Y(s), we can find the inverse Laplace transform to get the solution y(t) in the time domain.
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Using the rule of thirds to evaluate the composition of a well-crafted painting, the eye should be attracted to A. one or more points of interest. B. the top of the painting. C. the lighter values in the painting. D. the midpoint of the composition.
Answer: A: one or more points of interest.
Step-by-step explanation:
The rule says that an image should be divided into 9 equal squares by two horizontal and two vertical lines and that the important objects of the image should be placed between the lines (in the centers of the squares) or in the intersections of the lines.
In this way, the eye is dragged to the different points that the artist wants the viewer to focus on the image, then the correct option is option A: one or more points of interest.
Consider a triangle where A=30 degrees, a=1.5cm and b=2cm
How many possible triangles have the above properties? Write either 0 or 1 or 2.
I don't understand this question at all. ...?
With the given conditions of angle A being 30 degrees, side a being 1.5 cm, and side b being 2 cm in a triangle, only one possible triangle satisfies these properties according to the Law of Sines.
Explanation:The question relates to determining the number of possible triangles given certain conditions. Specifically, it refers to a triangle with an angle A of 30 degrees, a side a of 1.5 cm opposite to A, and another side b of 2 cm. To decide the number of possible triangles that fit these criteria, we make use of the Law of Sines. This law states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
In the given scenario, since side a (1.5 cm), opposite to angle A (30 degrees), is shorter than side b (2 cm), it is possible that two different triangles could exist, often referred to as the ambiguous case in the Law of Sines. However, the fact that side a is shorter than side b coupled with angle A being acute (30 degrees) leaves us with only one possible triangle. This is because the side opposite the smallest angle (A) cannot form a valid triangle if it's longer than the other given side. Therefore, there is just one solution to this problem.
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Find the point on the terminal side of θ = 3pi/4 that has a y coordinate of 1.
Answer:
B
Step-by-step explanation:
hope this helps
Jordan’s job is located 67 miles from his home. His car’s gas mileage is 21 miles to the gallon. Approximately how many gallons of gasoline does Jordan need each day?
I think the answer is 6.381 gallons or 3.2?
The number of gallons of gasoline Jordan need each day is:
3.190476 gallons.
Step-by-step explanation:It is given that:
Jordan’s job is located 67 miles from his home.
His car’s gas mileage is 21 miles to the gallon.
i.e. for 21 miles he need 1 gallon of gas.
This means in order to cover 1 mile he needs 1/21 gallon of gas.
Hence,
In order to cover 67 miles he needs 67/21 gallon of gas
i.e.
for 67 miles he needs 3.190476 gallons of gas.
A 16-inch cheese pizza at Fine Dough Pizza costs $10.00 and each additional topping is $1.50. The same size pizza at Johnson's Pizza costs $8.00 and each additional topping is $2.00. After how many toppings do the pizzas cost the same? Which pizza costs more with additional toppings?
To find the number of toppings at which the pizzas cost the same, we can set the total cost expressions for both pizzas equal to each other and solve for 'x'. After 4 additional toppings, the pizzas will cost the same. With 5 additional toppings, Johnson's Pizza costs more.
Explanation:To find the point at which the pizzas cost the same, we need to determine the number of toppings that will make the total cost equal for both pizzas. Let's denote the number of additional toppings as 'x', and the cost of each additional topping at Fine Dough Pizza as $1.50, and at Johnson's Pizza as $2.00. The total cost for Fine Dough Pizza will be $10.00 + $1.50x, and the total cost for Johnson's Pizza will be $8.00 + $2.00x. To find the number of toppings at which the pizzas cost the same, we can set the two total cost expressions equal to each other and solve for 'x':
$10.00 + $1.50x = $8.00 + $2.00x
To solve the equation, we can subtract $1.50x from both sides:
$10.00 = $8.00 + $0.50x
Next, we subtract $8.00 from both sides:
$2.00 = $0.50x
To isolate 'x', we divide both sides by $0.50:
x = 4
Therefore, after 4 additional toppings, the pizzas will cost the same.
To determine which pizza costs more with additional toppings, we can compare the total cost expressions for both pizzas for a specific number of toppings. For example, let's consider 5 additional toppings:
Total cost at Fine Dough Pizza = $10.00 + $1.50(5) = $10.00 + $7.50 = $17.50
Total cost at Johnson's Pizza = $8.00 + $2.00(5) = $8.00 + $10.00 = $18.00
Therefore, with 5 additional toppings, Johnson's Pizza costs more.
what is 72/60 in simplest form
In 5 years, Dad will be three times as old as his daughter Jill will be then. If the sum of their present ages is 50, how old are they now?
If x + 5 is Jill's age five years from now, which of the following equations could be used to solve the problem?
a.3x + 5 = 55 - x
b.3x + 15 = 55 - x
c.3x + 15 = 45 - x ...?
For the function f(x) = x2 -12, find (f o f-1)(4)
Solve for b. ab + c = d b = a + c/d b = a/(c - d) b = (d - c)/a
Answer: im sorry if im wrong but i think the answer is 0.3
Step-by-step explanation:
The price of a gallon of milk is 3.70 5 dollars what is the price that is rounded to the nearest dollar
Write the number21,989.098 in expanded form and word form
15 is what percent of 56
Find the circumference of the circle if the square has an area of 144 m2. Give your answer to the nearest tenth. Use 3.14 form
The circumference of the circle is found by first determining the side length of the square, then using that to calculate the diameter and radius of the circle. Using 3.14 as an approximation for pi, the circumference is calculated to be 37.7 meters when rounded to the nearest tenth.
To find the circumference of the circle when the square has an area of 144 m², we first determine the side length of the square using the formula for the area of a square, A = a², where A is the area and a is the side length.
A = 144m²
a = 12m
Since the area is 144 m², the side length of the square is "12 m (since 12 x 12 = 144).
Since the circle fits perfectly within the square, the diameter D of the circle is equal to the side length of the square. Therefore, D = 12 m. The radius r is half the diameter, so in this case, r = 6 m.
The formula for the circumference C of a circle is C = 2πr.
C = 2 x 3.14 x 6 m = 37.68 m
The circumference is 37.7 m.
Write the equation for 5x + 2y = 3 in slope-intercept form.
Answer:
Slope-Intercept Form states that in the equation of a straight line y=mx+b the slope is the number represented by m that is multiplied on the x, and b is the y intercept i.,e the point where line cut the vertical y-axis.
Given equation: 5x+2y=3 ......[1]
To convert this equation into slope intercept form;
Subtraction Property of equality states that you subtract the same number from both sides of an equation.
Subtract 5x from both sides in equation [1];
[tex]5x+2y-5x=3-5x[/tex]
Simplify:
[tex]2y=3-5x[/tex] or we can write this as;
[tex]2y=-5x+3[/tex] ......[2]
Division property of equality states that you divide the same number from both sides of an equation:
Divide by 2 from both sides of an equation in [2]
[tex]\frac{2y}{2} =\frac{-5x+3}{2}[/tex]
On simplify:
[tex]y=-\frac{5}{2}x+\frac{3}{2}[/tex]
Therefore, the given equation in slope intercept form is: [tex]y=-\frac{5}{2}x +\frac{3}{2}[/tex]
If 100 envelopes cost 70 cents, how much would 250 cost?
To find out how much 250 envelopes would cost, set up a proportion using the given information that 100 envelopes cost 70 cents. Cross-multiply and solve for the unknown cost, which is $1.75.
Explanation:To find out how much 250 envelopes would cost, you can use the concept of proportions. Since 100 envelopes cost 70 cents, you can set up a proportion to find the cost of 250 envelopes: 100/70 = 250/x, where 'x' represents the unknown cost. Cross-multiplying, you get 100x = 70 * 250. Dividing both sides by 100, you find that x = 175.
Therefore, 250 envelopes would cost 175 cents.
In decimal form, 175 cents is equal to $1.75.
If 7 units represent 98 liters find the value of 15 units
Final answer:
To find the value of 15 units when 7 units represent 98 liters, set up a proportion, cross multiply, and divide to find the value of 15 units, which is 210 liters.
Explanation:
To find the value of 15 units when 7 units represent 98 liters:
Set up a proportion: 7 units = 98 liters and determine the value of 15 units.Cross multiply to solve for the unknown value: 7 units * x liters = 15 units * 98 liters.Divide to find the value of 15 units: x = (15 * 98) / 7 = 210 liters. Before 1995, three-digit area codes for the united states had the following restrictions:
(a) neither 0 nor 1 could be used as the first digit.
(b) 0 or 1 had to be used for the second digit.
(c) there were no restrictions on the third digit.
How many different area codes were possible?
A convention center rents a meeting room for $250 per day. There is a $30 one-time fee to make a reservation.
What function equation represents the total cost to rent a meeting room for d days?
A. c(d) = 250d + 30
B. c(d) = 30d − 250
C. c(d) = 30d + 250
D. c(d) = 250d − 30
Alberto is making a budget. He owes a cell phone bill of $60 per month for 23 more months, plus the fees for any extra usage such as text messaging or extra minutes. Which of the following is the correct category for this expense?
a.
Essential (flexible) expense
b.
Emergency fund
c.
Essential (fixed) expense
d.
Non-essential expense
Answer:
a. Essential (flexible) expense
Step-by-step explanation:
Most people consider a cell phone an essential expense, since they are used in power outages or emergencies while traveling.
This is a flexible expense, since there are extra fees for extra usage.
The phone bill expense is an Essential (flexible) expense so option (A) will be correct.
What is the reasoning?The procedure of utilizing logical reasoning to analyze a condition to determine the best problem-solving approach for a particular issue, then using this approach to create and explain a resolution.
In another word, reasoning is a tricky but interesting problem in mathematics that required relations of variables to solve.
The mobile cost is an essential cost because it is can be used in an emergency situation or it is very useful for calling massaging or video calling.
Since the plan for mobile recharge varies higher or lower according to benefits so it will be a flexible expense.
Hence "The phone bill expense is an Essential (flexible) expense".
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solve.
5m + 7/2 = -2m + 5/2
What are the solutions of 4x2 + x = −3?
Select one:
Answer:
The correct answer is c.
Step-by-step explanation:
Given the equation
[tex]ax^{2}+bx+c=0[/tex]
The quadratic equation solves the values of ''x'' which satisfy the original equation (In the equation I replace ± by the symbol |):
[tex]\frac{-b|\sqrt{b^{2}-4ac}}{2a}[/tex]
The solutions of [tex]4x^{2}+x=-3[/tex] are the same that the solutions of [tex]4x^{2}+x+3=0[/tex]
With [tex]a=4\\b=1\\c=3[/tex] we replace this in the quadratic equation ⇒
[tex]\frac{-1|\sqrt{(1)^{2}-4(4)(3)}}{(2)(4)}[/tex] ⇒
[tex]\frac{-1|\sqrt{1-48}}{8}[/tex] ⇒
[tex]\frac{-1|\sqrt{-47}}{8}[/tex]
Given that [tex]i^{2}=-1[/tex] (imaginary numbers property)
[tex]\frac{-1|\sqrt{47i^{2}}}{8}=\frac{-1|i\sqrt{47}}{8}[/tex]
The solutions are [tex]\frac{-1+i\sqrt{47}}{8}[/tex]
and [tex]\frac{-1-i\sqrt{47}}{8}[/tex]
The correct answer is c.
Answer: C
Step-by-step explanation: I just took the test and got it right <3
How to factorise 4b squared + 19b
The diagram below shows a square inside a regular octagon. The apothem of the octagon is 13.28 units. To the nearest square unit, what is the area of the shaded region?
Area of shaded region is 463 square units.
What is area of octagon?Area of octagon = [tex]\frac{1}{2}[/tex] × [tex]8[/tex] × base × height
What is apothem of octagon?The apothem of any polygon is equal to the line that connects the center of the polygon with one of its sides perpendicularly. Using the apothem, we can calculate the area of the polygons in an easier way.
Given
Apothem of the octagon = 13.28 units
Base of octagon = 11 units
Area of octagon = [tex]\frac{1}{2}[/tex] × [tex]8[/tex] × base × height
= [tex]\frac{1}{2}[/tex] × [tex]11[/tex] × [tex]13.28[/tex] × [tex]8[/tex]
= 584.32
Area of square = side × side
= 11 × 11
= 121
Area of shaded region = Area of octagon - Area of square
= 84.32 - 121
= 463.32
≅ 463 square units.
Hence, area of shaded region is 463 square units.
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If t = -2π/3, evaluate (if possible) sin t, cos t, tan t
Mark wants to create a scale drawing of the poster shown below:
A rectangle is shown. The length of the rectangle is labeled 4 centimeters. The width of the rectangle is labeled 10 centimeters.
What are the dimensions of the poster at one-half its current size?
2 cm by 10 cm
2 cm by 5 cm
8 cm by 20 cm
8 cm by 5 cm
Factor completely: 5ab(x + 6) − 4(x + 6) ...?
Answer:
D is your answer :)
Step-by-step explanation:
Simplify as an Exponent: 10 •10• 3 • 3 • 3 =
what is 5√28 + √63 in simplest radical form
The simplest radical form of the expression 5√28 + √63 will be 29√7.
How do convert the exponent into the simplest radical form?If the power of the variable "a" is b/c.
Then the simplest radical form of the expression will be
[tex]\rm \left ( a \right )^{\rm \frac{b}{c}} = \rm \sqrt[\rm c]{\rm a^b}[/tex]
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.
The expression is given below.
⇒ 5√28 + √63
Simplify the expression, then we have
⇒ 5√28 + √63
⇒ 5√(4 x 7) + √(9 x 7)
⇒ 5 x 4√7 + 9√7
⇒ 20√7 + 9√7
⇒ 29√7
The simplest radical form of the expression 5√28 + √63 will be 29√7.
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A field test for a new exam was given to randomly selected seniors. The exams were graded, and the sample mean and sample standard deviation were calculated. Based on the results, the exam creator claims that on the same exam, nine times out of ten, seniors will have an average score within 6% of 80%.
Is the confidence interval at 90%, 95%, or 99%? What is the margin of error? Calculate the confidence interval and explain what it means in terms of the situation.
Answer:
Margin of error: 6%; Confidence Interval Percentage: 90%; Confidence Interval: 74% and 86%
Step-by-step explanation:
Margin of error: 6% -- It's already given in the question.
Confidence interval is at 90%. 9/10=.90
The exam creator is saying that they are 90% confident that the seniors test scores will be between 74% and 86%.
0.8 - 0.06 = 0.74
0.8 + 0.06 = 0.86
The confidence interval is (0.74, 0.86)
Find all the cube roots of -125i in polar form. Convert all results to a+bi form. ...?
Final answer:
To find all cube roots of -125i in polar form, convert them to a+bi form by expressing -125i in polar form, finding the cube roots, and converting them to a+bi form.
Explanation:
To find all the cube roots of -125i in polar form and convert them to a+bi form, follow these steps:
Express -125i in polar form: -125i = 125∠-90°.
Find the cube roots by taking the cube roots of the magnitude and adding multiples of 360°/3 to the argument.
The cube roots of -125i in a+bi form are 5∠-30°, 5∠90°, and 5∠210°, which correspond to -2.5-4.33i, 0+5i, and 2.5-4.33i respectively.