a) Find a solution to the initial value problem u_t + u_x = 0 ; u (1,x) = x/1 + x^2
b) Solve the initial value problem u_t + 2u_x = 1 ; u(0, x) = e^(-x^2)

The position of 3125 is 25 in the sequence that is the first one with a value greater than 3000.

\( \bf{Let \: a_n \: be \: the \: {n}^{th} \: term},\)

\( \bf{then \: a_n \: = {5n}^{2} }\)

Let m be the term which is the first one with a value greater than 3000.

\( \bf \longrightarrow{a_m > 3000} \\ \\\bf \longrightarrow{ 5 {m}^{2} > 3000} \\ \\\bf \longrightarrow{ {m}^{2} > \frac{3000}{5}} \\ \\\bf \longrightarrow{ {m}^{2} > 600 }\\ \\ \bf \longrightarrow{m > \sqrt{600} } \\ \\\bf \longrightarrow{ m > 24.494 }\\ \\\bf \longrightarrow{ m = 25}\)

Thus \(\bf{25}^{th}\) term is

\( \bf \longrightarrow{a_{25} = 5 {(25)}^{2} } \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf = 5(625\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \bf= 3125\)

(6, 5) a solution to the linear equation y = 2/3x + 1, while (9, 8) is not.

An ordered pair is a solution to a linear equation if the linear equation is true when we substitute the values into the linear equation.

Another method for determining the solution is to plot the graph of the linear equation and see if the point lies on the line of the linear equation or not. If a point lies on the line of the linear equation, it is a solution, if it is not on the line, it is therefore not a solution to the linear equation.

Plug in the point (6, 5) into the linear equation, y = 2/3x + 1:

5 = 2/3(6) + 1

5 = 5 [the linear equation is true, therefore, (6, 5) is a solution]

Plug in the point (9, 8) into the linear equation, y = 2/3x + 1:

8 = 2/3(9) + 1

8 = 7 [not true, it is not a solution]

The graph below, which represents the linear equation y = 2/3x + 1, shows that (6, 5) is a solution while (9, 8) is not.

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Answer:

y=14-8x

x= -2 + 3y/9x

Step-by-step explanation:

8x+y=14

isolate y so you get:

y=14-8x

-9x+3y=18

isolate x so you get:

-9x=18-3y

divide on both sides to get x by itself

-9x/-9x  18-3y/-9x

x= -2 + 3y/9x

Hope this helps :)

Ans:-

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Solution:-

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The value of x is 2 !

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hope it helps:)

The value of solution of expression is,

⇒ x = 2

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is,

⇒ 7 + 4x - 9 = 6

Now, We can simplify as;

⇒ 4x - 2 = 6

⇒ 4x = 4 + 2

⇒ 4x = 8

⇒ x = 2

Thus, The value of solution of expression is,

⇒ x = 2

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Answer: B -_-

Step-by-step explanation:

0 0=2

Answer:

Graph.

y=12

x>=0

x<=12

Answer:

Period is for a complete oscillation

\(period = 4\pi\)

Answer:

12 ft

Step-by-step explanation:

Hi there!

1) Find the length of the other leg in the right triangle

There are two ways we can go about this. The first way is to use trigonomic ratios:

We're given one of the legs and we're trying to solve for the other, so we can use the tangent ratio:

\(tan\theta=\frac{opp}{adj}\)

Let x be equal to the length of the other leg. Plug in the known values:

\(tan45=\frac{x}{7}\\7*tan45=x\\x=7\)

The other way to find this is to observe the angles. Because we're given that two of the angles are 45 and 90 degrees, we know that the third angle is 45 degrees. This makes this an isosceles triangle, meaning the other leg is 7 ft long.

2) Determine the height of the hoop

Add 7 ft and 5 ft (the person's height):

7+5=12 ft

I hope this helps!

\(\textit{arc's length}\\\\ s=\cfrac{r\theta \pi }{180}~~ \begin{cases} r=radius\\ \theta =\stackrel{in~degrees}{angle}\\[-0.5em] \hrulefill\\ r=4\\ \theta =50 \end{cases}\implies s=\cfrac{(4)(50)\pi }{180}\implies s=\cfrac{10\pi }{9}~mm\)

Answer:

12 sales

Step-by-step explanation:

a and b

65 per sale maximum commision 1300 so no base

40 per sale base 300

Set up equation:

\(65x=40x+300\)

Solve:

\(25x=300\)

\(x=12\)

So they must have had 12 sales.

Hope this helps :D

The objective of this research is to collect data on a specific component of the off-campus student population.The amount of time they spend travelling to courses each week.

a) The population for this study is all off-campus students at this college, and the sample is a random sample of 45 off-campus students who are polled. The sample is chosen at random to be representative of the entire population.

b) The data being gathered is the amount of time the surveyed students spend each week travelling to school. Because it reflects a numerical measurement, this is quantitative data. This information will be used to calculate the average commuting time for off-campus students and to identify any possible concerns or areas for improvement in commuting.

c) This is a descriptive research, since it seeks to characterise and summarise the features of the off-campus student population in terms of commuting time. The study's purpose is to better understand off-campus students' commuting patterns and gather ideas into how to enhance their commuting experience.

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Answer:

y=8

x=5

8=5+3

Step-by-step explanation:

Simple math, just switched around with variables. Anytime!

a = 13/2

a = 6.5 or 6 1/2

The equation of the horizontal asymptote of P isy = 22818.18.  This means that the insect population will approach a maximum value of approximately 22818 insects, no matter how many months have passed since they were introduced. The unit of measure for this value is insects per month.

a. The equation of the horizontal asymptote can be found by taking the limit of P(t) as t approaches infinity.

lim P(t) as t approaches infinity = lim (50+251)/(2+0.011t) as t approaches infinity

Since the denominator grows without bound as t approaches infinity, the limit of P(t) approaches 251/0.011 = 22818.18. Therefore, the equation of the horizontal asymptote is:

y = 22818.18

b. The horizontal asymptote tells us the maximum population that the protected area can sustain for this type of insect. In other words, as time goes on, the insect population will approach a maximum value of approximately 22818 insects. The unit of measure for the asymptote value is insects.

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The required ratio of  is (a + c) : c 11:8.

Given that a:b=1:4 and b:c=3:2.

We can simplify the ratio b:c by multiplying both sides by 4 to get b:c=12:8=3:2.

To find the ratio (a+c):c, we need to express a and c in terms of b. From the first ratio, we have \(a=\frac14 b$\). From the second ratio, we have \(c=\frac{2}{3}b$\). Substituting these values into the expression (a+c):c, we get:

\($$(a+c):c = \left(\frac{1}{4}b + \frac{2}{3}b\right):\frac{2}{3}b$$\)

Simplifying the expression inside the parentheses, we get:

\($\frac{1}{4}b + \frac{2}{3}b = \frac{3b}{12} + \frac{8b}{12} = \frac{11b}{12}$$\)

Therefore, the ratio \($(a+c):c$\) is:

\($(a+c):c = \frac{11b}{12}:\frac{2}{3}b = 11:8$$\)

Hence, the required ratio is 11:8$.

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a) Each side of the square is 9 cm long. b) The perimeter of the shape is 18 cm.

Perimeter refers to the total length of the boundary or outer edge of a two-dimensional shape.

a) The perimeter of a square is the sum of the lengths of all its sides. Since all sides of a square are equal, we can divide the perimeter by the number of sides to find the length of one side:

Perimeter of square = 4 x Length of one side

36 cm = 4 x Length of one side

Divide both sides by 4:

Length of one side = 36 cm / 4 = 9 cm

Therefore, each side of the square is 9 cm long.

b) To find the perimeter of the shape made by cutting an equilateral triangle from a rectangle, we need to add up the lengths of all its sides.

The length of the rectangle is equal to the length of the base of the equilateral triangle plus the length of one of its sides. Since the equilateral triangle has all sides equal, we can find its length by dividing the length of the rectangle by 2:

Length of equilateral triangle = Length of rectangle / 2

Length of equilateral triangle = 7.2 cm / 2 = 3.6 cm

The perimeter of the equilateral triangle is three times its side length:

Perimeter of equilateral triangle = 3 x Length of one side

Perimeter of equilateral triangle = 3 x 3.6 cm = 10.8 cm

The perimeter of the shape is the sum of the length of all its sides, which is the sum of the length of the rectangle and the perimeter of the equilateral triangle:

Perimeter of shape = Length of rectangle + Perimeter of equilateral triangle

Perimeter of shape = 7.2 cm + 10.8 cm = 18 cm

Therefore, the perimeter of the shape is 18 cm.

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a) If a square has a perimeter of 36 cm, how long is each side of the square?

b) The following shape is made by cutting an equilateral triangle from a rectangle. Find its perimeter.

7.2 cm

13 cm

18 cm

23 cm

The total area of the wrapping paper is 72.5 in².

In the given figure (attached below), we have two shapes one is a triangle and the other one is a rectangle. To find the total area of the wrapping paper we have to add the area of the rectangle part and the area of the trianglular part.

Total area = Area of the rectangular part + area of the triangular part.

Area of the rectangular part = length x breadth

from the below figure, length = 15 in

                                     breadth = 4 in

So, area of the rectangular part = 15 in x 4 in = 60 in²

Similarly, area of the triangular part = 1/2 x base x height

from the below figure, base of the triangle = 15 in -10 in = 5 in

                                      height of the triangle = 9 in - 4 in = 5 in

So, area of the triangular part = 1/2 x 5 in x 5in =  12.5 in²

Now, the total area of the wrapping paper = area of the rectangular part + area of the triangular part = 60 in² + 12.5 in² = 72.5 in².

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Answer:

4

Step-by-step explanation:

To find the degree of a polynomial, identify the term with the greatest exponent. The exponent of that term is the degree of the polynomial.

In the given polynomial 5x⁴+4x²+2x+1​, the term with the greatest exponent is 5x⁴. This term has an exponent of 4, so therefore, the degree of the polynomial is 4.

I hope this helps!

The graph of the exponential function is on the image at the end.

Here we want to graph the function:

f(x) = 0.5*(4)ˣ

To graph this (or any function) we can find some points on the function, and to do so, we need to evaluate it.

when x = 0:

f(0) =   0.5*(4)⁰ = 0.5

Then the point is (0, 0.5)

when x = 1

f(1) =   0.5*(4)¹ = 2

So we have the point (1, 2)

if x = 2

f(2) =   0.5*(4)² = 8

So we have the point (2, 8)

Now we can graph these points and connect them with a general exponential curve.

The graph of the exponential function is shown below.

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p(t) = 6 + C * e^(-8t) is the general solution to the given differential equation.

The given differential equation is dp/dt = 48 − 8p. To find its general solution, we need to separate the variables and integrate both sides.

dp/(48 - 8p) = dt

Integrating both sides, we get:

-1/8 ln|48 - 8p| = t + C

where C is the constant of integration.

To solve for p, we can isolate the absolute value:

ln|48 - 8p| = -8t - 8C

Taking the exponential of both sides, we get:

|48 - 8p| = e^(-8t-8C)

Since the absolute value of a quantity can be either positive or negative, we need to consider both cases:

48 - 8p = e^(-8t-8C)  or  8p - 48 = e^(-8t-8C)

Simplifying each equation, we get:

p = 6 - (1/8) e^(-8t-8C)  or  p = 6 + (1/8) e^(-8t-8C)

These are the general solutions to the given differential equation.
To find the general solution to the given differential equation dp/dt = 48 - 8p, we'll first recognize that it's a first-order linear differential equation. We can rewrite it as:

dp/dt + 8p = 48

Next, we'll find the integrating factor, which is e^(∫8dt) = e^(8t). Now, we'll multiply both sides of the equation by the integrating factor:

e^(8t)(dp/dt + 8p) = 48e^(8t)

Now the left side of the equation is the derivative of the product of p and the integrating factor:

d(p * e^(8t))/dt = 48e^(8t)

Integrate both sides with respect to t:

∫d(p * e^(8t))/dt dt = ∫48e^(8t) dt

This simplifies to:

p * e^(8t) = 6e^(8t) + C

Now, we'll solve for p by dividing both sides by e^(8t):

p(t) = 6 + C * e^(-8t)

This is the general solution to the given differential equation.

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Answer:

The relation is a FUNCTION

Step-by-step explanation:

as you can see the value of domain or the x there's no repeating number at all.

Answer:

Not a function

Step-by-step explanation:

(0,3),(3,5) and (0,3),(-6,-2) hit (3,5),(-2,6)

Answer:

1 whole mile

Step-by-step explanation:

1/4 x 4

Answer:

1 1/4 miles  = 5/4 miles

Step-by-step explanation:

Answer:

\(4km\)

Step-by-step explanation:

In order to find the answer to this question you must simply divide.

\(5m=8km\)

\(2.5m=4km\)

\(8\div2=4\)

\(5=2.5+2.5\)

\(=4km\)

Hope this helps

The equation does not have a solution.

Option D is the correct answer.

We have,

To find the solution of the equation (15z + 6.8) = 5(4 + 0.7z) using the graph, we need to determine the x-value (or z-value in this case) where the two lines intersect.

Based on the given options, we can observe that the equation does not have a solution.

This can be determined by looking at the graph of the two equations.

If we graph the equations y = (15z + 6.8) and y = 5(4 + 0.7z),

We will see that the lines do not intersect.

This means there is no common value of z that satisfies both equations simultaneously.

Therefore,

The equation does not have a solution.

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If f(x, y, z) = xyzi + yj + zk, the divergence of the curl of the vector field f is zero.

To find the divergence of the curl of the vector field f(x,y,z) = xyzi + yj + zk, we first need to compute the curl of f, which is given by:

curl f = (∇ × f) = ∂(zk)/∂y - ∂(yj)/∂z + (∂(yj)/∂x - ∂(xyzi)/∂y)k + (∂(xyzi)/∂z - ∂(zk)/∂x)j + (∂(zk)/∂x - ∂(yj)/∂y) i

Simplifying this expression, we get:

curl f = (-z)i + xj + yk

Next, we need to find the divergence of the curl of f, which is given by:

div(curl f) = ∇ · (∇ × f) = ∂(∂(zk)/∂y - ∂(yj)/∂z)/∂x + ∂(∂(yj)/∂x - ∂(xyzi)/∂y)/∂y + ∂(∂(xyzi)/∂z - ∂(zk)/∂x)/∂z

Substituting the values from the curl f expression, we get:

div(curl f) = ∂(-z)/∂x + ∂(x)/∂y + ∂(y)/∂z

Simplifying this expression, we get:

div(curl f) = 0

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y = e is the equation of the tangent line to the given curve at the specified point , y = \(e^{x}\) / x , (1, e)

Through the coordinate geometry formal of point-slope form, the equation of tangent and normal can be calculated.

The tangent has the equation (y - y1) = m(x - x1), and a normal travelling through this point and perpendicular to the tangent has the equation (y - y1) = -1/m (x - x1).

thus , y' =  \(\frac{e^{x}-xe^{x} }{x^{2} }\)

y'(1) = 0 = m

Our slope is indicated by the horizontal line.

y−\(y_{1}\)=m(x−\(x_{1}\))

Our line will simply be our y-coordinate because our slope(m) is 0:

e

Therefore:

The tangent line's equation is y=e.

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Answer:

5 and 11

Step-by-step explanation:

Make equations:

x+y = 16

3x-4 = y

Substitute 3x-4 as y in x+y = 16

You get x + 3x - 4 = 16

Simplify into 4x - 4 = 16

Add 4 on both sides

you get 4x = 20

Divide each side by 4,

You get x = 5. Plug 5 back into x+y = 16

Subtract 5 from both sides, and you get y = 11

The appropriate family of distributions to approximate the probability of rolling a 6 on a fair die is the discrete uniform distribution, which assumes equal probabilities for each outcome. In this case, the probability of rolling a 6 would be approximately 1/6 based on the assumption of fairness.

For the problem of approximating the probability of rolling a 6 on a fair die, an appropriate family of distributions to consider is the discrete uniform distribution.

The discrete uniform distribution is commonly used to model situations where each outcome has an equal probability of occurring. In the case of rolling a fair die, the die has six equally likely outcomes (numbers 1 to 6).

Each outcome has a probability of 1/6 of occurring, making the discrete uniform distribution a suitable choice.

By assuming a discrete uniform distribution, we can assign equal probabilities to each outcome (1/6 for rolling a 6) and approximate the probability of rolling a 6 based on the assumption of fairness.

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The purchase price in dollars of the television that has been marked up 25% of the retail price of $4,500 would be $5,625.

To find the purchase price of the television in dollars, follow these steps:

1. Determine the markup amount: 25% of the retail price ($4,500).
2. Subtract the markup amount from the retail price to find the purchase price.

Step 1: Calculate the markup amount:
25% of $4,500 = 0.25 × 4,500 = $1,125

Step 2: Subtract the markup amount from the retail price:
$4,500 (retail price) - $1,125 (markup) = $3,375

The purchase price of the television in dollars is $3,375.

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