Definition 7.1.1 laplace transform let f be a function defined for t ≥ 0. then the integral {f(t)} = ∞ e−stf(t) dt 0 is said to be the laplace transform of f, provided that the integral converges. to find {f(t)}. f(t) = cos t, 0 ≤ t < π 0, t ≥ π

To answer this item, we let x be the speed of the boat in still water. The speed of the current, we represent as y.

When the boat travels upstream or against the current, the speed is equal to x – y and x + y if it travels downstream or along with the current.

The time it takes for the an object to travel a certain distance is calculated by dividing the distance by the speed.

First Travel:    35 / (x – y)   + 55 / (x + y) = 12

Second travel: 30 / (x – y)   + 44 / (x + y) = 10

Let us multiply the two equations with the (x-y)(x+y)

This will give us,

              35(x + y) + 55(x – y) = 12(x-y)(x+y)

              30(x + y) + 44(x – y) = 10(x-y)(x+y)

Using dummy variables:

Let a = x + y and b be x – y

                35a + 55b = 12ab

                30a + 44b = 10ab

From the first equation,

                     b = 35a/(12a – 55)

Substituting to the second equation,

                30a + 44(35a/(12a – 55)) = 10a(35a/(12a-55))

The value of a is 11.

              b = 35(11)/(12(11) – 55))

              b = 5

Putting back the equations,

        x + y = 11

       x – y = 5

Adding up the equations give us,

  2x = 16

    x = 8 km/hr

The value of x, the speed of the boat in still water, is 8 km/hr. 

speed of the stream = 3 km/hr

and speed of boat in still water= 8 km/hr

Let s be the speed of the boat upstream

and s' be the speed of the boat downstream.

We know that:

[tex]Time=\dfrac{distance}{speed}[/tex]

Hence, we get:

  [tex]\dfrac{35}{s}+\dfrac{55}{s'}=12[/tex]

and

[tex]\dfrac{30}{s}+\dfrac{44}{s'}=10[/tex]

Now, let

[tex]\dfrac{1}{s}=a\ and\ \dfrac{1}{s'}=b[/tex]

Hence, we have:

[tex]35a+55b=12--------------(1)\\\\\\and\\\\\\30a+44b=10--------------(2)[/tex]

on multiplying equation (1) by 4 and equation (2) by 5 and subtract equation (1) from (2) we get:

[tex]a=\dfrac{1}{5}[/tex]

and by putting value of a in (2) we get:

[tex]b=\dfrac{1}{11}[/tex]

Hence, speed of boat in upstream= 5 km/hr

and speed of boat in downstream= 11 km/hr

and we know that:

speed of boat in upstream=speed of boat in still water(x)-speed of stream(y)

and speed of boat in downstream=speed of boat in still water(x)+speed of stream(y)

Hence, we get:

[tex]x-y=5\\\\\\and\\\\\\x+y=11[/tex]

Hence, on solving the equation we get:

[tex]x=8[/tex]

and y=3

Hence, we get:

speed of the stream = 3 km/hr

and speed of boat in still water= 8 km/hr

multiply 6400 x 6%

6% = 0.06

6400 x 0.06 = 384

 her commission was $384

Answer:

The commission amount of the salesperson is $384.

Step-by-step explanation:

A salesperson sold a total of $6,400.00.

The rate of commission is 6% or 0.06. Commissions are based on sales. These are some percentage of the sales amount.

So, here the amount will be = [tex]0.06\times6400=384[/tex] dollars

So, the commission amount of the salesperson is $384.

There was 3 answers.

Answer one is It has 4 zeros and at most 3 relative maximums or minimums.
Answer two is It is a translation of the percent function 4 units to the right.
Answer three is Both ends of the graph of the function go up.

:)

it is a transition of the parent function 4 units to the right, it has 4 zeros and at most 3 relative maximums and minimums, both ends of the graph of the function go up this is for apex

The value of s is greater or equal to 22.

It shows a relationship between two numbers or two expressions.

There are commonly used four inequalities:

Less than = <

Greater than = >

Less than and equal = ≤

Greater than and equal = ≥

We have,

2s + 5 greater than or equal to 49.

This can be written as,

(2s + 5) ≥ 49

Solve for s.

2s + 5 ≥ 49

2s ≥ 49 - 5

2s ≥ 44

s ≥ 22

Thus,

s is greater than or equal to 22.

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Answer:

4 of the 20 radio contest winners selected to try for the grand prize : C

5 friends waiting in line at the movies: C

6 students selected at random to attend a presentation: P

Final answer:

The scenarios illustrate combination when order does not matter (selecting contest winners and student attendees) and permutation when order matters (friends in line). The correct sequence is combination, permutation, combination.

Explanation:

In the context of the scenarios provided, we need to differentiate whether the situations are examples of combinations or permutations. A permutation is an arrangement of objects where order matters, while a combination is a selection of objects where order does not matter.

Thus, the correct answer to the sequence of scenarios is: combination, permutation, combination, which correlates with option c.

John's net pay is calculated by subtracting deductions for Medicare, Social Security, state and federal taxes, and insurance from his gross pay of $500. The total deductions amount to $168.25, resulting in a net pay of $331.75.

Calculation of John's Net Pay

To calculate John's net pay, we need to subtract all the deductions from his gross pay. Since his gross pay is $500, we will apply the following deductions:

Medicare tax: 1.45% of $500 = $7.25

Social Security tax: 6.2% of $500 = $31.00

State tax: 2% of $500 = $10.00

Federal income tax: 20% of $500 = $100.00

Insurance deduction: $20.00

Add up all deductions: $7.25 (Medicare) + $31.00 (Social Security) + $10.00 (State Tax) + $100.00 (Federal Tax) + $20.00 (Insurance) = $168.25

John's net pay is therefore calculated by subtracting the total deductions from his gross pay: $500.00 - $168.25 = $331.75.

Answer:

The answer is 5 in.

Step-by-step explanation:

Since the scale for the model on the left is 1 in = 12 ft, and the scale for the model on the right is 1 in = 3 ft, the model on the right is 4 times larger than the model on the left.

Multiply the side length of the model on the left by 4 to find the side length of the model on the right.

Final answer:

The algebraic expression for finding the average melting points of helium, hydrogen, and neon, using variables h, j, and k as their respective melting points, is (h + j + k) / 3.

Explanation:

The question asks for the algebraic expression that represents the average melting points of helium, hydrogen, and neon. The variables h, j, and k denote the individual melting points of these elements, respectively. To calculate the average melting point, you would add the melting points of each element and divide by the number of elements.

The algebraic expression for the average melting point is:

(h + j + k) / 3

Answer:

So the answer would be No fine is paid if the laptop is returned exactly at the time at which it is due

volume = (1/3)*PI*(r1^2+r1*r2+r2^2)*h

 h=10

r1=5

r2=2

= 408.41 cubic inches

 round off answer as needed.

David must deposit approximately $16,150.01 at the end of each year to accumulate $1 million by the end of 20 years at a 10 percent interest rate.

To calculate the equal annual end-of-year deposits that David must make to accumulate $1 million in 20 years at a 10 percent interest rate, we can use the formula for the future value of an ordinary annuity.

The formula for the future value of an ordinary annuity is given by:

[tex]FV = P * ((1 + r)^n - 1) / r[/tex]

where:

FV is the future value of the annuity (the desired $1 million in this case)

P is the annual deposit (what we need to find)

r is the annual interest rate (10% or 0.10 as a decimal)

n is the number of years (20 years in this case)

Substituting the known values:

[tex]$1,000,000 = P * ((1 + 0.10)^{20} - 1) / 0.10[/tex]

Now, we can solve for P:

$1,000,000 = P * (6.1917364224) / 0.10

$1,000,000 = P * 61.917364224

P = $1,000,000 / 61.917364224

P ≈ $16,150.01

So, David must deposit approximately $16,150.01 at the end of each year to accumulate $1 million by the end of 20 years at a 10 percent interest rate.

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we know that

The x-intercept is the value of the coordinate x when the value of the function is equal to zero

so

In this problem we have that the x-intercepts of the graphs are the points

[tex](-2,0)\\(-1,0)\\(1,0)\\(2,0)[/tex]

therefore

the answer is the option

B)-1,0

The rational zero -1 is a root of f(x). Synthetic division yields [tex]\(x^3 + 2x^2 - 7x - 2\)[/tex]. Further factorization or testing other rational roots finds the remaining zeros.

To find a rational zero of the polynomial function [tex]\(f(x) = x^4 + 3x^3 - 5x^2 - 9x - 2\)[/tex], we can use the Rational Root Theorem. According to this theorem, any rational zero of the polynomial function must be of the form ±p/q, where p is a factor of the constant term (-2 in this case) and q is a factor of the leading coefficient (1 in this case).

The factors of -2 are ±1, ±2, and the factors of 1 are ±1. Therefore, the possible rational zeros are:

±1, ±2

We can try these values to see if they are roots of the polynomial.

Let's start by trying x = 1:

[tex]\[f(1) = (1)^4 + 3(1)^3 - 5(1)^2 - 9(1) - 2\]\[= 1 + 3 - 5 - 9 - 2\]\[= -12\][/tex]

So, x = 1 is not a root.

Next, let's try x = -1:

[tex]\[f(-1) = (-1)^4 + 3(-1)^3 - 5(-1)^2 - 9(-1) - 2\]\[= 1 - 3 - 5 + 9 - 2\]\[= 0\][/tex]

Therefore, x = -1 is a root of the polynomial.

To find the other zeros, we can perform polynomial division or synthetic division by dividing f(x) by (x + 1). Let's use synthetic division:

-1       1      3       -5       -9      -2  

         1      2       -7       -2      ↓

The result is [tex]\(x^3 + 2x^2 - 7x - 2\)[/tex]. Now, we can factor this cubic polynomial or continue using the Rational Root Theorem to find additional roots. Let's try x = 1 again:

[tex]\[f(1) = (1)^3 + 2(1)^2 - 7(1) - 2\]\[= 1 + 2 - 7 - 2\]\[= -6\][/tex]

x = 1 is not a root, so we continue to try the other possible rational zeros. However, to save time, let's check if any of the values of [tex]\(x = \pm 2\)[/tex] are roots using synthetic division:

For x = 2:

2     1      2        -7     -2

      1      4          1      ↓

For \(x = -2\):

-2     1      2      -7       -2

         1     0      -7        ↓

Since none of these values result in a remainder of 0, [tex]\(x = \pm 2\)[/tex] are not roots.

Therefore, the zeros of the polynomial function [tex]\(f(x) = x^4 + 3x^3 - 5x^2 - 9x - 2\) are \(x = -1\),[/tex] and the other zeros can be found by further factoring the reduced cubic polynomial.

To minimize unit cost, 160 machines must be made.

Step-by-step explanation:

Find the first derivative of the cost function:  C'(x) = 1.6x - 256.

Set the derivative equal to 0 and solve for x to find the critical point: 1.6x - 256 = 0. x = 160.

Check the nature of the critical point using the second derivative test to confirm that x = 160 gives the minimum cost.

Therefore, the number of machines that must be made to minimize the unit cost is 160 machines.

The number of machines that must be made to minimize the unit cost is 160.

To find the number of machines that must be made to minimize the unit cost, we need to find the minimum point of the function [tex]\(C(x) = 0.8x^2 - 256x + 25,939\).[/tex]

The function \(C(x)\) represents a quadratic equation, and the vertex of a quadratic equation represents its minimum or maximum point. The x-coordinate of the vertex of a quadratic function in the form [tex]\(ax^2 + bx + c\) is given by \(-\frac{b}{2a}\).[/tex]

[tex]For the function \(C(x) = 0.8x^2 - 256x + 25,939\), we have \(a = 0.8\) and \(b = -256\).Now, let's calculate the x-coordinate of the vertex:\[ x_{\text{vertex}} = -\frac{b}{2a} = -\frac{-256}{2 \times 0.8} = -\frac{-256}{1.6} = 160 \][/tex]

So, the number of machines that must be made to minimize the unit cost is 160.

Answer:

Value of the expression is 0

Step-by-step explanation:

[tex]\frac{1}{3} m -1-\frac{1}{2} n[/tex]

Given the value of m and n

m= 21  and n= 12

We plug in the value of m  and n in the given expression

[tex]\frac{1}{3} m -1-\frac{1}{2} n[/tex]

[tex]\frac{1}{3}(21) -1-\frac{1}{2}(12)[/tex]

[tex]\frac{21}{3} -1-\frac{12}{2}[/tex]

[tex]7-1-6= 0[/tex]

So the value of given expression is 0 when we plug in the values of m  and n

Answer:

Step-by-step explanation:

Given that A cone is placed inside a cylinder. The cone has half the radius of the cylinder, but the height of each figure is the same

Whatever position cone is placed, the space remaining will have volume as

volume of the cylinder - volume of the cone

Let radius of cylinder be r and height be h

Then volume of  cylinder  = [tex]\pi r^2 h[/tex]

The cone has height as h and radius as r/2

So volume of cone = [tex]\frac{1}{3} \pi (\frac{r}{2} )^2h\\=(\pi r^2 h)\frac{1}{24}[/tex]

the volume of the space remaining in the cylinder after the cone is placed inside it

=[tex]\pi r^2 h (1-\frac{1}{24} )\\=\frac{23 \pi r^2 h}{24}[/tex]

Answer:

11/12 pie r^2 h

Step-by-step explanation: