Answer:
a. 37 rooms are need to break even.
b. 73 rooms are required to make twice as much profit as you spent.
Step-by-step explanation:
Revenue of the company from each additional room is $1200.
Total construction cost = $44,000.
At break even condition, total revenue is equal to total cost. In other words, the profit of the firm is zero at break even.
Let x be the number of rooms that are need to break even.
Total revenue of x room is
[tex]TR=1200x[/tex]
At break even,
Total revenue = Total cost
[tex]1200x=44000[/tex]
Divide both sides by 1200.
[tex]x=\frac{44000}{1200}=36.6667\approx 37[/tex]
Therefore, 37 rooms are need to break even.
Let y be the number of rooms to make twice as much profit as you spent.
Total revenue of y room is
[tex]TR=1200y[/tex]
Total revenue = 2 × Total cost
[tex]1200x=2\times 44000[/tex]
[tex]1200x=88000[/tex]
Divide both sides by 1200.
[tex]y=\frac{88000}{1200}=73.33\approx 73[/tex]
Therefore 73 rooms are required to make twice as much profit as you spent.
Consider the following statement: ∀ a, b, c ∈ Z, if a − b is even and b − c is even, then a − c is even.
Write the converse and inverse of this statement. Indicate (by formal reasoning) which among the statement, its converse and its inverse, are true and which are false. Give a counterexample for each that is false.
Answer Step-by-step explanation:
Given statement if a-b is even and b-c is even then a-c is even .
Let p: a-b and b-c are even
q: a-c is even.
Converse: If a-c is even then a-b and b-c are both even.
Inverse:If a-b and b-c are not both even then a-c is not even.
If a= Even number
b= Even number
c=Even number
If a-c is even then a-b and b-c are both even..Hence, the converse statement is true.
If a=Odd number
b=Odd number
c= Odd number
If a-c is even then a-b and b-c are both even number .Hence, the converse statement is true.
If a=Even number
b= Even number
c= Odd number
a-b and b-c are both odd not even number but a-c is even number
a=8,b=6 c=3
a-b=8-6=2
b-c=6-3=3
a-c=8-3=5
If a-c is odd then a-b even but b-c is odd .Hence , the converse statement is false.But the inverse statement is true.
If a= Odd number
b=Even number
c= Even number
If a-b is odd and b-c is even then a-c is odd not even . Hence, the inverse statement is true.
If a= Odd number
b=Eve number
c=Odd number
a=9,b=6,c=5
a-b=9-6=3
b-c=6-5=1
a-c=9-5=4
Here, a-b and b-c are not both even but a-c is even .Hence, the inverse statement is false.
Explain why p -> q is false, only when p is true and q is false. Let p: It is raining q: I will give you a ride home after class and the meaning of the truth for p-> q (If it is raining, then I will give you a ride home)
Answer:
Step-by-step explanation:
Let p : It is raining
Truth value of this statement is True.
q : I will give you a ride home after class
Truth value of the statement q is True.
So for the statement p→q (If it is raining, then I will give you a ride home)
truth value will be True.
Therefore, when statement p is True and q is False, p→q will be False.
A metal bar is to be divided into two pieces so that one piece is 3 centimeters longer than twice the length of the other. If the sum of the squares of the two lengths is 109 square centimeters, find the two lengths
Answer:
The two lengths are 3.4303 and 9.8606 centimeters
Step-by-step explanation:
The metal bar is divided into two pieces, so we are going to call X the length of the first piece and Y the length of the second piece.
From the phrase: one piece is 3 centimeters longer than twice the length of the other, we can separate and rewrite as:
One piece - is - 3 centimeters - longer than - twice the length of the other
X = 3 + 2 * Y
So, X=3+2Y is our first equation
From the phrase: the sum of the squares of the two lengths is 109 square centimeters, we can rewrite as:
The sum of the squares of the two lengths - is - 109 square centimeters
[tex]X^{2} +Y^{2}[/tex] = 109
So, X^2+Y^2=109 is our second equation
Replacing the first equation on the second question we get:
[tex]X^{2} +Y^{2} =109\\(3+2Y)^{2} +Y^{2} =109[/tex]
[tex]9+(2*3*2Y)+4Y^{2} +Y^{2} -109=0[/tex]
[tex]5Y^{2} +12Y-100=0[/tex]
Solving this equation we find two solutions:
Y=3.4303 and Y= -5.8303
Since the question is talking about the length there is no sense use Y=-5.8303, then our first length is 3.4303
So replacing this value on the first equation we get:
X= 3 + 2*Y
X= 3 + 2*3.4303
X= 9.8606
Finally the two length are 3.4303 and 9.8606 centimeters
What is the contrapositive of the following: "If I form a study group then I raise my grades." A. If I work alone then I lower my grades. B. If I raise my grades then I form a study group. C. If I lower my grades then I work alone D. If I form a study group then I lower my grades. E. If I form a study group then I raise my grades. F. If I raise my grades then I work alone.
Answer:
C. If I lower my grades then I work alone
Step-by-step explanation:
In logic, the contrapositive of a conditional statement, [tex]P\to Q[/tex] is [tex]\neg Q\to \neg P[/tex].
In other words, the contrapositive of "If P then Q" is "If not Q, then not P".
The given conditional statement is:
"If I form a study group then I raise my grades."
We can break this conditional statement into the following propositional statements:
P : I form a study group
¬P : I work alone
Q : I raise my grades
¬Q : I lower my grades
The contrapositive of "If I form a study group then I raise my grades." becomes:
" If I lower my grades then I work alone"
The correct choice is C
To find the contrapositive of the given conditional statement "If I form a study group then I raise my grades," we have to negate both the hypothesis (the "if" or P part) and the conclusion (the "then" or Q part) and also reverse their order.
The original statement is in the form: If P then Q.
P: I form a study group.
Q: I raise my grades.
The contrapositive is: If not Q then not P.
Not Q: I do not raise my grades.
Not P: I do not form a study group.
So, the contrapositive of the original statement would be: "If I do not raise my grades then I do not form a study group."
Now let's match this to one of the choices given:
A. If I work alone then I lower my grades. (This is not the contrapositive; it introduces new concepts that weren't part of the original statement.)
B. If I raise my grades then I form a study group. (This is the converse, not the contrapositive.)
C. If I lower my grades then I work alone. (This is the contrapositive: "lower my grades" is equivalent to "do not raise my grades," and "work alone" is equivalent to "do not form a study group.")
D. If I form a study group then I lower my grades. (This is not the contrapositive; it's the inverse of the original statement.)
E. If I form a study group then I raise my grades. (This is the original statement.)
F. If I raise my grades then I work alone. (This is not the contrapositive; it introduces the concept of working alone.)
The correct choice that represents the contrapositive of the original statement is C. If I lower my grades then I work alone.
) A motor boat weighs 32,000 lb and its motor provides a force of 5000 lb. Assume that the water resistance is 100 pounds per foot per second of the velocity of the boat. Then 1000 dv dt = 5000 − 100v. If the boat starts from rest, what is the maximum velocity of the boat?
Answer:
Maximum velocity, v = 50 ft/s
Step-by-step explanation:
Given
1000[tex]\frac{dv}{dt}=5000-100v[/tex] -----------(1)
Dividing (1) by 1000, we get
[tex]\frac{dv}{dt}=5-\frac{v}{10}[/tex]
[tex]\frac{dv}{dt}+\frac{v}{10}=5[/tex] -----------------(2)
Now we can solve the above equation using method of integrating factors
[tex]u(t)=e^{\int \frac{1}{10}dt}[/tex]
[tex]u(t)=e^{\frac{1}{10}t}[/tex]
Now multiplying each side of (2) by integrating factor,
[tex]e^{\frac{1}{10}t}(\frac{dv}{dt})+\frac{v}{10}e^{\frac{1}{10}t}=5e^{\frac{1}{10}t}[/tex]
Combining the LHS into one differential we get,
[tex]\frac{d}{dt}\left ( e^{\frac{1}{10}t}v \right ) = \int 5e^{\frac{1}{10}t}.dt[/tex]
[tex]e^{\frac{1}{10}t}v = 50e^{\frac{1}{10}t}[/tex] + c
v(t)=50+ce
Appltying the initial condition v(0)=0, we get
[tex]0=50+ce^{-\frac{1}{10}(0)}[/tex]
0=50+c
c=-50
So the particular solution is
[tex]v(t)=50-50e^{-\frac{1}{10}t}[/tex]
[tex]v(t)=50\left (1-e^{-\frac{1}{10}t}\right)[/tex]
Therefore, the maximum velocity is 50 ft/s
The forces balance out at 50 ft/s, which is the maximum velocity the boat can achieve.
To determine the maximum velocity of the boat, we need to consider the forces acting on it. The equation given is:
1000 [tex]\frac{dv}{dt}[/tex] = 5000 - 100v
At maximum velocity, the acceleration of the boat will be zero, which means [tex]\frac{dv}{dt}[/tex] = 0. Therefore, setting the left-hand side of the equation to zero, we get:
0 = 5000 - 100v
Solving for v, we have:
5000 = 100vv = [tex]\frac{5000}{100}[/tex]v = 50 ft/s
Thus, the maximum velocity of the boat is 50 ft/s. This is where the force provided by the motor equals the resistive force from the water.
If A is a 6 × 9 matrix what is the maximum possible rank of A? What is the minimum possible rank of A ?
Answer:
Largest possible rank =6
Minimum possible rank=0
Step-by-step explanation:
We are given that A is a matrix of order [tex] 6\times 9 [/tex]
We have to find the maximum possible rank and minimum possible rank of A.
In given matrix we have 6 rows and 9 columns.
Rank: Rank is defined as the number of non zero rows or columns of matrix and any row or column is not a linear combination of other two or more rows or columns
Rank of matrix of order [tex] m\times n [/tex]
Where m< n
Then largest possible rank is m
We are given a matrix of order[tex]6\times 9[/tex] where 6< 9
Therefore, the largest possible rank is 6.
The minimum possible rank is zero because a given matrix can be zero matrix therefore, the rank of zero matrix is zero.
Largest possible rank =6
Minimum possible rank=0
In the Midpoint Rule for triple integrals we use a triple Riemann sum to approximate a triple integral over a box B, where f(x, y, z) is evaluated at the center (xi, yj, zk) of the box Bijk. Use the Midpoint Rule to estimate the value of the integral. Divide B into eight sub-boxes of equal size. (Round your answer to three decimal places.)triple integrals of cos(xyz) dV,where B = {(x, y, z) | 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2}
To approximate the volume with 8 boxes, we have to split up the interval of integration for each variable into 2 subintervals, [0, 1] and [1, 2]. Each box will have midpoint [tex]m_{i,j,k}[/tex] that is one of all the possible 3-tuples with coordinates either 1/2 or 3/2. That is, we're sampling [tex]f(x,y,z)=\cos(xyz)[/tex] at the 8 points,
(1/2, 1/2, 1/2)
(1/2, 1/2, 3/2)
(1/2, 3/2, 1/2)
(3/2, 1/2, 1/2)
(1/2, 3/2, 3/2)
(3/2, 1/2, 3/2)
(3/2, 3/2, 1/2)
(3/2, 3/2, 3/2)
which are captured by the sequence
[tex]m_{i,j,k}=\left(\dfrac{2i-1}2,\dfrac{2j-1}2,\dfrac{2k-1}2\right)[/tex]
with each of [tex]i,j,k[/tex] being either 1 or 2.
Then the integral of [tex]f(x,y,z)[/tex] over [tex]B[/tex] is approximated by the Riemann sum,
[tex]\displaystyle\iiint_B\cos(xyz)\,\mathrm dV\approx\sum_{i=1}^2\sum_{j=1}^2\sum_{k=1}^2\cos m_{i,j,k}\left(\frac{2-0}2\right)^2[/tex]
[tex]=\displaystyle\sum_{i=1}^2\sum_{j=1}^2\sum_{k=1}^2\cos\frac{(2i-1)(2j-1)(2k-1)}8[/tex]
[tex]=\cos\dfrac18+3\cos\dfrac38+3\cos\dfrac98+\cos\dfrac{27}8\approx\boxed{4.104}[/tex]
(compare to the actual value of about 4.159)
The Midpoint Rule for triple integrals approximates the integral of a function by taking the function's value at the midpoint of sub-boxes into which the domain is divided. In this problem, B is divided into eight 1x1x1 sub-boxes, and cos(xyz) is evaluated at each sub-box's center.
Explanation:The Midpoint Rule for triple integrals is a method for approximating the value of a triple integral over a box B by dividing the box into smaller sub-boxes and evaluating the function at the midpoint of each sub-box.
For the given integral of cos(xyz) over the box B defined by the ranges 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, and 0 ≤ z ≤ 2, we are asked to divide B into eight sub-boxes of equal size.
This involves calculating the midpoints for each sub-box (xi, yj, zk) and then evaluating the function at these points.
The size of each sub-box would be 1 x 1 x 1 since the original box dimensions are 2 x 2 x 2.
The midpoints for the sub-boxes will be at (0.5, 0.5, 0.5), (0.5, 0.5, 1.5), (0.5, 1.5, 0.5), (0.5, 1.5, 1.5), (1.5, 0.5, 0.5), (1.5, 0.5, 1.5), (1.5, 1.5, 0.5), and (1.5, 1.5, 1.5).
The volume of each sub-box is 1, so the Riemann sum becomes the sum of cos(xyz) evaluated at these midpoints times the volume of each sub-box.
After calculating the Riemann sum, it provides an approximate value for the triple integral.
The actual calculation would require substituting each midpoint into the function cos(xyz) to get the respective value for each sub-box, then summing these values and finally multiplying by the volume of a single sub-box to get an approximation for the entire integral.
Vernon’s work for finding the value of x is shown below.

Step 1: 16x + 8 = 76
Step 2: 16x = 68
Step 3: x = 4.25
Did Vernon solve for the correct value of x? If not, explain where he made his error.
Yes, he solved for the correct answer.
No, he should have set the sum of ∠AED and ∠DEC equal to 180°, rather then setting ∠AED and ∠DEC equal to each other.
No, he should have added 8 to both sides rather than subtracting 8 from both sides.
No, he should have multiplied both sides by 16 rather than dividing both sides by 16.
Answer:
No, he should have set the sum of ∠AED and ∠DEC equal to 180°, rather then setting ∠AED and ∠DEC equal to each other.
Step-by-step explanation:
He was using the measurements of m∠AED & m∠CED, which are supplementary angles, not vertical angles (therefore making them, when combined, equal to 180°).
If they were vertical angles (at the case of m∠AED & m∠BEC, or the other pair), then yes, they will be congruent. But in this case, they are not, so you don't solve it like they are vertical angles.
~
Answer:
B is the answer.
Step-by-step explanation:
Yes, he solved for the correct answer.
No, he should have set the sum of ∠AED and ∠DEC equal to 180°, rather than setting ∠AED and ∠DEC equal to each other.
No, he should have added 8 to both sides rather than subtracting 8 from both sides.
No, he should have multiplied both sides by 16 rather than dividing both sides by 16.
help please, really confused
Answer:
1a. y = 5x +3
1b. y = -1/7x + 2
1c. y = 2x +3
2a. y = -1/2x +3; slope = -1/2, y-intercept = +3
2b. y = 3/2x +6; slope = 3/2, y-intercept = +6
Step-by-step explanation:
Apparently the "y-intercept form" referred to is the one more commonly called "slope-intercept form." That form is ...
y = mx + b . . . . . . . . m is the slope; b is the y-intercept
You arrive at this form by solving each equation for y. You do that the same way you solve any equation for any variable: undo what is done to the variable of interest.
In these general form equations, the y-variable term has a value multiplying y and some other terms added to that. First of all, you subtract the added terms (from both sides of the equation). This transforms ...
ax +by +c = 0
into
by = -ax -c
Next, you divide by the constant that is multiplying y. Of course all terms on both sides of the equation are divided by that, so you now have ...
y = (-a/b)x -c/b
The slope is -a/b, and the y-intercept is -c/b.
___
1a. 5x -y +3 = 0
-y = -5x -3 . . . . . subtract non-y terms
y = 5x +3 . . . . . . divide by -1
__
1b. x +7y -14 = 0
7y = -x +14 . . . . . subtract non-y terms
y = -1/7x +2 . . . . .divide by 2
__
1c. 6x -3y +9 = 0
-3y = -6x -9 . . . . subtract terms not containing y
y = 2x +3 . . . . . . divide by -3
__
2a. x +2y -6 = 0
2y = -x +6 . . . . . subtract non-y terms
y = -1/2x +3 . . . . divide by 2 -- this is graphed as the red line below
__
2b. 3x -2y +12 = 0
-2y = -3x -12 . . . . subtract non-y terms
y = 3/2x +6 . . . . . divide by -2 -- this is graphed as the blue line below
_____
Comment on the 2nd problem
Of course, the y-intercept (constant term in slope-intercept form) is the point on the y-axis where the line crosses. The slope tells you the ratio of "rise" to "run". That is, a slope of -1/2 means the line drops one unit for each 2 units it goes to the right. A slope of 3/2 means the line increases (rises) by 3 units for each 2 units it goes to the right.
Compute the total and annual return on the following investment Five years after paying $2900 for shares in a startup company, you sell the shares for $1900 (at a loss). The total return is (Do not round until the final answer Then round to the nearest tenth as needed) The annual return is (Do not round until the final answer. Then round to the nearest tenth as needed)
Answer:
total return is -34.4828 % and The annual return is - 8.1094 %
Step-by-step explanation:
given data
principal amount of share = $2900
selling amount pay = $1900
time period (t) = 5 years
to find out
total return and The annual return
solution
we know interest formula
amount = principal [tex](1 + r/100)^{t}[/tex]
put all these value amount principal, time n we get rate
1900 = 2900 [tex](1 + r/100)^{5}[/tex]
[tex](1 + r/100)^{5}[/tex] = 1900/2900
1 + r/100 = 0.9189
r = −8.1094
The annual return is - 8.1094 %
and total return is ((selling amount pay - principal amount ) / principal amount ) × 100
put these value and we get total return
total return = ((1900 - 2900) / 2900 ) × 100
total return = -0.344828 × 100
total return = -34.4828 %
Among 8846 cases of heart pacemaker malfunctions, 375 were found to be caused by firmware, which is software programmed into the device. If the firmware is tested in 3 different pacemakers randomly selected from this batch of 8846 and the entire batch is accepted if there are no failures, what is the probability that the firmware in the entire batch will be accepted? Is this procedure likely to result in the entire batch being accepted?
Step-by-step explanation:
Probability, P = [tex]\frac{no. of favourable outcomes}{Total no. of possible outcomes}[/tex]
Out of 8846 heart pacemaker malfunctions cases, caused by firmware cases are 375
then, no. of cases not caused by firmware are:
8846 - 375 = 8471
Probability for three different pacemakers respectively is given by:
P(1) = [tex]\frac{8471}{8846}[/tex]
we select the next malfunctioned pacemaker from the remaining i.e., out of 8471, excluding the chosen malfunctioned pacemaker
P(2|1) = [tex]\frac{8470}{8845}[/tex]
Therefore, the events are not independent of each other
Now, if the selection is without replacement, then
P(3|1 & 2) = [tex]\frac{8469}{8844}[/tex]
Now, by general multiplication rule(as the events are not independent):
P(none of the 3 are caused by malfunction) =
[tex]\frac{8471}{8846}\times\frac{8470}{8845}\times\frac{8469}{8844}[/tex]
= 0.9019
P(none of the 3 are caused by malfunction) = 90.19%
The probability is high, therefore the whole batch will be accepted.
Probability of an event represents the chances of that event to occur.
The probability of the entire batch to get selected is 0.879Yes, this procedure is likely to result in the entire batch being accepted.What is the multiplication rule of probability for independent events?Suppose there are n mutually independent events.
The probability of their simultaneously occurrence is given as
[tex]P(A_1 \cap A_2 \cap ... \cap A_n) = P(A_1) \times P(A_2) \times ... \times P(A_n)[/tex]
(This is true only if all those events are mutually independent).
The probability of getting a failure(malfunction) in a randomly selected pacemaker for the given case is
[tex]\dfrac{375}{8846} \approx 0.042[/tex]
The probability of a pacemaker working fine = 1- its failure probability = 1 - 0.042 = 0.958
Since each of the 3 selected pacemakers are independent for their failure or success of each other, thus, if we have:
[tex]A_1[/tex] = Event of properly working of first pacemaker
[tex]A_2[/tex] = Event of properly working of second pacemaker
[tex]A_3[/tex] = Event of properly working of third pacemaker
Then, as all three are independent for their working from each other, thus,
[tex]P(A_1 \cap A_2 \cap A_3) = P(A_1) \times P(A_2)\times P(A_3) = (0.958)^3 \approx 0.879\\\\\begin{aligned}{P(\text{batch getting selected}) &= P(\text{All three pacemaker working})\\& = P(A_1 \cap A_2 \cap A_3) \\&\approx 0.879\\\end{aligned}[/tex]
Thus, as this probability is high, thus, this procedure is likely to result in the entire batch being accepted.
Thus,
The probability of the entire batch to get selected is 0.879Yes, this procedure is likely to result in the entire batch being accepted.Learn more about multiplication rule of probability here:
https://brainly.com/question/14399918
How much should you invest now at 7 1/4% simple annual interest in order to earn $190 interest in 4 months?
Answer:
so principal amount is $7862.07
Step-by-step explanation:
Given data
rate = 7*1/4 % = 29/4 %
interest = $190
time = 4 months = 4/12 year
to find out
principal
solution
we know the simple interest formula i.e.
interest = ( principal × rate × time ) /100 ..................1
now put all value rate time and interest in equation 1 we get interest here
interest = ( principal × rate × time ) /100
190 = ( principal × 29/4 × 4/12 ) /100
principal = 190 × 12 ×100 / 29
principal = 7862.068966
so principal amount is $7862.07
Working together, two secretaries can stuff the envelopes for a political fund-raising letter in 3 hours. Working alone, it takes the slower worker 8 hours longer to do the job than the faster worker. How long does it take each to do the job alone?
Answer:
Faster worker takes 4 hours and slower worker takes 12 hours.
Step-by-step explanation:
Let x be the time ( in hours ) taken by faster worker,
So, according to the question,
Time taken by slower worker = (x+8) hours,
Thus, the one day work of faster worker = [tex]\frac{1}{x}[/tex]
Also, the one day work of slower worker = [tex]\frac{1}{x+8}[/tex]
So, the total one day work when they work together = [tex]\frac{1}{x}+\frac{1}{x+8}[/tex]
Given,
They take 3 hours in working together,
So, their combined one day work = [tex]\frac{1}{3}[/tex]
[tex]\implies \frac{1}{x}+\frac{1}{x+8}=\frac{1}{3}[/tex]
[tex]\frac{x+8+x}{x^2+8x}=\frac{1}{3}[/tex] ( Adding fractions )
[tex]3(2x+8)=x^2+8x[/tex] ( Cross multiplication )
[tex]6x+24=x^2+8x[/tex] ( Distributive property )
[tex]x^2+2x-24=0[/tex] ( Subtraction property of equality )
By quadratic formula,
[tex]x=\frac{-2\pm \sqrt{100}}{2}[/tex]
[tex]x=\frac{-2\pm 10}{2}[/tex]
[tex]\implies x=4\text{ or }x=-6[/tex]
Since, hours can not negative,
Hence, time taken by faster worker = x hours = 4 hours,
And, the time taken by slower worker = x + 8 = 12 hours.
To solve this work rate problem, we set up an equation with combined work rates and find that the faster worker takes 3 hours alone, while the slower worker takes 11 hours alone.
Explanation:The question states that two secretaries can stuff envelopes together in 3 hours. The slower worker takes 8 hours more than the faster worker to complete the job alone. To find how long it takes each secretary to complete the job alone, we can set up an equation using the reciprocal of their work rates.
Let x be the time it takes for the faster worker to stuff the envelopes alone. Then, the slower worker will take x + 8 hours. The work rate of the faster worker is 1/x and the slower worker's rate is 1/(x + 8). Working together, their combined work rate is 1/3 per hour (since they complete the task in 3 hours).
The combined work rate equation will be:
1/x + 1/(x + 8) = 1/3
To solve this equation:
Multiply every term by the common denominator, which is 3x(x + 8).This gives us 3(x + 8) + 3x = x(x + 8).Simplify and solve the resulting quadratic equation.This results in x² + 8x - 3x - 24 = 0, which simplifies to x² + 5x - 24 = 0.Factor the quadratic equation to find the values of x.The factors of 24 that add up to 5 are 8 and -3, so (x + 8)(x - 3) = 0.Therefore, x = 3 or x = -8. Since time cannot be negative, we disregard x = -8.The faster worker takes 3 hours and the slower worker takes 3 + 8 = 11 hours.As a result, the faster worker takes 3 hours to complete the job alone, and the slower worker takes 11 hours to complete the job alone.
PROBLEM 1 Verify that the differential equation (2x y)dx +(x - 6y)dy = 0 is exact and then solve the associated differential equation
Answer with explanation:
⇒(2 x y )d x+(x-6 y) d y=0
P= 2 x y
Q=x-6 y
[tex]P_{y}=2 x\\\\Q_{x}=1[/tex]
So this Differential Equation is exact.
To solve this, we will first evaluate,[tex]\varphi (x,y)[/tex].
[tex]\varphi_{x}=P\\\\\varphi_{y}=Q\\\\\varphi=\int P d x\\\\= \int 2 x y dx\\\\\varphi=x^2 y\\\\\varphi(x,y)=x^2y+k(y)------(1)[/tex]
Differentiating with respect to , y
[tex]\varphi'(x,y)=x^2+k'(y)=Q=x-6 y\\\\\rightarrow x-6 y-x^2=k'(y)\\\\ k(y)=\int (x-6 y -x^2) dy\\\\k(y)=x y-3 y^2-x^2 y+f\\\\\varphi(x,y)=x^2y+x y-3 y^2-x^2 y+f\\\\\varphi(x,y)=x y-3 y^2+f[/tex]
Substituting the value of , k(y) in equation 1.
This is required Solution of exact differential equation.
Thorium 234 is a radioactive substance that decays at a rate proportional to the amount present. 1 gram of this material is reduced to 0.8 grams in one week
(a) Find an expression that expresses the amount of Thorium 234 present at any time
(b) Find half life of Thorium 234
(c) Find amount of Thorium 234 present after 10 weeks
Answer: a) [tex]t=\frac{2.303}{k}\log\frac{a}{a-x}[/tex]
b) 3.15 weeks.
c) 0.11 grams
Step-by-step explanation:
a) radioactive decay follows first order kinetics and thus Expression for rate law for first order kinetics is given by:
[tex]k=\frac{2.303}{t}\log\frac{a}{a-x}[/tex]
where,
k = rate constant = ?
t = time taken for decomposition = 1 week
a = initial amount of the reactant = 1 g
a - x = amount left after decay process = 0.8 g
Now put all the given values in above equation, we get
[tex]k=\frac{2.303}{1}\log\frac{1}{0.8}[/tex]
[tex]k=0.22weeks^{-1}[/tex]
2) To calculate the half life, we use the formula :
[tex]t_{\frac{1}{2}=\frac{0.693}{k}[/tex]
[tex]t_{\frac{1}{2}=\frac{0.693}{0.22}=3.15weeks[/tex]
Thus half life of Thorium 234 is 3.15 weeks.
3) amount of Thorium 234 present after 10 weeks:
[tex]10=\frac{2.303}{0.22}\log\frac{1}{a-x}[/tex]
[tex](a-x)=0.11g[/tex]
Thus amount of Thorium 234 present after 10 weeks is 0.11 grams
Which of the following functions from R rightarrow R are one-to-one (injective), onto (surjective), or both (bijective)? Prove your answers. a. f(x) = 5x + 4 b. g(x) = 2x^2 - 2 c. h(x)= 1 + 2/x, x notequalto 0.
Answer:
a) bijective
b) neither surjective or injective
c) injective
Step-by-step explanation:
Since we are looking from real numbers to real numbers, we want the following things
1) We want every real number y to get it. (surjective)
2) We want every y that gets hit to be hit only once. (injective).
---
If we have both things then the function is bijective.
a) f(x)=5x+4 this is a line with a positive slope.
That means it is increasing left to right. Every increasing or even decreasing line is going to hit every real number y. This is a bijection.
b) g(x)=2x^2-2 is a parabola. Parabola functions always have y's that get hit more than once and not all y's get hit because the parabola is either open up or down starting from the vertex. This function is neither injective or surjective.
c) h(x)=1+(2/x) x is not 0.
1+(2/x) is never 1 because 2/x is never 0 for any x. This means the real number y=1 will never be hit and is therefore not surjective. This function is injective because every 1 that is hit is only hit once. If you want use the horizontal line test to see this.
Please help me with this
Answer:
m<1 = 45°
m<2 = 76°
m<3 = 80°
Step-by-step explanation:
Points to remember
1). Vertically opposite angles are equal
2). Sum of angles of a triangles is 180°
To find the measures of given angles
From the figure we get,
m<1 = 45° [Vertically opposite angles]
By using angle sum property <2 + 59 + 45 = 180
m<2 = 180 - 104
m<2 = 76°
Also m<1 + m<3 + 55 = 180
m<3 = 180 - (m<1 + 55)
= 180 - (45 + 55)
= 180 - 100
m<3 = 80°
A client has an order for 1000 mL 0.45 Saline to be infused over 8 hours. How many mL/hr should the IV pump be programmed for?
Answer:
125 mL/h
Step-by-step explanation:
Intravenous pumps are used when there is a need of constant pumping of fluids into the body. Injecting every hour with the required volume causes damage.
Volume of 0.45 saline = 1000 mL
Time to be infused = 8 hours
The Intravenous pump should be programmed in the following way
[tex]\frac{Volume}{time}=\frac{1000}{8}\\\Rightarrow flow\ rate=125\ mL/hr[/tex]
Hence, the IV pump should be programmed for 125 mL/hr.
Write an equation of the circle with center (-3, 5) and radius 4.
Answer:
(x+3)^2 + (y-5)^2 = 4^2
or
(x+3)^2 + (y-5)^2 = 16
Step-by-step explanation:
The equation of a circle can be written in the form
(x-h)^2 + (y-k)^2 = r^2
where (h,k) is the center and r is the radius
(x--3)^2 + (y-5)^2 = 4^2
(x+3)^2 + (y-5)^2 = 4^2
or
(x+3)^2 + (y-5)^2 = 16
Let A, B R be non-empty sets. Suppose that A B. Suppose that B has a least upper bound. Prove that A has a least upper bound, and lub A lessthanorequalto lub B.
Answer with explanation:
Given → A, B ⊆R, are non-empty sets.
Also, A ⊆ B.
It is also given that , B has least upper bound.
To Prove:→A has a least upper bound,
and → lub A ≤ lub B.
Proof:
A and B are non empty sets.
Also, A ⊆ B
it means there are some elements in B which is not in A and there are some elements in set B which may be greater or smaller than or equal to set A.
The meaning of least upper bound is the the smallest member of the set from which all members are greater.
As, A⊆B
So, there are two possibilities ,either
→ Least upper bound set A = least upper bound set B------(1)
→ Least upper bound set A < least upper bound set B-------(2)
Combining (1) and (2)
→→→l u b(A) ≤ l u b (B)
The graph is a transformation of one of the basic functions. Find the equation that defines the function.
Answer:
f(x) = -logx + 4.
Step-by-step explanation:
y = -log x approaches the point (0, 1) but this approaches (0, 5)
Looks like the graph of - log x + 4.
Answer:
[tex]y = 5-\sqrt{x}[/tex]
Step-by-step explanation:
Note that the graph shown is a transformation of the parent function
[tex]y = \sqrt{x}[/tex]
The main function cuts the x-axis at x = 0 and has no negative values of y. In addition, the main function is always growing.
Note that the function shown is decreasing and intercepts the y-axis at y = 5.
Therefore, the graph of the main function has been reflected on the x-axis and displaced 5 units upwards.
If we do y = f(x) then to reflect the main function on the x-axis and move it upwards 5 units we make the following transformation
[tex]y = -f(x) + 5[/tex].
Then the graphed function is:
[tex]y = 5-\sqrt{x}[/tex]
Dopamine is available as 400 mg in 250 mL of D5W. A 2 year old weighing 12 kg is receiving 10 mcg/kg/min. How many hours will the infusion last?
Answer:
55.5556 hours.
Step-by-step explanation:
Let's solve the problem.
The amount of dopamine rate applied to a person is based on the formula: 10mcg/kg/min. Such relation can be express as follows:
(10mcg/kg/min)=
(10mcg/kg)*(1/min)
Now by multiplying by the weight (12 kg) of the 2 years old person, we have:
(10mcg/kg)*(1/min)*(12kg)=
(10mcg*12kg/kg)*(1/min)=
(120mcg)*(1/min)=
120mcg/min, which is the rate of dopamine infusion, which can be express as:
(120mcg/min)*(60min/1hour)=
(120mcg*60min)/(1hour*1min)=
7200mcg/hour=
1hour/7200mcg, which means that for each hour, 7200mcg dopamine are infused.
Because the D5W product has 400 mg of dopamine, then we need to convert 400 mg to X mcg of dopamine in order to use the previous obtained rate. This means:
Because 1mcg=0.001mg then:
(400mg)*(1mcg/0.001mg)=
(400mg*1mcg)/(0.001mg)=
400000mcg, which is the amount of dopamine in D5W.
Now, using the amount of dopamine in D5W and the applied rate we have:
(rate)*(total amount of dopamine)=hours of infusion
(1hour/7200mcg)*(400000mcg)=hours of infusion
(1hour*400000mcg)/(7200mcg)=hours of infusion
(55.5556 hours) =hours of infusion
In conclusion, the infusion will last 55.5556 hours.
Use De Morgan's laws to write an equivalent statement for the sentence. The bus does not have an engine or the people do not have money.
Which statement is equivalent to the given sentence?
A. It is false that the bus has an engine and the people have money.
B. The bus has an engine and the people have money.
C. It is false tht the bus has an engine or the people have money.
D. The bus has an engine and the people do not have money.
Please show work legibly as I do not see well.
Answer:
A. It is false that the bus has an engine and the people have money.
Step-by-step explanation:
Given statement is -
The bus does not have an engine or the people do not have money.
The De Morgan's laws states that the complement of the union of two given sets is the intersection of their complements.
Also, the complement of the intersection of these two sets is the union of their complements.
negative (p ∨ q)⇔ p ∧ negative q
So, here option A is correct. It is false that the bus has an engine and the people have money.
Final answer:
Using De Morgan's laws, the equivalent statement for 'The bus does not have an engine or the people do not have money' is 'It is not the case that the bus has an engine and the people have money,' which corresponds to answer choice A.
Explanation:
The question asks to use De Morgan's laws to write an equivalent statement for the sentence: "The bus does not have an engine or the people do not have money." Let's denote 'the bus has an engine' by A and 'the people have money' by B. The original statement can be written in logical form as ¬A ∨ ¬B, which using De Morgan's laws, translates to ¬(A ∧ B). This means that the equivalent statement is: "It is not the case that the bus has an engine and the people have money." Therefore, the correct answer is A.
Find the area of the region trapped between LaTeX: y=1-2x^2 y = 1 − 2 x 2 and LaTeX: y=\left|x\right| y = | x | , shown above. The answer is LaTeX: \frac{A}{12} A 12 . Below, enter only the whole number LaTeX: A A .
The area is given by the integral,
[tex]\displaystyle\int_{-1/2}^{1/2}(1-2x^2-|x|)\,\mathrm dx[/tex]
The integrand is even, so we can simplify the integral somewhat as
[tex]\displaystyle2\int_0^{1/2}(1-2x^2-|x|)\,\mathrm dx[/tex]
When [tex]x\ge0[/tex], we have [tex]|x|=x[/tex], so this is also the same as
[tex]\displaystyle2\int_0^{1/2}(1-2x^2-x)\,\mathrm dx[/tex]
which has a value of
[tex]2\left(x-\dfrac23x^3-\dfrac12x^2\right)\bigg|_0^{1/2}=2\left(\dfrac12-\dfrac1{12}-\dfrac18\right)=\boxed{\dfrac7{12}}[/tex]
so that A = 7.
Pierre sold 340 tickets for a concert. Balcony tickets cost $5 while tickets for the lower floor cost $10. If Pierre sold $2,700 worth of tickets, how many lower floor tickets did he sell?
Answer:
He sold 200 lower floor tickets.
Step-by-step explanation:
Consider the provided information.
Pierre sold 340 tickets for a concert.
Let Balcony tickets represents by x and lower floor ticket represents by y.
As we know he sold 340 tickets in total, i.e.
x + y = 340 ......(1)
Balcony tickets cost $5 while tickets for the lower floor cost $10.
If he sold x balcony tickets, so the money he got after selling x balcony tickets is 5x. Similarly, If he sold y lower floor tickets, so the money he got after selling y lower floor tickets is 10y.
Pierre sold $2,700 worth of tickets, i.e.
5x + 10y = 2700 ......(2)
Now use elimination method to solve the system of equations:
Multiply the equation (1) with 5 and subtract them as shown below:
5x + 5y = 1700
5x + 10y = 2700
_____________
-5y = -1000
y = 200
Thus, he sold 200 lower floor tickets.
Studies have documented that 75% of college business students cheat on final exams. A business professor monitored a random sample of 40 students via web cameras as they took their final exam.The probability that exactly 30 of the students monitored cheat on the final exam is:(Round your answer to 4 decimal places)
Answer:
The probability is 0.1444.
Step-by-step explanation:
Let X be the event that college business students cheated on final exams,
Since, the probability that student cheats on exam, p = 75 % = 0.75,
So, the probability that student does not cheat on exam, q = 1 - p = 0.25,
The binomial distribution formula,
[tex]P(x)=^nC_r p^r q^{n-r}[/tex]
Where, [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
The probability that exactly 30 of the students monitored cheat on the final exam out of 40 students is,
[tex]P(X=30)=^{40}C_{30} 0.75^{30} 0.25^{40-30}[/tex]
[tex]=847660528\times 0.75^{30} \times 0.25^{10}[/tex]
[tex]=0.144364346356[/tex]
[tex]\approx 0.1444[/tex]
The correct probability that exactly 30 of the 40 students monitored cheat on the final exam is 0.0256.
To solve this problem, we can use the binomial probability formula, which is given by:
[tex]\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \][/tex]
Given:
- ( n = 40 ) (the number of students monitored),
- ( k = 30 ) (the number of students who cheat),
- ( p = 0.75 ) (the probability of a student cheating on the final exam).
We can plug these values into the binomial probability formula:
[tex]\[ P(X = 30) = \binom{40}{30} \cdot (0.75)^{30} \cdot (1-0.75)^{40-30} \][/tex]
First, calculate the binomial coefficient:
[tex]\[ \binom{40}{30} = \frac{40!}{30!(40-30)!} \][/tex]
Next, calculate the powers of p and (1-p) :
[tex]\[ (0.75)^{30} \] \[ (0.25)^{10} \][/tex]
Now, multiply these values together:
[tex]\[ P(X = 30) = \binom{40}{30} \cdot (0.75)^{30} \cdot (0.25)^{10} \][/tex]
Using a calculator or a software tool, we can compute the exact value of the probability:
[tex]\[ P(X = 30) \approx 0.0256 \][/tex]
Therefore, the probability that exactly 30 of the 40 students monitored cheat on the final exam is approximately 0.0256 when rounded to four decimal places.
Evaluate ∫ C → F ⋅ → d r where → F ( x , y , z ) = ⟨ 1.75 y z cos ( x y z ) , 1.75 x z cos ( x y z ) , 1.75 x y cos ( x y z ) ⟩ and C is the line segment joining the point ( 5 , π 2 , 1 5 ) to the point ( 6 , 1 6 , π 6 ) . g
[tex]\vec F(x,y,z)=\dfrac74\cos(xyz)\langle yz,xz,xy\rangle[/tex]
Computing the line integral directly is cumbersome, if not impossible by elementary means. Let's instead try to determine if [tex]\vec F[/tex] is conservative. We look for a scalar function [tex]f(x,y,z)[/tex] such that [tex]\nabla f=\vec F[/tex]. We should have
[tex]\dfrac{\partial f}{\partial x}=yz\cos(xyz)[/tex]
(ignoring the 7/4 for a moment). Integrating both sides wrt [tex]x[/tex] gives
[tex]\displaystyle\int\cos(xyz)yz\,\mathrm dx=\sin(xyz)+g(y,z)[/tex]
Then differentiating wrt [tex]y[/tex] gives
[tex]\dfrac{\partial(\sin(xyz))}{\partial y}=xz\cos(xyz)=xz\cos(xyz)+\dfrac{\partial g}{\partial y}[/tex]
[tex]\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)[/tex]
Differentiating wrt [tex]z[/tex] gives
[tex]\dfrac{\partial(\sin(xyz))}{\partial z}=xy\cos(xyz)=xy\cos(xyz)+\dfrac{\mathrm dh}{\mathrm dz}[/tex]
[tex]\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C[/tex]
So we have (and here we re-introduce the 7/4)
[tex]f(x,y,z)=\dfrac74\sin(xyz)+C[/tex]
and by the fundamental theorem of calculus,
[tex]\displaystyle\int_C\nabla f\cdot\mathrm d\vec r=f(\vec b)-f(\vec a)[/tex]
where [tex]\vec a[/tex] and [tex]\vec b[/tex] are vectors representing the start- and endpoints of [tex]C[/tex]. So
[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\frac74\sin\frac\pi6-\frac74\sin\frac\pi2=\boxed{\frac78}[/tex]
The problem asks to evaluate a line integral of a given function over a line segment. The process involves parameterization of the line segment, substitution into the vector function, and finally integration. However, due to complexity of the function, an exact solution might be difficult to achieve.
Explanation:The problem asks us to evaluate a line integral, specifically ∫ C → F ⋅ → d r, where the function → F ( x , y , z ) = ⟨ 1.75 y z cos ( x y z ) , 1.75 x z cos ( x y z ) , 1.75 x y cos ( x y z ) ⟩, and C is the line segment joining two points.
To solve this problem, we need to take the vector field through each of the points of the line segment C. We then cross multiply these results, and calculate the magnitude to get the final answer. However, due to the complex nature of the function → F, an exact analytical solution might be difficult to attain and numerical methods might be required.
Please take note that this is a simplified explanation and actual calculations would be much more complex involving steps like parameterization of the line segment, substituting it into the vector function, and then taking integral.
Learn more about Line Integral here:https://brainly.com/question/34566674
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Madison is picking out some movies to rent, and she has narrowed down her selections to 7 foreign films, 3 horror films, 6 action movies, and 4 dramas. How many different combinations of 9 movies can she rent if she wants all 4 dramas?
Final answer:
Madison can choose from 4368 different combinations of 9 movies, including all 4 dramas and 5 out of the 16 non-drama movies.
Explanation:
Madison is picking out movies to rent and wants all 4 dramas out of her 9 selections. With 7 foreign films, 3 horror films, 6 action movies, and 4 dramas, the total number of non-drama movies is 16 (7+3+6). Since she must select all 4 dramas, she needs to choose the remaining 5 movies from the 16 non-drama movies available.
To find how many different combinations of 9 movies she can rent, we use the combination formula C(n, r) = n! / [r!(n-r)!], where n is the total number of items to choose from, r is the number of items to choose, and ! denotes factorial, representing the product of all positive integers up to that number.
The question asks us to calculate C(16, 5) because Madison is choosing 5 movies out of the 16 non-drama movies. This calculation yields:
C(16, 5) = 16! / [5!(16-5)!] = 4368
Therefore, there are 4368 different combinations of 9 movies Madison can rent if she wants all 4 dramas.
In a circle with a diameter of 25.2 ft, an arc is intercepted by a central angle of 168º.
What is the arc length?
Use 3.14 for π and round your final answer to the nearest hundredth.
Enter your answer as a decimal in the box.
Answer:
[tex]36.93\ ft[/tex]
Step-by-step explanation:
step 1
Find the circumference of the circle
The circumference is equal to
[tex]C=\pi D[/tex]
we have
[tex]D=25.2\ ft[/tex]
substitute
[tex]C=25.2\pi\ ft[/tex]
step 2
we know that
The circumference of a circle subtends a central angle of 360 degrees
so
using proportion
Find out the arc length by a central angle of 168 degrees
[tex]\frac{25.2\pi}{360}=\frac{x}{168}\\\\x=25.2\pi*168/360\\\\x=(25.2*3.14)*168/360\\\\x=36.93\ ft[/tex]
Determine whether the statement is true or false. If it is false, rewrite it as a true statement. The method for selecting a stratified sample is to order a population in some way and then select members of the population at regular intervals. Choose the correct answer below. A. False. The method for selecting a cluster sample is to order a population in some way and then select members of the population at regular intervals. B. False. The method for selecting a systematic sample is to order a population in some way and then select members of the population at regular intervals. C. True.
Answer:
The method for selecting a stratified sample is to order a population in some way and then select members of the population at regular intervals. - FALSE.
This can be re written as :The method for selecting a systematic sample is to order a population in some way and then select members of the population at regular intervals.
Explanation for stratified sampling:
In stratified sampling, the members of a population are divided into two or more strata with similar characteristics and then a random sample is selected from each strata. This way ensures that members of each group within a population will be sampled.
Explanation for cluster sampling:
The method for selecting a cluster sample is to order a population in some way and then select members of the population at regular intervals.
In cluster sampling, the population is divided into clusters, and all of the members of one or more clusters are selected.
Final answer:
The statement provided is false as it describes the method of systematic sampling instead of stratified sampling. Stratified sampling requires dividing the population into strata and randomly selecting individuals from each stratum, while systematic sampling selects members at regular intervals.
Explanation:
The statement is false. The correct method for selecting a stratified sample is to divide the population into groups known as strata, and then use simple random sampling to identify a proportionate number of individuals from each stratum. The statement describes the method for selecting a systematic sample, not a stratified sample. A systematic sample is obtained by ordering a population and then selecting members at regular intervals, not by stratifying them.
On the other hand, a cluster sample involves dividing the population into clusters (groups), often geographically defined, and then randomly selecting some of these clusters. All the members from these selected clusters are included in the sample. Hence, the method of selecting individuals at regular intervals is not used in cluster sampling either.