Answer:
Lenny's tax payment for the year will be $7,752
(The third quarter payment would be $1,938)
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Problems.
Taxes are calculated once a year based on your yearly income. That being said we can look at Lenny's income for the year ($40,800) and multiply that by 0.19 (19%) to calculate his tax payment for the year.
[tex]$40,800 * 0.19 = $7,752[/tex]
So Lenny's tax payment for the year will be $7,752 . The question asks for payment for the third quarter but as mentioned above tax is calculated yearly.
That being said if you would like to know how much it would be quarterly we can simply divide the tax payment by 4.
[tex]7,752 / 4 = 1,938[/tex]
So the third quarter payment would be $1,938
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Find the value of 715×211 Although these numbers aren't quite as nice as the ones from the example, the procedure is the same, so the difficulty is the same same excepting the ability to perform the calculation in your head. You may choose to use a calculator.
To calculate the value of 715 × 211, you can use the standard multiplication method by multiplying each digit of the two numbers and summing up the results.
Explanation:To find the value of 715 × 211, you can use the standard multiplication method. Start by multiplying the ones digit of 715 (5) by each digit of 211 (1, 1, and 2), and write down the results. Then, multiply the tens digit of 715 (1) by each digit of 211, and write down the results one place to the left of the previous results. Finally, multiply the hundreds digit of 715 (7) by each digit of 211 and write down the results two places to the left. Sum up the columns and you will get the final product.
Here's how it looks:
715
× 211
--------
715
1430
+1425
--------
150665
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Customer account "numbers" for a certain company consist of 3 letters followed by 2 numbers.Step 1 of 2 : How many different account numbers are possible if repetitions of letters and digits are allowed?
Final answer:
If 3 letters are followed by 2 numbers and repetitions are allowed, there are a total of 1757600 possible different account numbers. Each letter position has 26 choices and each number position has 10 choices.
Explanation:
The student has asked to determine the number of different customer account "numbers" a company can have if the accounts consist of 3 letters followed by 2 numbers, with repetitions allowed for both letters and numbers.
To calculate the total number of possible account numbers, we can use the multiplication principle of counting. The number of options for each position of the account number is multiplied together to get the total number of combinations.
For the 3 letters, each position can contain any letter from A-Z, which gives us 26 choices per position. Since repeats are allowed, each of the 3 positions has 26 possible choices.
For the 2 numbers, each position can contain any digit from 0-9, which gives us 10 choices per position.
Therefore, to find the total number of possible account numbers, we calculate:
26 × 26 × 26 × 10 × 10 = 1757600 possible account numbers.
Which is the solution to the equation 3/5(x+4/3)= 1.04 ?
Please help me with this
Answer:
Definition of mid-point
Step-by-step explanation:
Midpoint is the center point of a line segment
At midpoints the line segment is divided into two equal lengths
Point K is the midpoint of segment MJ and length of segment MK = length of segment KJ
Point K is the midpoint of segment OL and length of segment OK = length of segment KL
Automobile sales in a country were 20.6 million thisâ year, a 4.9â% increase over last year. Find the number of auto sales in the country last year.?
number â(Round to the nearestâ tenth.)?
Answer: 19.6 million
Step-by-step explanation:
The exponential growth function is given by :-
[tex]A=A_0(1+r)^x[/tex], where A is the initial amount , r is rate of interest and x is time period.
Given : The automobile sales in a country this year : A= 20.6 million
The rate of increase : r = 4.9 %=0.049
For last year , we take x = 1 , then the required exponential equation will be :-
[tex]20.6=A_0(1+0.049)^1\\\\\Rightarrow\ A_0=\dfrac{20.6}{1.049}=19.63775\approx19.6[/tex]
Hence, the number of auto sales in the country last year = 19.6 million.
Final answer:
To find last year's auto sales, the formula original amount = final amount / (1 + rate of increase) is used. The sales last year, before a 4.9% increase to 20.6 million, were approximately 19.6 million when rounded to the nearest tenth.
Explanation:
To find the number of automobile sales last year before the increase, we can use the formula: original amount = final amount / (1 + rate of increase).
Given that the sales this year were 20.6 million and the rate of increase was 4.9%, the calculation for last year's sales would be as follows:
Original sales = 20.6 million / (1 + 0.049) = 20.6 million / 1.049
After performing the division, we get:
Original sales = 19.638 million
Rounding to the nearest tenth, the number of auto sales in the country last year was 19.6 million.
2. A multiple-choice test contains 10 questions. There are four possible answers for each question lo how many ways can a student answer the questions on the test if the student answers every question? the test if the studest
Answer:
The number of ways are 1,048,576.
Step-by-step explanation:
Consider the provided information.
Product rule: If one event occurs in n contexts and the second event occurs in m contexts, then the number of ways in which the two events happen is n×m.
There are 10 questions and each question has 4 choices.
Therefore, for first question we have 4 choices, for second question we have 4 choices similarly for 10th question we have 4 choices which can be represented as:
4×4×4×4×4×4×4×4×4×4 = [tex]4^{10}[/tex]
4×4×4×4×4×4×4×4×4×4 = 1048576
Thus, the number of ways are 1,048,576.
Find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the series solutions with the solutions of the differential equation obtained using the method of Section 4.3. Try to explain any differences between the two forms of the solution. y'' − y' = 0
To find power series solutions of the differential equation y'' − y' = 0, we can assume a power series solution and find the recurrence relation. Two power series solutions are found by choosing different initial conditions. The power series solutions are equivalent to the exponential solutions obtained using another method.
Explanation:To find power series solutions of the differential equation y'' − y' = 0, we can assume a power series solution of the form y(x) = ∑(n=0)∞ a_nx^n. Substituting this into the differential equation and simplifying, we find that the power series satisfies the recurrence relation a_{n+2} = a_{n+1} in terms of a_0 and a_1.
By letting a_0 = 0 and a_1 = 1, we obtain the power series solution y_1(x) = x. Alternatively, by letting a_0 = 1 and a_1 = 0, we obtain the power series solution y_2(x) = 1.
Comparing these power series solutions with the solutions obtained using the method of Section 4.3, we see that the power series solutions are polynomials. In this case, the power series solutions are equivalent to the solutions obtained using the method of Section 4.3, which are exponential functions.
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An 80 kg Rottweiler needs 40 mL/kg over 12 hours. What's the flow rate per hour?
A. 3,200 mL/hr
B. 267 mL/hr
C. 3.3 mL/hr
D. 133 mL/hr
Answer:
The flow rate is 267ml/hour
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Equations.
To solve this we first need to find out how many ml the Rottweiler needs over 12 hours. We do this by using the Rule of Three property.
[tex]\frac{40ml}{1kg} = \frac{x}{80kg}[/tex]
[tex]\frac{40ml*80kg}{1kg} =x[/tex]
[tex]3200ml = x[/tex]
So the Rottweiler needs 3200 ml over a 12 hour period. We now need to find the flow rate per hour. We can solve this by simply dividing 3200 ml by 12 hours.
[tex]3200ml / 12hours = 266.67ml/hour[/tex]
So the flow rate is 267 ml/hour (rounded to the nearest whole number)
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y) = x2 − y2; x2 + y2 = 81 maximum value minimum value
The Lagrangian is
[tex]L(x,y,\lambda)=x^2-y^2+\lambda(x^2+y^2-81)[/tex]
with critical points where the partial derivatives are identically zero:
[tex]L_x=2x+2\lambda x=0\implies 2x(1+\lambda)=0\implies x=0\text{ or }\lambda=-1[/tex]
[tex]L_y=-2y+2\lambda y=0\implies-2y(1-\lambda)=0\implies y=0\text{ or }\lambda=1[/tex]
[tex]L_\lambda=x^2+y^2-81=0\implies x^2+y^2=81[/tex]
If [tex]x=0[/tex], then [tex]L_\lambda=0\implies y=\pm9[/tex]If [tex]\lambda=-1[/tex], then [tex]L_y=0\implies y=0[/tex] and [tex]L_\lambda=0\implies x=\pm9[/tex]If [tex]y=0[/tex], we get the previous conclusion of [tex]x=\pm9[/tex]If [tex]\lambda=1[/tex], then [tex]L_x=0\implies x=0[/tex] and we again get [tex]y=\pm9[/tex]So we have four critical points to consider: (0, -9), (0, 9), (-9,0), and (9, 0). We have
[tex]f(0,-9)=-81[/tex]
[tex]f(0,9)=-81[/tex]
[tex]f(-9,0)=81[/tex]
[tex]f(9,0)=81[/tex]
So the maximum value is 81 and the minimum value is -81.
The extreme values of a function are the minimum and the maximum values of the function.
The extreme values are -81 and 81
The function is given as:
[tex]\mathbf{f(x,y) = x^2 - y^2}[/tex]
[tex]\mathbf{x^2 + y^2 = 81}[/tex]
Subtract 81 from both sides of [tex]\mathbf{x^2 + y^2 = 81}[/tex]
[tex]\mathbf{x^2 + y^2 - 81 = 0}[/tex]
Using Lagrange multiplies, we have:
[tex]\mathbf{L(x,y,\lambda) = f(x,y) + \lambda(0)}[/tex]
Substitute [tex]\mathbf{f(x,y) = x^2 - y^2}[/tex] and [tex]\mathbf{x^2 + y^2 - 81 = 0}[/tex]
[tex]\mathbf{L(x,y,\lambda) = x^2 - y^2 + \lambda(x^2 + y^2 - 81)}[/tex]
Differentiate
[tex]\mathbf{L_x = 2x + 2\lambda x}[/tex]
[tex]\mathbf{L_y = -2y + 2\lambda y}[/tex]
[tex]\mathbf{L_{\lambda} = x^2 + y^2 -81}[/tex]
Equate to 0
[tex]\mathbf{2x + 2\lambda x = 0}[/tex]
[tex]\mathbf{-2y + 2\lambda y = 0}[/tex]
[tex]\mathbf{x^2 + y^2 -81 = 0}[/tex]
So, we have:
[tex]\mathbf{2\lambda x = -2x}[/tex]
[tex]\mathbf{2\lambda y = 2y}[/tex]
Divide both sides of [tex]\mathbf{2\lambda x = -2x}[/tex] by -2x
[tex]\mathbf{\lambda = -1}[/tex]
Divide both sides of [tex]\mathbf{2\lambda y = 2y}[/tex] by 2y
[tex]\mathbf{\lambda = 1}[/tex]
The above means that:
[tex]\mathbf{\lambda = -1\ or\ x = 0}[/tex]
[tex]\mathbf{\lambda = 1\ or\ y = 0}[/tex]
Recall that: [tex]\mathbf{x^2 + y^2 = 81}[/tex]
When x = 0, we have:
[tex]\mathbf{0^2 + y^2 = 81}[/tex]
Take square roots of both sides
[tex]\mathbf{y = \±9}[/tex]
When y = 0, we have:
[tex]\mathbf{x^2 + 0^2 = 81}[/tex]
Take square roots of both sides
[tex]\mathbf{x = \±9}[/tex]
To determine the critical points, we consider:
[tex]\mathbf{\lambda = -1\ or\ x = 0}[/tex] or [tex]\mathbf{y = \±9}[/tex]
[tex]\mathbf{\lambda = 1\ or\ y = 0}[/tex] or [tex]\mathbf{x = \±9}[/tex]
So, the critical points are:
[tex]\mathbf{(x,y) = \{ (0, -9), (0, 9), (-9,0), (9, 0)\}}[/tex]
Substitute the above values in [tex]\mathbf{f(x,y) = x^2 - y^2}[/tex]
[tex]\mathbf{f(0,-9) = 0^2 - (-9)^2 = -81}[/tex]
[tex]\mathbf{f(0,-9) = 0^2 - (9)^2 = -81}[/tex]
[tex]\mathbf{f(-9,0) = (-9)^2 - 0^2 = 81}[/tex]
[tex]\mathbf{f(9,0) = (9)^2 - 0^2 = 81}[/tex]
Considering the above values, we have:
[tex]\mathbf{Minimum= -81}[/tex]
[tex]\mathbf{Maximum= 81}[/tex]
Hence, the extreme values are -81 and 81, respectively.
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Riding a bike a person takes 20 minutes to go to work. The trip back home takes 30 minutes. If the rate back is 8 mph slower than the trip to work, find the rates (speeds) each way and the distance to work.
We know that speed is defined as the ratio of distance to time.
i.e.
[tex]Speed=\dfrac{Distance}{Time}[/tex]
Let the distance traveled to work be: x m.
Now, while going to work it takes a person 20 minutes.
This means that the speed of the person while going to work is:
[tex]S_1=\dfrac{x}{20}[/tex]
Also, the time taken to come back home is: 30 minutes.
This means that the speed of person while riding to home is:
[tex]S_2=\dfrac{x}{30}[/tex]
Also, it is given that the rate back is 8 mph slower than the trip to work.
This means that:
[tex]S_1-S_2=8[/tex]
i.e.
[tex]\dfrac{x}{20}-\dfrac{x}{30}=8\\\\i.e.\\\\\dfrac{30x-20x}{600}=8\\\\i.e.\\\\\dfrac{10x}{600}=8\\\\i.e.\\\\\dfrac{x}{60}=8\\\\i.e.\\\\x=480\ \text{m}[/tex]
Hence, the distance to work is: 480 m.
Also, the rate while going to work is:
[tex]=\dfrac{480}{20}\\\\=24\ \text{mph}[/tex]
and the trip back to home is covered with the speed:
[tex]=\dfrac{480}{30}\\\\=16\ \text{mph}[/tex]
Which of the following is not listed as natural causes for our annihilation? OA. Super Volcano B. Asteroid Impact C. Gamma Rays D. Epidemic E. None of the above
Answer:
its E none of the above
Step-by-step explanation:
The price of gasoline purchased varies directly with the number of gallons of gas purchased. If 19 gallons are purchased for $22.50, what is the price of purchasing 14 gallons? Let x represent the number of gallons purchased, and let y represent the total price. Round your answer to the nearest hundredth.
Answer:
[tex]\text{16.57 dollars}[/tex]
Step-by-step explanation:
A proportion is needed to find the value of x
[tex]$\frac{22.50}{19} =\frac{x}{14}\Longrightarrow 19x = 315 \Longrightarrow x = 16.57$ \\ \\ \text{It would cost 16.57 dollars to purchase 14 gallons.}[/tex]
The cost of 14 gallons of gasoline is $16.58.
What is proportion?Proportions are of two types one is the direct proportion in which if one quantity is increased by a constant k the other quantity will also be increased by the same constant k and vice versa.
In the case of inverse proportion if one quantity is increased by a constant k the quantity will decrease by the same constant k and vice versa.
Given, The price of gasoline purchased varies directly with the number of gallons of gas purchased and 19 gallons are purchased for $22.50.
Let k be the proportionality constant.
∴ y = kx.
22.50 = 19k.
k = 22.50/19.
k = 1.184.
So, the cost of 14 gallons of gasoline is (14×1.184) = $16.58.
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Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficients. (a) x4 − 2x3 + x2 + 3x − 2 x2 − 2x + 1
[tex]\dfrac{x^4-2x^3+x^2+3x-2}{x^2-2x+1}[/tex]
The degree of the numerator exceeds the degree of the denominator, so first you have to divide:
[tex]x^2+\dfrac{3x-2}{x^2-2x+1}[/tex]
Now, [tex]x^2-2x+1=(x-1)^2[/tex], so the remainder term can be expanded to get
[tex]\boxed{x^2+\dfrac a{x-1}+\dfrac b{(x-1)^2}}[/tex]
The tread life of tires mounted on light duty trucks follows the normal probability distribution with a mean of 60,000 miles and a standard deviation of 4,000 miles. Suppose you bought a set of four tires, what is the likelihood the mean tire life of these four tires is more than 66,000 miles?
Answer: 0.0013
Step-by-step explanation:
Given : The test scores are normally distributed with
Mean : [tex]\mu=\ 60,000[/tex]
Standard deviation :[tex]\sigma= 4,000[/tex]
Sample size : [tex]n=4[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x = 66,000
[tex]z=\dfrac{66000-60000}{\dfrac{4000}{\sqrt{4}}}=3[/tex]
The p-value = [tex]P(z>3)\=1-P(z<3)=1- 0.9986501\approx0.0013[/tex]
Hence, the likelihood the mean tire life of these four tires is more than 66,000 miles = 0.0013
Find the remainder when dividing 2^2013 by 15.
*Answer should be in modulo. Example: Find the remainder when dividing 2^100 by 21 and the answer is 2^100 = 16mod(21).*
[tex]2^{2013}=2^{4\cdot503+1}\\\\2^4=16\equiv 1\pmod{15}\\2^{4\cdot 503}\equiv 1\pmod{15}\\2^{4\cdot 503+1}\equiv 2\pmod{15}\\\\2^{2013}\equiv 2\pmod{15}[/tex]
3. For each of the following lists of integers, provide a simple formula or rule.. Assuming that your formula or rule is correct, determine the next three term of the sequence. 15, 20, 25, 30, 35,... a. b. 5,9, 13, 17, 21, ...
Step-by-step explanation:
Consider the first sequence:
15, 20, 25, 30, 35,...
Note that each term is increased by 5 from its previous term.
Therefore,
[tex]a_n=a_{n-1}+5[/tex]
If the pattern continue, the next three term of the sequence will be:
[tex]a_6=a_{6-1}+5[/tex]
[tex]a_6=a_{5}+5[/tex]
[tex]a_6=35+5[/tex]
[tex]a_6=40[/tex]
Similarly,
[tex]a_7=a_{7-1}+5[/tex]
[tex]a_7=a_{6}+5[/tex]
[tex]a_7=40+5[/tex]
[tex]a_7=45[/tex]
Similarly,
[tex]a_8=a_{8-1}+5[/tex]
[tex]a_8=a_{7}+5[/tex]
[tex]a_8=45+5[/tex]
[tex]a_8=50[/tex]
Thus, the next three term of the sequence 15, 20, 25, 30, 35,... is 40, 45, and 50.
Now, consider the second sequence:
5, 9, 13, 17, 21,...
Note that each term is increased by 4 from its previous term.
Therefore,
[tex]a_n=a_{n-1}+4[/tex]
If the pattern continue, the next three term of the sequence will be:
[tex]a_6=a_{6-1}+4[/tex]
[tex]a_6=a_{5}+4[/tex]
[tex]a_6=21+4[/tex]
[tex]a_6=25[/tex]
Similarly,
[tex]a_7=a_{7-1}+4[/tex]
[tex]a_7=a_{6}+4[/tex]
[tex]a_7=25+4[/tex]
[tex]a_7=29[/tex]
Similarly,
[tex]a_8=a_{8-1}+4[/tex]
[tex]a_8=a_{7}+4[/tex]
[tex]a_8=29+4[/tex]
[tex]a_8=33[/tex]
Thus, the next three term of the sequence 5, 9, 13, 17, 21,... is 25, 29, and 33.
The time it takes to ring up a customer at the grocery store follows an exponential distribution with a mean of 3.5 minutes. What is the probability density function for the time it takes to ring up a customer?
The probability density function for the time it takes to ring up a customer at the grocery store, following an exponential distribution with a mean of 3.5 minutes, is (1/3.5) × e[tex]^{(-x/3.5)[/tex].
Explanation:The probability density function for the time it takes to ring up a customer at the grocery store, following an exponential distribution with a mean of 3.5 minutes, is given by:
f(x) = (1/3.5) × e[tex]^{(-x/3.5)[/tex]
Where x is the time it takes to ring up a customer.
In this case, the exponential distribution models the time between events, which in this context is the time between customer arrivals at the grocery store.
The exponential distribution is a continuous probability distribution that is often used to model random events that occur independently and exponentially over time.
A loan of $1000 is to be paid back, with interest, at the end of 1 year. Aft er 3 months, a partial payment of $300 is made. Use the US Rule to determine the balance due at the end of one year, considering the partial payment. Assume a simple interest rate of 9%.
Answer:
total balance due at the end of 1 year is $769.75
Step-by-step explanation:
Given data
loan amount = $1000
time period = 1 year
return = $300
rate = 9%
to find out
balance due at the end of one year
solution
we know in question $300 return after 3 month so we first calculate interest of $1000 for 3 month and than we after 3 month remaining 9 month we calculate interest for $700
interest for first 3 month = ( principal × rate × time ) / 100 .............1
here time is 3 month so = 3/12 will take and rate 9 % and principal $1000
put all these value in equation 1 we get interest for first 3 month
interest for first 3 month = ( principal × rate × time ) / 100
interest for first 3 month = ( 1000 × 9 × 3/12 ) / 100
interest for first 3 month = $22.5
now we calculate interest for remaining 9 months i.e.
interest for next 9 months = ( principal × rate × time ) / 100
here principal will be $700 because we pay $300 already
interest for next 9 months = ( 700 × 9 × 9/12 ) / 100
interest for next 9 months = $47.25
now we combine both interest that will be
interest for first 3 months +interest for next 9 months = interest of 1 year
interest of 1 year = $22.5 + $47.25
interest of 1 year = $69.75
so amount will be paid after 1 year will be loan amount + interest
amount will be paid after 1 year = 1000 + 69.75
amount will be paid after 1 year is $1069.75
so total balance due at the end of 1 year = amount will be paid after 1 year - amount paid already
total balance due at the end of 1 year = $1069.75 - $300
total balance due at the end of 1 year is $769.75
Find i (the rate per period) and n (the number of periods) for the following loan at the given annual rate.
Annual payments of $3,600 are made for 12 years to repay a loan at 5.7% compounded annually.
i=
n=
Answer:
i = 5.7%
n = 12
Step-by-step explanation:
Compounded annually means once per year. So the rate per period is 5.7%, and the number of periods is 12.
Answer:
i = 5.7%
n = 12
Step-by-step explanation:
i (the rate per period) and n (the number of periods) for the following loan at the given annual rate.
Annual payments of $3,600 are made for 12 years to repay a loan at 5.7% compounded annually.
Therefore,
i = 5.7%
n = 12
Write the linear system of differential equations in matrix form then solve the system.
dx/dt = x + y
dy/dt = 4x + y
x(0) = 1, y(0) = 2
In matrix form, the system is
[tex]\dfrac{\mathrm d}{\mathrm dt}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}1&1\\4&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}[/tex]
First find the eigenvalues of the coefficient matrix (call it [tex]\mathbf A[/tex]).
[tex]\det(\mathbf A-\lambda\mathbf I)=\begin{vmatrix}1-\lambda&1\\4&1-\lambda\end{vmatrix}=(1-\lambda)^2-4=0\implies\lambda^2-2\lambda-3=0[/tex]
[tex]\implies\lambda_1=-1,\lambda_=3[/tex]
Find the corresponding eigenvector for each eigenvalue:
[tex]\lambda_1=-1\implies(\mathbf A+\mathbf I)\vec\eta_1=\vec0\implies\begin{bmatrix}2&1\\4&2\end{bmatrix}\begin{bmatrix}\eta_{1,1}\\\eta_{1,2}\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}[/tex]
[tex]\lambda_2=3\implies(\mathbf A-3\mathbf I)\vec\eta_2=\vec0\implies\begin{bmatrix}-2&1\\4&-2\end{bmatrix}\begin{bmatrix}\eta_{2,1}\\\eta_{2,2}\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}[/tex]
[tex]\implies\vec\eta_1=\begin{bmatrix}1\\-2\end{bmatrix},\vec\eta_2=\begin{bmatrix}1\\2\end{bmatrix}[/tex]
Then the system has general solution
[tex]\begin{bmatrix}x\\y\end{bmatrix}=C_1\vec\eta_1e^{\lambda_1t}+C_2\vec\eta_2e^{\lambda_2t}[/tex]
or
[tex]\begin{cases}x(t)=C_1e^{-t}+C_2e^{3t}\\y(t)=-2C_1e^{-t}+2C_2e^{3t}\end{cases}[/tex]
Given that [tex]x(0)=1[/tex] and [tex]y(0)=2[/tex], we have
[tex]\begin{cases}1=C_1+C_2\\2=-2C_1+2C_2\end{cases}\implies C_1=0,C_2=2[/tex]
so that the system has particular solution
[tex]\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}e^{3t}\\2e^{3t}\end{bmatrix}[/tex]
Final answer:
The linear system of differential equations can be written in matrix form as [dx/dt, dy/dt] = [1, 1; 4, 1] * [x, y]. By solving the system with the given initial conditions x(0) = 1 and y(0) = 2, the values of x and y at different time points can be determined.
Explanation:
To write the linear system of differential equations in matrix form, we can express the given equations as:
[dx/dt, dy/dt] = [1, 1; 4, 1] * [x, y]
Using the initial conditions x(0) = 1 and y(0) = 2, we can solve the system of equations to find the values of x and y at different time points.
A normal distribution has a mean 20 and standard deviation 5. What is the z score that corresponds to the value x=222
Answer: The z score that corresponds to the value x=22 is 0.4 .
Step-by-step explanation:
Given : A normal distribution has a mean 20 and standard deviation 5.
i.e. [tex]\mu=20[/tex]
[tex]\sigma=5[/tex]
Let x be the random selected variable.
We know that to find the z-score corresponds to the value x is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x = 22, we have
[tex]z=\dfrac{22-20}{5}=\dfrac{2}{5}\\\\\Rightarrow\ z=0.4[/tex]
Hence, the z score that corresponds to the value x=22 is 0.4
A z-score in a normal distribution measures the number of standard deviations a value is from the mean. To calculate it, use the formula z = (x - μ) / σ for the specific values provided, such as half a standard deviation below the mean, 5 points above the mean, three standard deviations above the mean, and 22 points below the mean.
The calculation of a z-score within a normal distribution is a common task in statistics, allowing one to determine how many standard deviations a particular value, x, is from the mean, μ, of the distribution. The z-score is calculated using the formula:
z = (x - μ) / σ
where x is the value in question, μ is the mean, and σ is the standard deviation. Now, we will calculate the z-scores for the given situations:
One-half of a standard deviation below the mean:5 points above the mean:Three standard deviations above the mean:22 points below the mean:Remember, when you use these calculations for specific numerical values, you need to insert the actual values of mean and standard deviation into the formula.
A vacuum cleaner dealership sold 370 units in 2011 and 411 units in 2012. Find the percent increase or decrease in the number of units sold.
The number of units sold increased or decreased? by about what percent?
Answer:
The percent of Increase is of 11.08% (0.1108)
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.
Since we have two different values for two different years, we can use the following Algebraic Expression to calculate the percent difference of sales between both years. The Expression would be the following,
[tex]370 * (x+1) = 411[/tex]
Where x is the percent difference. Now we solve for x,
[tex]370 * (x+1) = 411[/tex]
[tex]x+1 = 1.1108[/tex]
[tex]x = 0.1108[/tex]
so now we see that the percent of Increase is of 11.08% (0.1108)
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
The number of units sold increased by about 11.08%.
Explanation:To find the percent increase or decrease in the number of units sold, we need to calculate the difference between the number of units sold in 2012 and 2011, and then divide that difference by the number of units sold in 2011.
The amount of increase or decrease is calculated as: (Number of units sold in 2012 - Number of units sold in 2011)/Number of units sold in 2011 x 100
In this case, the calculation is: (411 - 370)/370 x 100 = (41/370) x 100 = 11.08%
Therefore, the number of units sold increased by about 11.08%.
Prove that (AB)−1=B−1A−1
Answer with Step-by-step explanation:
Consider,
[tex](AB)^{-1}(AB)=I[/tex] (Identity rule)
Multiplying by B⁻¹ on the both the sides, we get that
[tex](AB)^{-1}(AB)B^{-1}=IB^{-1}\\\\(AB)^{-1}A(BB^{-1})=B^{-1}[/tex]
And we know that BB⁻¹ = I
So, it becomes,
[tex](AB)^{-1}A=B^{-1}[/tex]
Now, multiplying by A⁻¹ on both the sides, we get that
[tex](AB)^{-1}AA^{-1}=B^{-1}A^{-1}\\\\(AB)^{-1}=B^{-1}A^{-1}[/tex] (AA⁻¹=I)
Hence, proved.
Trucks in a delivery fleet travel a mean of 120 miles per day with a standard deviation of 22 miles per day. The mileage per day is distributed normally. Find the probability that a truck drives between 100 and 157 miles in a day. Round your answer to four decimal places.
The probability that a truck drives between 100 and 157 miles in a day within a normal distribution can be calculated using z-scores. The z-scores for 100 and 157 miles are computed relative to the mean and standard deviation, and the corresponding probabilities are obtained from the standard normal distribution table. The final probability is the difference of these two probabilities.
Explanation:Given that the distribution of trucks' daily mileage is normally distributed, we can approach this problem by using the principles of normal distribution and z-scores. The z-score is a measure of how many standard deviations an element is from the mean.
First, we calculate the z-scores for both 100 miles and 157 miles:
Z1 =(100 - 120) / 22 = -0.9091 Z2 = (157 - 120) / 22 = 1.6818
Next, we look up these z-scores in the standard normal distribution table (or use a calculator with a normal distribution function), which will give us the probabilities P(Z To arrive at four decimal places precision, this process typically involves using a statistical calculator or software.
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Eliminate the parameter.
x = 3 cos t, y = 3 sin t
Answer:
x^2+y^2 = 3^2
Step-by-step explanation:
We need to eliminate the parameter t
Given:
x = 3 cos t
y = 3 sin t
Squaring the above both equations
(x)^2=(3 cos t)^2
(y)^2 =(3 sin t)^2
x^2 = 3^2 cos^2t
y^2=3^2 sin^2t
Now adding both equations
x^2+y^2=3^2 cos^2t+3^2 sin^2t
Taking 3^2 common
x^2+y^2=3^2 (cos^2t+sin^2t)
We know that cos^2t+sin^2t = 1
so, putting the value
x^2+y^2=3^2(1)
x^2+y^2 = 3^2
Hence the parameter t is eliminated.
To eliminate the parameter in the given equations x = 3 cos t and y = 3 sin t, we can substitute cos(t) and sin(t) in terms of x and y to eliminate the parameter. The resulting equations represent the line y = x.
Explanation:To eliminate the parameter in the given equations x = 3 cos t and y = 3 sin t, we need to express x and y in terms of each other without the parameter 't'. Using the identity cos^2(t) + sin^2(t) = 1, we can solve for cos(t) and sin(t), and substitute them into the equations to eliminate the parameter.
Using the fact that cos(t) = x/3 and sin(t) = y/3, we can rewrite the equations as x = 3 cos(t) = 3(x/3) = x and y = 3 sin(t) = 3(y/3) = y. Therefore, eliminating the parameter results in x = x and y = y, which simply means that the equations represent the line y = x.
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A customer brings a check of 2,941. he wants 100 in cash, put 20% of the remaining into her savings account then the rest into a checking account. How much will ge be putting in his checking account
Answer:
Amount theta she is putting in Checking account is 2272.80
Step-by-step explanation:
Given:
Amount on check = 2941
Amount that he want in cash = 100
Amount she put in saving account = 20% of remaining after getting cash
Remaining Amount she put in checking account.
To find: Amount in her Checking Account.
Amount left after taking cash = 2941 - 100 = 2841
Amount that she put in saving account = 20% of 2841 = [tex]\frac{20}{100}\times2841[/tex] = 568.20
Amount in her checking account = 2941 - 100 - 568.20 = 2272.8
Therefore, Amount theta she is putting in Checking account is 2272.80
How many primes are between 2^31 and 2^32? What is the approximate ratio of primes to
all numbers between 2^31 and 2^32?
Final answer:
There are approximately 132489 prime numbers between 2³¹and 2³², with a ratio of primes to all numbers being approximately 0.1156.
Explanation:
To find the number of primes between 2³¹and 2³², we can use the Sieve of Eratosthenes algorithm. With this algorithm, we can mark all the multiples of each prime number, and the remaining unmarked numbers will be prime.
Using this method, we can calculate that there are approximately 132489 primes between 2³¹ and 2³². The ratio of primes to all the numbers between 2³¹and 2³²is approximately 0.1156.
You would like to make a salad that consists of lettuce, tomato, cucumber, and onions. You go to the supermarket intending to purchase one variety of each of these ingredients. You discover that there are nine varieties of lettuce, four varieties of tomatoes, two varieties of cucumbers, and three varieties of onions for sale at the supermarket. How many different salads can you make?
Answer: 216
Step-by-step explanation:
Given : We like to make a salad that consists of lettuce, tomato, cucumber, and onions.
The number of varieties of lettuce = 9
The number of varieties of tomatoes = 4
The number of varieties of cucumbers = 2
The number of varieties of onions = 3
Now, the number of different salads we can make is given by :-
[tex]9\times4\times2\=216[/tex]
Hence, we can make 216 different types of salads.
A medical equipment industry manufactures X-ray machines. The unit cost C (the cost in dollars to make each X-ray machine) depends on the number of machines made. If X machines are made, then the unit cost is given by the function C (x) = 1.2x^2 -600x + 89,966. How many machines must be made to minimize the unit cost?
Do not round your answer.
Answer:
x = 250 units
Step-by-step explanation:
We can easily solve this problem by using a graphing calculator or any plotting tool.
We must find the minimum point in the graph. This corresponds to the number of machines that produce the minimum cost.
The equation is
C (x) = 1.2x^2 -600x + 89,966
Please see attached image below
By producing x = 250 units, we obtain the minimum cost
An english reading list has 9 american novels and 7 english novels. A student must read 5 from the list and at least 3 must be english novels. In how many different ways can the five books be selected combination
[tex]_7C_3\cdot {_{13}C_2}=\dfrac{7!}{3!4!}\cdot\dfrac{13!}{2!11!}=\dfrac{5\cdot6\cdot7}{2\cdot3}\cdot\dfrac{12\cdot13}{2}=2730[/tex]
The number of ways the student can select 5 books such that at least 3 are English novels can be calculated as the sum of combinations of 3 English and 2 American, 4 English and 1 American, and all 5 being English.
Explanation:The subject matter of this question is based in the mathematics field, specifically combinatorics. To tackle this problem, we will utilize the concept of combination, which is a way of selecting items from a larger set where order does not matter.
The student has to select 5 books out of 16 (9 American and 7 English novels). But at least 3 should be English novels. It means the student can pick 3, 4 or all 5 novels as English novels. Let's calculate each possibility:
3 English novels and 2 American novels: Ways = C(7,3) * C(9,2) 4 English novels and 1 American novel: Ways = C(7,4) * C(9,1) All 5 English novels: Ways = C(7,5)
So, the total number of ways = [C(7,3)*C(9,2)] + [C(7,4)*C(9,1)] + C(7,5). Here C(n,r) denotes combination and is equal to n! / [(n-r)!*r!], where '!' denotes factorial.
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