(a) The eigenvalues of the coefficient matrix is [-1,3] and for λ=42, we get the eigenvector [1,5].
Itcan be found by solving the characteristic equation |A-λI|=0, where A is the coefficient matrix and λ is the eigenvalue. Solving for λ, we get λ=0 and λ=42.
o find the eigenvectors, we substitute each eigenvalue into the equation (A-λI)x=0 and solve for x. For λ=0, we get the eigenvector [-1,3]. For λ=42, we get the eigenvector [1,5].
(b) The solution is y(t) = c1e^(0t)[-1,3] + c2e^(42t)[1,5].
To find the real-valued solution to the initial value problem, we can use the eigenvectors and eigenvalues to diagonalize the coefficient matrix. We have A = PDP^-1, where P is the matrix whose columns are the eigenvectors and D is the diagonal matrix with the eigenvalues on the diagonal.
Using the initial condition y2(0) = 15, we can solve for the constants c1 and c2.
The solution is y(t) = c1e^(0t)[-1,3] + c2e^(42t)[1,5]. Solving for c1 and c2 using the initial condition, we get
y(t) = [-15e^(42t) + 3e^(0t), 15e^(42t) + 5e^(0t)].
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How many pounds of eroded material does the verde riveer carry past given point in 1/4 of a day
The Verde River's discharge and sediment load determine how much eroded material it can move beyond a given site in 6 hours, or 1/4 of a day. We cannot accurately determine the amount of eroded material carried by the river without knowing these numbers.
The amount of water that flows past a specific place in a river per unit of time is known as its discharge. The amount of sediment (including eroded material) that a river carries is referred to as its sediment load.
Both of these numbers can fluctuate considerably based on a number of variables, including the time of year, regional geology, and weather patterns.
In order to precisely determine the amount of eroded debris carried by the Verde River past a specific point in 1/4, thus we need further information.
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In a study of the effects of marijuana during pregnancy, measurements on babies of mother who used marijuana during pregnany were compared to measurements on babies of mothers who did not. A 95% confidence interval for the difference in mean head circumference (nonuse minus use) was .61 to 1.19 cm. What can be said from this statement about a p-value for the hypothesis that the mean difference is zero?
The 95% confidence interval of .61 to 1.19 cm provides strong evidence that there is a difference in mean head circumference between babies of mothers who used marijuana during pregnancy and those who did not. Furthermore, the small p-value suggests that the mean difference is statistically significant, and is not likely to be zero.
The 95% confidence interval for the difference in mean head circumference between babies of mothers who used marijuana during pregnancy and those who did not is .61 to 1.19 cm. This implies that the true mean difference is likely to be between .61 and 1.19 cm. The p-value is a measure of how likely it is that the difference in means is zero, and can be used to assess the statistical significance of the difference in means.
Therefore, the p-value for the hypothesis that the mean difference is zero is very small and can be considered statistically significant.
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The small p-value suggests that the mean difference is statistically significant, and is not likely to be zero.
A 95% confidence interval (CI) is a range of values that is expected to include the true population mean with a probability of 0.95. The CI for the difference in mean head circumference between non-users and users of marijuana during pregnancy is 0.61 to 1.19 cm.
This means that the sample mean difference (nonuse minus use) falls within this interval.
If the true population mean difference were zero, the sample mean difference would fall within the range of random sampling error, and the null hypothesis that there is no difference between the groups would be supported.
However, since the 95% CI does not include zero, we can conclude that the difference is statistically significant at the 0.05 level.
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From the 30 pictures I have of my daughter's first birthday, my digital picture frame will only hold 3 at a time. How many different groups of 3 pictures can I put on the frame?
There are 4060 different groups of 3 pictures that can be displayed on the frame.
What is combination?A combination is a method of choosing a subset of items from a bigger set when the order of the items is irrelevant. For instance, the two-letter possibilities for a set of three letters A, B, and C are AB, AC, and BC. Combinations are employed in many different mathematical and practical situations, including combinatorial optimization, statistics, and probability theory. Choosing a password with a specific amount of characters, picking a combination of things from a menu, and assembling a team from a group of players are all examples of how they are utilised in daily life.
Given that, from 30 pictures we need to form groups of 3.
The combination of selecting groups is given as:
nCk = n! / (k! * (n - k)!)
Substituting the values we have:
30C3 = 30! / (3! * (30 - 3)!)
= 30! / (3! * 27!)
= (30 * 29 * 28) / (3 * 2 * 1)
= 4060
Hence, there are 4060 different groups of 3 pictures that can be displayed on the frame.
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if a data set is symmetrical, which statistical measure best represents the entire spread of the data set?
The best statistical measure to represent the entire spread of the data set when a data set is symmetrical is the standard deviation.
The standard deviation is a measure of how much the individual data points vary from the mean of the data set.
It is calculated by taking the square root of the variance.
It is the average of the squared differences of each data point from the mean.
In a symmetrical data set, the mean, median, and mode are all equal and located at the center of the data set.
This represents that the data points are evenly distributed around the center.
The standard deviation can provide a good measure of the spread of the data points from this center.
Other measures of spread representing the range or interquartile range, may not be as useful in a symmetrical data set .
As they are more affected by outliers and skewness in the data.
Therefore, the symmetrical data set in best way represented by statistical measure known as standard deviation.
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Suppose that the Quick-Fix Car Service franchise finds that the income generated by its stores can be modeled by assuming that the income is a continuous stream with a monthly rate of flow at time t given by. f(t) = 10,000e0.02t (dollars per month)
Find the total income from a Quick-Fix store for the first 2 years of operation.
Suppose that the Quick-Fix Car Service franchise finds that the income generated by its stores can be modeled by assuming that the total income is a continuous stream with a monthly rate of flow at time t given by f(t) = 10,000e^0.02t (dollars per month).
The given function is f(t) = 10,000e^0.02t (dollars per month)So, the monthly rate of flow of income is given by f(t) = 10,000e^0.02t (dollars per month)The total income generated in the first 2 years of operation can be found using integration.
So, the integral of f(t) with respect to t from 0 to 24 months will give us the total income generated in the first 2 years of operation.∫₀²₄f(t)dt = ∫₀²₄10,000e^0.02tdt= [500,000e^0.02t] from 0 to 24= 500,000(e^0.02*24 - e^0)≈ $223,443.95Thus, the total income generated from a Quick-Fix store for the first 2 years of operation is approximately $223,443.95.
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Solve, give all possible values of x
|2x+3| = 2
Answer:
-½ and -⁵/₂
Step-by-step explanation:
solve for both 2x +3 = 2 and 2x +3 = -2
An electronics company currently has 278 stores throughout the U. S. Because of the popularity of online shopping, many of the company's stores are closing. The total number of stores is expected to decline by about 5% this year. Assuming the same decline continues, you can use a function to approximate the total number of stores remaining after x years
We can estimate that after 5 years, there will be approximately 215 stores remaining.
Assuming a 5% decline in the number of stores each year, the function to approximate the total number of stores remaining after x years is exponential, and can be written in the form:
g(x) = a *[tex](0.95)^x[/tex]
where g(x) is the total number of stores remaining after x years, and a is the initial number of stores (278 in this case).
Therefore, the equation for the function is:
g(x) = 278 * [tex](0.95)^x[/tex]
This function takes into account the compounding effect of the 5% decline each year on the initial number of stores and can be used to estimate the number of stores that will remain after any number of years. For example, after 5 years, the number of stores would be approximate:
g(5) = 278 *[tex](0.95)^5[/tex]
g(5) ≈ 214.5
Therefore, we can estimate that after 5 years, there will be approximately 215 stores remaining.
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Complete Question:
An electronics company currently has 278 stores throughout the U.S. Because of the popularity of online shopping, many of the company's stores are closing. The total number of stores is expected to decline by about 5% this year. Assuming the same decline continues, you can use a function to approximate the total number of stores remaining after x years. Write an equation for the function. If it is linear, write it in the form gx=mx+b . If it is exponential, write it in the form gx=abx. gx=___?
The following question has two parts. First, answer part A. Then, answer part B.
A fortune cookie has these dimensions.
A purple rectangle is shown. Its height is labeled as five-eighths, and its length is labeled as 2 and one-half.
Part A
Jonathan says he knows that
1
2
of
2
1
2
is
1
1
4
. And he knows that
5
8
is almost one half if you use estimation. But he is not sure whether
5
8
×
2
1
2
will be more or less than
1
1
4
.
Select the correct option.
A.
5
8
×
2
1
2
is equal to
1
1
4
B.
5
8
×
2
1
2
is less than
1
1
4
C.
5
8
×
2
1
2
is more than
1
1
4
D.
It is not possible to say using estimation.
Part B
Calculate the area of the fortune cookie.
A.
1
5
8
square inches
B.
1
25
16
square inches
C.
1
9
16
square inches
D.
1
25
16
square inches
1) 5/8 × 2 1/2 is slightly more than 1/4. Therefore, the correct option is:
C.
2) The area of the fortune cookie is: B. 1 25/16 square inches.
What is area?In mathematics, area is the measure of the size or extent of a two-dimensional shape or surface, such as a rectangle, triangle, circle, or any other closed figure. It is the amount of space inside the boundary of a shape, usually measured in square units.
For example, the area of a rectangle can be found by multiplying its length and width. The formula is:
Area = length × width
The area of a circle can be found by multiplying the square of its radius by pi (π). The formula is:
Area = πr²
where r is the radius of the circle.
Part A:
To estimate whether 5/8 × 2 1/2 is more or less than 1/4, we can round 5/8 to 1/2 and round 2 1/2 to 2. Then we have:
1/2 × 2 = 1
So, 5/8 × 2 1/2 is slightly more than 1/4. Therefore, the correct option is:
C. 5/8 × 2 1/2 is more than 1/4.
Part B:
The area of the fortune cookie can be found by multiplying its length by its height. The length is given as 2 1/2, which can be written as an improper fraction:
2 1/2 = 5/2
The height is given as 5/8. So, the area can be devised as:
Area = length × height
Area = (5/2) × (5/8)
Area = 25/16
Therefore, the area of the fortune cookie is:
B. 1 25/16 square inches.
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Can someone help me with this please??
Answer:
Angle CAB= 28°
Angle ABC= 45°
Angle DCA= 73°
Angle DCE=107°
Step-by-step explanation:
Enter the correct answer in the box.
Write this expression in simplest form.
Don’t include any spaces or multiplication symbols between coefficients or variables in your answer.
16h^(10/2) *remove the root sign
16h^5 *simplify the exponent
Answer: 16h^5
Step-by-step explanation: im correct
Mrs Smith walks a half a mile a day after work. She works five days a week. How many yards will she have walked for the week by Friday morning?
The distance Mrs. Smith covers is 3520 yards during the duration of the week by Friday morning.
One week has seven days in total.
Mrs. Smith walks half a mile each day after work, she walks a total of
0.5 miles/ day × 7 days/ week = 3.5 miles/ week
Now, if we calculate the distance on Friday morning, she must have walked four times till Friday morning since she has to walk after her work.
Therefore,
0.5 miles/ day × 4 days = 2 miles
To convert miles to yards, we can use the fact that there are 1760 yards in one mile:
2 miles/week × 1760 yards/mile = 3520 yards/week
Therefore, by Friday morning, Mrs. Smith will have walked 3520 yards.
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Solve the following simultaneous equations algebraically
xy = 6
y-x=1
Step-by-step explanation:
xy=6
x=6/y (1)
now,
XY=6
6/y-y=6
in 2015,the annual rate of population growth in the us was about 1.2%.find the growth factor forthe us
The annual rate of population growth in the US was about 1.2% in 2015, then the growth factor will be 1.012. The growth factor depends upon annual growth rate.
What is the growth factor?Growth factor typically refers to a constant multiplier that is used to calculate the growth or decay of a quantity over time. It is given as:
Growth factor = (1 + annual growth rate)
Therefore, the growth factor for the US in 2015 can be calculated as follows:
Growth factor (1+0.012) = 1.012
Therefore, the growth factor for the population of the US in the year 2015 was about 1.012.
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1) Adam wants to buy a home priced at $215,000. The bank requires him to make a 5% down payment and
he will finance the rest for 30 years at 4.5% interest. He has to also pay the closing costs below. Find the
a) the down payment b) the amount of the mortgage c) the closing costs d) the amount financed with
closing costs e) the monthly payment f) the total amount repaid g) the amount paid to interest.
Application Fee
Borrower's Credit check
Points
Appraisal Fee
Title Search
Title Insurance
Attorney Fee
Documentation stamp
Processing fee
$ 25
65
1.5% of Mortgage
350
215
450
400
0.30% of Mortgage
1.25% of Mortgage
can someone explain interval and set notation (algebra 2)
Interval notation is a way to represent an interval of real numbers on the number line. Set notation is a way to represent a set of elements.
What is interval and set notation?Interval notation is a way to represent an interval of real numbers on the number line.
The notation uses parentheses, brackets, and infinity symbols to indicate whether the endpoints of the interval are included or excluded from the set of numbers.
For example, [3, 8) represents the interval of real numbers from 3 (included) to 8 (excluded), while (-∞, 4) represents the interval of real numbers less than 4 (excluding 4), and extending to negative infinity.
Set notation is a way to represent a set of elements. It uses curly braces to enclose the elements of the set and can include various symbols to indicate properties of the set.
For example, {2, 3, 5, 7, 11} represents the set of prime numbers less than 12, while {x | x is an even number} represents the set of even numbers.
The vertical bar | is used to separate the variable (x in this case) from the condition that must be met for elements to be included in the set (x is an even number).
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Solve the following equation using the graphing method. Write answers in set notation. Round irrational solutions to the nearest tenths place (1 decimal).
Equation: 5x^2+2x-6=0
Thus, from the attached graph the, solution of the equation in set notation is found as: Domain = (-∞, +∞) and Range: [6, ∞).
Explain about the graphing method?The visual method actually only requires three easy actions to complete:
Slope-Intercept Form should be applied to both equations.Each linear equation's graph should appear within the similar coordinate plane.Identify the system's solution.
If the lines cross, the system's solution can be found by using the coordinates of such intersection point.The problem cannot be solved if the lines are parallel.There are an endless number of solutions if the lines meet, which means they form a single line.The given equation is-
5x^2+2x-6=0
Since it is a quadratic equation it will have 1 value on y axis for every two values for x.
The graph will be quadratic with 2 degree.
Using the graphing tool, draw the graph.
Thus, from the attached graph the, solution of the equation in set notation is found as:
Domain = (-∞, +∞)
Range: [6, ∞)
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g a random sample of 100 automobile owners in the state of alabama shows that an automobile is driven on average 23,500 miles per year with a standard deviation of 3900 miles. assume the distribution of measurements to be approximately normal. a) construct a 99% confidence interval for the average number of miles an automobile is driven annually in alabama.
We can be 99% confident that the average number of miles an automobile is driven annually in Alabama is between 21,342.6 and 24,637.4 miles
To answer this question, we need to use the following formula for a confidence interval for the mean: CI = (μ - z*(σ/√n), μ + z*(σ/√n)), Where μ is the population mean, z is the z-score for the given confidence level, σ is the population standard deviation, and n is the sample size. Using the given information, we can calculate the confidence interval for the mean:CI = (23500 - 2.575*(3900/√100), 23500 + 2.575*(3900/√100)), CI = (21342.6, 24637.4)
To summarize, we used the formula for a confidence interval for the mean and the given information to calculate the confidence interval for the average number of miles an automobile is driven annually in Alabama. This confidence interval is (21342.6, 24637.4), which means we can be 99% confident that the average number of miles an automobile is driven annually in Alabama is between 21,342.6 and 24,637.4 miles.
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using quicksort, if the pivot is [ 28 ], which values are found to be in the wrong partition after the first pass across the array? 28 17 45 51 35 14 88 24
There are no values in the wrong partition after the first pass across the array using quicksort when the pivot is [28].
The given problem states that using quicksort, if the pivot is [28], which values are found to be in the wrong partition after the first pass across the array? {28, 17, 45, 51, 35, 14, 88, 24}.Quicksort is a sorting algorithm that uses the divide and conquer approach. Here’s how to find out which values are found to be in the wrong partition after the first pass across the array .Step-by-step The first thing to do is to pick the pivot value. It can be any element in the array. The pivot value is 28.Then, the partition function is performed to place the pivot element in its final position in the sorted array.
All the values smaller than the pivot will be on the left side, and all the values greater than the pivot will be on the right side of the pivot. Here’s the array after the partition: 17 24 14 [28] 35 45 51 88Now, the elements to the left of the pivot are 17 24 14, and the elements to the right of the pivot are 35 45 51 88. Since the pivot is placed in its final position, there are no values in the wrong partition after the first pass across the array. Therefore, there are no values in the wrong partition after the first pass across the array using quicksort when the pivot is [28].
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Find a particular solution Find a particular solution to the differential equation dạy 4. dy +5 dt + 3y = edit dt2 In the form y = Aedit, where A is a complex constant. Here i = -1 is the square root of -1. g(t) = symbolic expression 2
A particular solution to the differential equation is [tex]y = Ae^(iwt) =(1/4)e^({i\sqrt{3t} )}.[/tex]
A particular solution to the differential equation,
[tex]\frac{d^4y}{dt^4} +5\frac{d^2y}{dy^2} +3y=2[/tex],
we assume that y has the form [tex]y = Ae^(iwt)[/tex], where A is a complex constant and w is a real constant to be determined. Substituting this form of y into the differential equation, we get:
[tex](-w^4 + 5w^2 + 3)Ae^(iwt) = 2.[/tex]
we solve the quadratic equation [tex]-w^4 + 5w^2 + 3 = 0[/tex], which can be factored as:
[tex](w^2 - 3)(-w^2 + 1) = 0.[/tex]
The roots are w = ±√3 and w = ±i. Since w must be real, we choose w = √3. Substituting this value of w and solving for A, we get:
[tex]A= \frac{2}{\sqrt{(-\sqrt{3} )^4 + 5(\sqrt{3})^2 +3 } }[/tex] [tex]=\frac{2}{5+3} = \frac{1}{4}[/tex]
Therefore, the particular solution is:
[tex]y = Ae^(iwt) =(1/4)e^({i\sqrt{3t} )}.[/tex]
A differential equation is a mathematical equation that relates a function to its derivatives. It is a powerful tool used in many fields of science and engineering to describe how physical systems change over time. The equation typically includes the independent variable (such as time) and one or more derivatives of the dependent variable (such as position, velocity, or temperature).
Differential equations can be classified based on their order, which refers to the highest derivative present in the equation, and their linearity, which determines whether the equation is a linear combination of the dependent variable and its derivatives. Solving a differential equation involves finding a function that satisfies the equation. This can be done analytically or numerically, depending on the complexity of the equation and the available tools.
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ABCD is a quadrilateral in which BD = 15 cm., perpendiculars from A and Con BD are 6 cm and 8 cm respectively. Calculate the area of the quadrilaterals
The area of the quadrilateral is 161.24 cm².
How to deal with quadrilateral?We can see that we can divide the quadrilateral into two triangles: ABD and CBD. We know that the height of ABD is 6 cm and the height of CBD is 8 cm. We also know that BD is 15 cm. To find the area of each triangle, we need to find the base of each triangle. We can do this using the Pythagorean theorem.
For triangle ABD:
AB² = AD² + BD²
AB² = (6 cm)² + (15 cm)²
AB² = 261 cm²
AB = [tex]\sqrt(261) cm[/tex]
For triangle CBD:
BC² = CD² + BD²
BC² = (8 cm)² + (15 cm)²
BC² = 289 cm²
BC = 17 cm
Now we can find the areas of the triangles:
Area of ABD =[tex]\frac{1}{2}[/tex] * AB * 6 cm
Area of ABD = [tex]\frac{1}{2}[/tex] * [tex]\sqrt(261) cm[/tex] * 6 cm
Area of ABD = 93.24 cm^2
Area of CBD = [tex]\frac{1}{2}[/tex] * BC * 8 cm
Area of CBD = [tex]\frac{1}{2}[/tex] * 17 cm * 8 cm
Area of CBD = 68 cm²
Finally, we can find the area of the quadrilateral by adding the areas of the triangles:
Area of ABCD = Area of ABD + Area of CBD
Area of ABCD = 93.24 cm² + 68 cm²
Area of ABCD = 161.24 cm²
Therefore, the area of the quadrilateral is 161.24 cm².
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A lumber company sells two types of wood: mahogany and black walnut. The company has 260 boards of mahogany and 320 boards of black walnut. The lumber company expects to sell at most 380 boards of wood. The profit for selling the wood is $20 per board for mahogany and $6 per board for black walnut.Write and graph a system of inequalities to represent the constraints of this problem situation.
The system of equation are
x ≤ 260 y ≤ 320 x + y ≤ 380The profit function is = 20x + 6y
How to write the system of equationsLet's define the following variables:
x: the number of boards of mahogany sold
y: the number of boards of black walnut sold
Based on the problem statement, we can write the following system of inequalities to represent the constraints:.
x ≤ 260 (there are only 260 boards of mahogany available)
y ≤ 320 (there are only 320 boards of black walnut available)
x + y ≤ 380 (the total number of boards sold can't exceed 380)
The profit function is:
P(x, y) = 20x + 6y
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55 of Ronnie's shots went in the basket when she took 125 shots? What percent of her shots went in?
Answer:
44%
Step-by-step explanation:
the answer is 44% because 55/125 can be simplifed by dividing both sides by 5 to get 11/25
Classify the polynomial
X+2
Urgent !!!
Answer: Quadratic polynomial.
Step-by-step explanation: Polynomials with 2 as the degree.
91, 95 and 97 which one is the prime number
97
Step-by-step explanation:
a prime number is a number that can only divide by itslef and by 1 and the result is a whole number (no fractions or dicemals)
91 can be divided by 13 and 7
which makes it not a prime number
95 can be divided by 5 and 19 so it's not a prime number either
Please help me on this!
Answer:
One solution
Step-by-step explanation:
I added a photo of my solution
Answer:
The system has one solution: (0, 4).
Angela made 23 cards for her friends. She wants to make 19 more cards. How many cards will she make in all?
By using addition calculation, we determine that Angela will end up making 42 cards in total.
Angela made 23 cards for her friends, but she wants to make even more to share with others. To determine how many cards she will make in total, we need to add the number of cards she has already made with the number of cards she plans to make.
So, we add 23 (the number of cards she has made) and 19 (the number of cards she plans to make) so in total, she will make:
23 + 19 = 42
Therefore, Angela will make 42 cards in all.
By using simple arithmetic calculation of basic addition, we find that Angela will make 42 cards in all.
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Determine whether each of the following conditional statements is true or false. (a) If 10<7,10<7, then 3=43=4. (c) If 10<7,10<7, then 3+5=83+5=8. (b) If 7<10,7<10, then 3=43=4. (d) I…Determine whether each of the following conditional statements is true or false.(a) If 10<7,10<7, then 3=43=4.(c) If 10<7,10<7, then 3+5=83+5=8.(b) If 7<10,7<10, then 3=43=4.(d) If 7<10,7<10, then 3+5=83+5=8.
The given conditional statements are false, true, false, True.
They are determined by following:
(a) False - The statement "If 10<7,10<7, then 3=43=4" is false, since 10 is not less than 7.
(b) True - The statement "If 7<10,7<10, then 3=43=4" is true, since 7 is less than 10.
(c) False - The statement "If 10<7,10<7, then 3+5=83+5=8" is false, since 10 is not less than 7.
(d) True - The statement "If 7<10,7<10, then 3+5=83+5=8" is true, since 7 is less than 10.
Conditional statements are used in mathematics and logic to express relationships between events and conditions. These statements consist of an "if-then" structure, where the "if" clause is the antecedent or condition, and the "then" clause is the consequent or outcome.
The truth value of the conditional statement depends on whether the condition is true or false. If the condition is true, then the outcome is also true, and the statement is considered true.
If the condition is false, then the outcome may be true or false, and the statement is considered false. Conditional statements are widely used in mathematical proofs, programming, and reasoning to establish logical connections between different events and conditions.
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5 A bathtub contains 35.5 liters of water.
When the plug is removed, 5 liters of wate
drain from the bathtub each minute. How
many minutes will it take all of the water to
drain from the bathtub?
A 177.5
B 7.1
C 17.75
D 6.95. i cant find my answer
Answer:
Option B
Step-by-step explanation:
Using Proportionality, we get
Let x be the quantity of water in bathtub. Y be time it be removed.
By formula, x2/y2= k = x1/y1
x/y = 5/1(x1=5, y1= 1)[Given]
5x2= 35.5
35.5 = 5y2
=> y = 35.5/5 = 7.1 minutes
-2 (x - 5) plssss helppp
Answer:
-2x + 10
Step-by-step explanation:
-2(x - 5)
-2(x) -2(-5)
-2x + 10
Helping in the name of Jesus.
I NEED HELP ON THIS ASAP!!
a) Graph this system of inequalities, we can plot the lines x = 0, y = 0, x = 260, y = 320, and x + y = 380
b) The maximum profit of $5200 achieved.
Define the term selling profit?Selling profit is the profit that a business makes on the sale of its products or services. It is the difference between the selling price and the cost of product.
a) Let x be the number of boards of mahogany sold and y be the number of boards of black walnut sold. Then, the constraints of the problem can be represented by the following system of inequalities:
x ≥ 0 (non-negative constraint)
y ≥ 0 (non-negative constraint)
x ≤ 260 (maximum number of mahogany boards available)
y ≤ 320 (maximum number of black walnut boards available)
x + y ≤ 380 (maximum number of boards that can be sold)
To graph this system of inequalities, we can plot the lines x = 0, y = 0, x = 260, y = 320, and x + y = 380 on a coordinate plane and shade the feasible region that satisfies all of the constraints. The feasible region is the area that is bounded by these lines and includes the origin (0, 0).
b) The profit function P(x, y) can be defined as follows:
P(x, y) = 20x + 6y
To maximize the profit, we need to find the values of x and y that satisfy all of the constraints and maximize the profit function P(x, y).
One way to do this is to use the corner-point method. We can evaluate the profit function at each of the corners of the feasible region and find the corner that gives the maximum profit.
The corners of the feasible region are (0, 0), (0, 320), (260, 0), and (120, 260).
P(0, 0) = 0
P(0, 320) = 6(320) = 1920
P(260, 0) = 20(260) = 5200
P(120, 260) = 20(120) + 6(260) = 4720
Therefore, the maximum profit of $5200 can be achieved by selling 260 boards of mahogany and 0 boards of black walnut.
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