Answer:
(a) three angles: 62°, 34°, and 84°
Step-by-step explanation:
You want to know which specification can result in an infinite number of triangles.
Angles onlyIf only the angles are specified, the sides can have an infinite number of (related) lengths.
three angles: 62°, 34°, and 84° will create an infinite number of triangles
Sides onlyAssuming the side lengths can form a triangle, they will specify exactly one triangle.
Lengths 3 cm, 5 cm, and 10 cm cannot form a triangle.
Sides and angle betweenWhen two sides are specified along with the angle between them, exactly one triangle can be formed.
How many yards are in two and three-fourths miles
There are 4840 yards in two and three-fourths miles.
What is a unitary method?A unitary method is a mathematical way of obtaining the value of a single unit and then deriving any no. of given units by multiplying it with the single unit.
We know 1 mile is equal to 1760 yards.
Therefore, The number of yards in two and three-fourths miles can be obtained by multiplying the unit equivalent of one mile with two and three-fourths miles which is,
= 1760×[tex]2\frac{3}{4}[/tex] yards.
= 1760×(11/4) miles.
= 440×11 yards.
= 4840 yards.
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Answer: 4840
Step-by-step explanation:
a square garden is 33 1/10 what is the length of each of its sides
Answer:
Step-by-step explanation:
if you mean 33 1/10cm2 then the side is square root of the number which is 5.753 but if the gardens side is 33 1/10 then all the sides are 33 1/10.maybe you typed the question wrongly or some
^^
Using f(x) = 3x - 3 and g(x) = -x, find g(f(x)).
Answer:
Given f(x) = 3x - 3 and g(x) = -x, to find g(f(x)), we first need to substitute f(x) into g(x).
So, g(f(x)) = g(3x - 3) = -(3x - 3) = -3x + 3.
The function S(t) = 4.5.3 models the growth of a tumor where t is the number of months since the tumor was discovered and Sis the size of the tumor in cubic millimeters. The size of the tumor when it was discovered was 4.5 cubic millimeters. Find the total change in the size of the tumor in the first 3 months and find the average rate of change in the size of the tumor in the first 3 months The total change in size of the tumor in the first 3 months was cubic millimeters. The average rate of change of the tumor in the first 3 months was cubic millimeters per month.
The total change in size of the tumor in the first 3 months is 13.5 cubic millimeters. The average rate of change of the tumor in the first 3 months is 4.5 cubic millimeters per month.
The total change in size of the tumor in the first 3 months can be calculated by subtracting the size at the beginning (4.5 cubic millimeters) from the size at the end of the 3-month period (S(3) = 4.5 * 3 = 13.5 cubic millimeters). Therefore, the total change in size of the tumor in the first 3 months is 13.5 cubic millimeters.
To calculate the average rate of change in the size of the tumor in the first 3 months, we need to divide the total change in size (13.5 cubic millimeters) by the time (3 months).
Therefore, the average rate of change of the tumor in the first 3 months is 4.5 cubic millimeters per month.
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An art class is making a mural for their school which has a triangle drawn in the middle. The length of the bottom of the triangle is x. Another side is 14 more than three times the length of the bottom of the triangle. The last side is 15 more than the bottom of the triangle. Write and simplify an expression for the perimeter of the triangle.
Answer:
P = 5x + 29
Step-by-step explanation:
Let's call the length of one side of the triangle "a".
According to the problem, another side is 14 more than three times the length of the bottom of the triangle, so it can be expressed as 3x + 14.
The last side is 15 more than the bottom of the triangle, so it can be expressed as x + 15.
The perimeter of the triangle is the sum of all three sides, so:
P = x + (3x + 14) + (x + 15)
Simplifying:
P = 5x + 29
There are several guidelines to follow when constructing graphs that summarize statistical data. Which of the following statements is least accurate?
A. Axes should be clearly labeled
B. Axes that are numerical should be to the appropriate scale
C. Graphs should have a lot of adornments
D. The simplest graph that effectively communicates the data should be used
The correct option: C. Graphs should have a lot of adornments is the statement which is least accurate.
Rules for constructing graphs for statistical data?Data that are gathered and/or generated through statistics during the course of statistical observations through statistical data processing are known as statistical data.
The guidelines below apply to vertical bar charts. Change "vertical" to "horizontal" and "horizontal" to "vertical" in these rules for horizontal bar charts.
Add a title to the graph's top.The scale on the vertical axis needs to be uniform and set to zero.Use gridlines that are horizontal.The meaning of each bar should indeed be written on the label.The qualitative variable which that bar chart depicts should be described in the label for the horizontal axis. A horizontal axis marker is not required if the labels on each individual bar are sufficiently illustrative.Put "Frequency" or "Count" on the vertical axis.Thus, the statement which is least accurate is "Graphs should have a lot of adornments"
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make a factorial story
suppose a random sample of 40 houses are selected from the city. estimate the probability that the mean value of the 40 houses is less than $500,000 . show your work
a. Thus, the probability that a randomly selected house has a value less than $500,000 is 0.71
b. the probability that the mean value of the 40 houses is less than $500,000 is 0.9863
a. The mean of the distribution is given as $403,000 and the standard deviation is $278,000.
To estimate the probability that a randomly selected house has a value less than $500,000:
P( x < 500,000)=P(x=0)+P(x=500)
=0.34+0.37+
=0.71
Thus, the probability that a randomly selected house has a value less than $500,000 is 0.71
b. -since 40 is larger than or equal to 30, we assume a normal distribution.
-The probability can therefore be calculated as:
P(x')=P( z < (x' -u)/σ.sqrt(n)))
P(z< (500-403)/(578sqrt(40)))
=P(z<2.2068)
=0.986336
Hence, the probability that the mean value of the 40 houses is less than $500,000 is 0.9863
The complete question is -
a) Suppose one house from the city will be selected at random. Use the histogram to estimate the probability that the selected house is valued at less than $500,000. Show your work.
(b) Suppose a random sample of 40 houses are selected from the city. Estimate the probability that the mean value of the 40 houses is less than $500,000. Show your work.
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suppose a family has 3 children of di erent ages. we assume that all combinations of boys and girls are equally likely.a) Formulate precisely the sample space and probability measure that describes the genders of three children in the order in which they born.b) Supose we see the parents with two girls. Assuming we have no other information beyond that at least two of the children are girls, what is the probability that the child we have not yet seen is a boy?c). Suppose we see the parents with two girls, and the parents tell us that these are the two youngest children. What is probability that the oldest child we have not yet seen is a boy?
If a family has 3 children of different ages and all combinations of boys and girls are considered equally likely:
a) the sample space and probability measure that describes the gender of the three children in their birth order is: 1/8.
b) Assuming we see the parents with two girls and there is no other information beyond that, the probability that the child we have not yet seen is a boy is: 1/4.
c). If we see the parents with two girls, and the parents tell us that these are the two youngest children, the probability that the oldest child we have not yet seen is a boy is: 1/8
a) The sample space for the genders of three children can be represented as a set of strings of length 3, each consisting of B (for boy) or G (for girl).
The total number of combinations of boys and girls is 2^3 = 8. Each combination has the same likelihood of occurring, so each combination can be assigned a probability of 1/8. So the sample space and probability measure can be described as:
Sample space: {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}
Probability measure: P(BBB) = P(BBG) = P(BGB) = P(BGG) = P(GBB) = P(GBG) = P(GGB) = P(GGG) = 1/8
b) Assuming we see the parents with two girls, the possible combinations for the genders of the three children are:
GGB, GGBThe probability of the third child being a boy is 1/2. So, given two girls, the probability that the child we have not yet seen is a boy is:
P(third child is a boy) = P(GGB) + P(GGB) = 1/8 + 1/8 = 1/4c) Assuming we see the parents with two girls and they tell us that these are the two youngest children, the possible combinations for the genders of the three children are:
GBGThe only possible combination is GBG, so the probability of the oldest child being a boy is 1. So, given two girls and that they are the youngest children, the probability that the oldest child we have not yet seen is a boy is:
P(oldest child is a boy) = P(GBG) = 1/8 = 1/8Complete Question:
"Suppose a family has 3 children of different ages and all combinations of boys and girls are considered equally likely.
a) Precisely formulate the sample space and probability measure that describes the gender of the three children in their birth order.
b) Given that the parents have two girls and no other information is available, what is the probability that the remaining child is a boy?
c) If the parents have two girls and they inform us that they are the two youngest children, what is the probability that the oldest child is a boy?"
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Help pls quickly it’s a answer I need fast thanks for help
Answer:
Step-by-step explanation:
D
We have two geometric sequences of positive real numbers:
6,a,b and 1/b,a,54
Solve for a
The value of a is 3√2 when the two geometric sequences of positive real numbers of 6, a,b and 1/b, a,54.
A sequence of the numbers in which all the numbers are arranged in such a manner so that the ratio of two successive numbers is always constant,
The geometric progression and the constant ratio are known as the common ratio (r).
A general G.P. can be represented as:
a,ar,ar²,ar³,...............,arⁿ⁻¹,........
Where a=First term, r is the Common ratio and the n=Number of terms.
The nth term of a G.P.:
an=arⁿ⁻¹
Some of the infinite terms of a G.P.:
S∞=a/1−r
Here r<1
Let:
[tex]a_n=a_1q^n^-^1\\a_n_-_1=a_1q^n^-^2\\a_n_+_1=a_1q^n[/tex]
Now multiply the last two equations:
[tex]a_n_-_1 a_n_+_1=a_1q^n^-^2a_1q^n\\\\a_n_-_1 a_n_+_1=a_1^2+q^n^-^2q^n\\\\a_n_-_1 a_n_+_1=a_1^2+q^n^+^n^-^2\\\\a_n_-_1 a_n_+_1=a_1^2+q^2^n^-^2\\\\a_n_-_1 a_n_+_1=a_1^2+q^2^(^n^-^1^)\\\\a_n_-_1 a_n_+_1=a_1^2+q^(^n^-^1^)^2\\\\\sqrt{a_n_-_1 a_n_+_1} =a_1^2+q^(^n^-^1^)\\\\\sqrt{a_n_-_1 a_n_+_1}=a_n[/tex]
Formula:
[tex]a_n=\sqrt{a_n_-_1a_n_+_1}[/tex]
a=√6b-----(1)
a=√1/b*54----(2)
By Equalating both equations:
6b=√1/b*54
b²=54/6
b²=9
b=3
substitute the b value in equation(1)
a=√6b
a=√6*3
a=√18
a=3√2
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Given (x – 7)2 = 36, select the values of x. x = 13 x = 1 x = –29 x = 42
The values of x are 13 and -1. Option A
How to factorize the expressionAlgebraic expressions are defined as expressions that have terms, variables, coefficients, constants and factors.
They are made up of arithmetic operations.
Given that;
(x - 7)²= 36
expand the bracket
x² - 7x - 7x + 49 = 36
collect like terns
x² - 14x + 49 = 36
x² - 14x + 13
Factorize the expression
(x² - 13x) + (x - 13)
Factor the common terms
x(x - 13) + 1(x - 13)
x = 13. x = -1
Hence, the values are 13 and -1
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Shannon manages a small zoo and she has been analyzing the attendance data. Shannon finds that the number of visitors increases exponentially as the temperature increases, and this situation is represented by the function f(x) = 3x. Shannon also finds a linear equation that models the number of people who leave the park early depending on the change in temperature, and it is represented by g(x) = −x + 4. The graph of the two functions is below. Find the solution to the two functions and explain what the solution represents. Graph increasing exponentially from the left, crosses the y axis at one and goes to infinity and intersects a linear graph that decreases and crosses the y axis at four and the x axis at four.
The solution to the two functions is the ordered pair (x, y) = (1, 3).
What is Function?A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
Given that,
f(x) = 3ˣ and g(x) = -x + 4
The solution to the both functions is the intersection of both graphs.
That is, f(x) = g(x)
3ˣ = -x + 4
When x = 1, then the value of both the functions is 3.
So the solution is (1, 3).
This indicates that number of visitors at temperature 1 is 3, which is equal to the number of people who leave the park at temperature 1.
Hence the solution to the functions f(x) and g(x) is the intersection of both the functions at the point (1, 3).
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A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at the focus will be reflected to the other focus. The focus of a hyperbolic mirror (see figure) has coordinates (17, 0). Find the vertex of the mirror if the mount at the top edge of the mirror has coordinates (17, 17). (Round your answers to two decimal places.) (x, y) = ()
The vertex of the hyperbolic mirror is (17, 8.5). To find the vertex of a hyperbolic mirror, you need to find the midpoint between its two foci.
In this case, the focus has coordinates (17, 0) and the mount at the top edge of the mirror has coordinates (17, 17). To find the midpoint, you average the x-coordinates and the y-coordinates.
The x-coordinate of the midpoint is: (17 + 17) / 2 = 17.
The y-coordinate of the midpoint is: (0 + 17) / 2 = 8.5
Therefore, the vertex of the hyperbolic mirror is (17, 8.5).
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A town’s population of children increased from 376 to 421 during the past year.
Which equation shows how to find the percent increase?
p = 421 - 376 / 376
p = 421 - 376 / 421
p = 376 / 421 - 376
p = 376 + 421 / 376
Answer:
the 3 answer is the right one
Which line best models line l?
y
F. x = 4
G. y = 4
H. x = 4y
x
J. y = x + 4
The line best models line l is x=4, the correct option is A.
How to find the function which was used to make graph?There are many tools we can use to find the information of the relation which was used to form the graph.
A graph contains data of which input maps to which output.
Analysis of this leads to the relations which were used to make it.
For example, if the graph of a function is rising upwards after a certain value of x, then the function must be having increasingly output for inputs greater than that value of x.
If we know that the function crosses x axis at some point, then for some polynomial functions, we have those as roots of the polynomial.
Given;
The graph shown in the figure.
Now,
On the graph the x coordinate is 4 throughout
So, x=4
On the graph the y coordinate is continous
Therefore, the graph function will be x=4
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Ben originally filled out 8 more applications than
Henry. Then each boy filled out 3 additional
applications, bringing the total to 28. How many
applications did each boy originally fill out?
The number of applications filled by Henry is 7 and the number of applications filled by Ben is 15.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Given that, Ben originally filled out 8 more applications than Henry.
Let the number of applications filled by Henry be x.
Now, x+8
Then each boy filled out 3 additional applications, bringing the total to 28.
Here, x+3+x+8+3=28
2x+14=28
2x=14
x=7
So, x+8=15
Therefore, the number of applications filled by Henry is 7 and the number of applications filled by Ben is 15.
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What is the value of x?
Enter your answer in the box.
x =
The angle x in the heptagon is 98 degrees.
How to find the interior angles of a polygon?The polygon above has 7 sides. Therefore, the name of the polygon is heptagon. Therefore, the sum of interior angles of a polygon can be represented as follows:
sum of interior angles of a polygon = 180(n - 2)
where
n = number of sidesTherefore,
sum of interior angles of the heptagon = 180(7- 2)
sum of interior angles of the heptagon = 180(5)
sum of interior angles of the heptagon = 900 degrees
Hence, let's find x as follows:
122 + 146 + 142 + 140 + 110 + 142 + x = 900
x = 900 - 802
x = 98 degrees
Hence,
x = 98 degrees
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Find the compound interest and the amount after twelve seconds if the interest is compounded every three seconds if the principal is $20,000 and the rate of interest is 800% per minute
The compound interest and the amount are $2349980000 and $2.35 * 10⁹
Finding the compound interest and the amountFrom the question, we have the following parameters that can be used in our computation:
Principal = $20,000
Rate = 800% per minute = 8
Duration = 12 seconds
n = 20 i.e. every 3 seconds
The amount is calculated as
A = P(1 + r/n)^(nt)
So, we have
A = 20,000(1 + 8/20)^(20*12)
A = 2.35 * 10⁹
So, we have
Interest = 2.35 * 10⁹ - 20000
Interest = 2349980000
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Find the missing length indicated:
8
4
2
10
The missing length is 6.
How do we calculate?8
4
2
10
From the numbers shown above, it is found that the difference from one length to the other is 2.
Therefore, the missing length is 6.
Length is described as a measure of distance.
In the International System of Quantities, length is also a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived.
In conclusion, the base unit for length is the meter in the International System of Units system the base unit for length is the meter.
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irene and her three sister plan to buy a gift for their mother birthday and split the cost equally the total cost of the gift must be less than 60$
Ver en español
At a berry farm, Clayton picks 1 1 4
cups of blackberries and twice as many raspberries. He takes the berries home and uses them to make mixed-berry muffins. If Clayton needs
3 4 of a cup of mixed berries for each batch of muffins, how many batches can he make?
The number of batches of muffins she can make will be 5.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that at a berry farm, Clayton picks 1¹/₄ cups of blackberries and twice as many raspberries. He takes the berries home and uses them to make mixed-berry muffins.
The number of the batches of muffins can be calculated as:-
Number = ( 1¹/₄ + 2 x 1 ¹/₄ ) / ( 3 / 4 )
Number = [ ( 5 / 4 ) + ( 5 / 2 ) ] / ( 3 / 4 )
Number = [ ( 15 / 4 ) / ( 3 / 4 ) ]
Number = 5
Therefore, the number of batches of muffins will be 5.
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Match each equation on the left to its solution on the right.
Some answer choices on the right will be used more than once.
The right expression will be -3x + 3 = 3( 1 - x ) for the given expressions.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that there are different expressions the correct expression choice is calculated s below:-
-3x + 3 = 3( 1 - x )
-3x + 3 = 3 - 3x
-3x + 3 = -3x + 3
Hence, the correct ex[ression would be -3x + 3 = 3( 1 - x ).
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find the ratio in which 2x 3y-5 duvudes tge kine segment joining (8, -9) and (2, 1). also, find the coorinates of the point of division
The ratio in which the line 2x + 3y – 5 = 0 divides the line segment joining the points (8, –9) and (2, 1) is 8:1, and the coordinates of the point are 8/3 and -1/9
The given data is, the line 2x+3y-5= 0 and the points are (8, -9) and (2, 1)
Let us consider the ratio to be α: 1
To find the values of a and b we need to use the midpoint formula:
[tex]X=\frac{m_1x_1+m_2x_2}{m_1+m_2} ,\frac{m_1y_1+m_2y_2}{m_1+m_2}[/tex]
For Points (8,–9) and (2, 1) by applying the section formula we get,
[tex]X=\frac{2\alpha +8}{\alpha +1} ,y=\frac{\alpha -9}{\alpha -1}[/tex]
Given equation of line is 2x + 3y–5 = 0 substitute the values of x,y in the equation of line:
[tex]X=2(\frac{2\alpha +8}{\alpha +1} )+3(\frac{\alpha -9}{\alpha -1})-5=0[/tex]
4α+16+3α-27-5α-5
2α=16
α=8
Hence the ratio is 8:1.
Also, the coordinates are
[tex]X=\frac{2\alpha +8}{\alpha +1}=\frac{2(8) +8}{8+1}=\frac{24}{9}=\frac{8}{3} \\\\ y=\frac{\alpha -9}{\alpha -1}=\frac{8-9}{8-1}=\frac{-1}{9}[/tex]
∴ The coordinates are (x,y) = (8/3,-1/9).
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Andre and Noah started tracking their savings at the same time. Andre started with $15 and deposits $5 per week. Noah started with $2.50 and deposits $7.50 per week. The graph of Noah's Savings is given and his equation is y = 7.5x + 2.5, where x represents the number of weeks and y represents his savings.
Write the equations for Andre's savings and graphs it alongside Noah's. What does the intersection point mean in this situation?
6 is the intersection point mean in the given situation.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Given that Andre started with $15 and deposits $5 per week.
Noah started with $2.50 and deposits $7.50 per week.
Noah's Savings is given and his equation is y = 7.5x + 2.5, where x represents the number of weeks and y represents his savings.
The equations for Andre's savings is y = 5x + 15
The intersection point of the two equations is the point at which both Andre and Noah have the same amount in their savings.
5x+15=7.5x + 2.5
Subtract 7.5x on both sides
-2.5x + 15 = 2.5
Add 2.5x to both sides
Divide both sides by 2.5
x = 6
Hence, 6 is the intersection point mean in this situation.
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suppose a and b are integers that divide the integer c. if a and b are relatively prime, show that ab divides c. show, by example, that if a and b are not relatively prime, then ab need not divide c.
If a and b are integers that divide c and are relatively prime, then ab divides c. If a and b are not relatively prime, then ab need not divide c.
If a and b are integers that divide c, this means that c is divisible by both a and b. Additionally, if a and b are relatively prime, this means that they have no common factors besides 1. In this case, since a and b both divide c, then it follows that their product, ab, also divides c.
However, if a and b are not relatively prime, this means that they have common factors besides 1. In this case, the common factors of a and b may not divide c, so it is not necessarily true that ab divides c.
For example, if a = 6, b = 8, and c = 24, then 6 and 8 both divide 24, but 6 and 8 are not relatively prime (they have a common factor of 2). In this case, 6 * 8 = 48 does not divide 24.
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Help please which graph shows…. Look at image
The correct graph is "Option 3". The solution is −11 < x < − 4 or x > 2.
Which graph shows the solution set of [tex]\frac{ x^{2}+9x-22}{x+4} \geq 0[/tex]?Let's work through your inequality piece by piece.
[tex]\frac{x^2+9x-22}{x+4} > 0[/tex]
Let's locate the inequality's crucial points.
[tex]\frac{x^2+9x-22}{x+4} = 0[/tex]
[tex]x^2+9x-22=0[/tex]
(Add x+4 to both sides)
(x−2)(x+11)=0
(Left side of the equation factor)
x−2=0 or x+11=0
(Set all variables to 0)
x=2 or x=−11
Verify any potential weak points.
x=2(Works in original equation) (Works in original equation)
x=−11(Works in original equation) (Works in original equation)
Important points:
x=2 or x=−11
(Equalizes both sides)
x=−4
(Sets the left denominator to 0)
Check the gaps between important points. Test the interval values to see if they are effective.
(Works incorrectly in the original inequality) [tex]x < -11[/tex]
(Works in original inequality)[tex]-11 < x < -4[/tex]
(Not applicable to the original inequality) x > 2 (Works in original inequality)
The answer is [tex]-11 < x < -4 $ or $ x > 2~[/tex]
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4. Rachel has 3 types of ornament. She has 6 different wooden animals, 4 different sea-shells and 3 different pottery ducks. (i) She lets her daughter Cherry choose 5 ornaments to play with. Cherry chooses at least 1 of each type of ornament. How many different selections can Cherry make? [5]
There are a total of 13 different ornaments to choose from, with 3 types of ornaments.
What is total number?Total number is a term used to describe the sum of all the values or items within a set. It is usually used to calculate an aggregate or grand total, such as the sum of all the numbers in a list or the total number of people in a population. Total number can also be used to describe the absolute value or quantity of something, such as the total number of days in a year.
When Cherry chooses at least 1 of each type, she will have to select a total of 5 ornaments. This means that Cherry can make a total of 13C5 (13 choose 5) different selections. This is equal to 715 different selections.
This can also be calculated by first selecting 1 ornament from each of the 3 types, which gives 3 different selections. Then, for the 2 remaining ornaments, Cherry can select any of the 13 ornaments, giving a total of 3 x 13 = 39 different selections. In total, this gives 3 + 39 = 42 different selections.
To conclude, Cherry can make a total of 715 different selections when she chooses at least 1 of each type of ornament.
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Determine:
A) Graph the solution of each SIL inequality (separately) using a test point and a tabulation. (line graph)
B) The vertices that make up the solution region:
- Justify the process to find the coordinates of the vertices of the solution region.
- Determine in a table the coordinates of all the vertices that make up the solution region of the SIL.
The intersection point is the solution of inequalities is (15, 7.5).
What are inequalities?Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.
The given system of inequalities are 13x+37y≤481 -----(i) and 27x+21y≤567 -----(ii)
Here, x≥4 and y≥2
From the graph we can see that, inequalities are intersecting at (15, 7.5)
Therefore, the intersection point is the solution of inequalities is (15, 7.5).
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A super large tank was initially filled with brine solution of 20 liters. At some point, fresh water starts to flow in at a rate of 8 liters per minute, and the well mixed solution flows out at half of the rate.Part I Set-up an Initial Value Problem to describe the above problem using: y(t) to represent the amount of salt in the tank t minutes after the process started. There was certain amount of salt in the tank initially,y0 to represent the initial amount.Part II Solve the above IVP to get y(t). Part III How much time (in minutes) will it take for the amount of salt in the tank be reduced to 20% of the initial amount?
The y(t) represents the amount of salt in the tank t minutes after the process started, and y(0) = 20 represents the initial amount. The y(t) is [tex]y(t) = 4 + 8t - \frac{1}{2} y^2[/tex]. It will take approximately 6.875 minutes for the amount of salt in the tank to be reduced to 20% of the initial amount.
From the given data,
A super large tank was initially filled with brine solution of 20 liters.
At some point, fresh water starts to flow in at a rate of 8 liters per minute, and the well mixed solution flows out at half of the rate.
The initial value problem (IVP) to describe the above problem can be set up as follows:
⇒[tex]\frac{d}{dt} y(t)=\frac{1}{2} (8-y(t))[/tex]
⇒y(0)=20
Where y(t) represents the amount of salt in the tank t minutes after the process started, and y(0) = 20 represents the initial amount.
Part II:
To solve the IVP, we need to find y(t) for a given t. This can be done using numerical methods or analytical methods, such as separation of variables. Using separation of variables, we have:
⇒[tex]\frac{d}{dt} y(t)=\frac{1}{2} (8-y(t))[/tex]
⇒[tex]\frac{dy}{dt}=\frac{1}{2}(8-y)[/tex]
⇒2dy = (8 - y) dt
⇒2dy = (8 - y)dt
Integrating both sides with respect to t, we get:
⇒∫2dy = ∫(8 - y)dt
⇒2y = 8t - (1/2)y^2 + C
where C is a constant of integration. Using the initial condition y(0) = 20, we can find the value of C:
⇒[tex]2y=8t-\frac{1}{2}y^{2} +C[/tex]
⇒[tex]2y=8t-\frac{1}{2}y^{2} +20[/tex]
Solving for y, we get:
⇒[tex]y(t) = 4 + 8t - \frac{1}{2} y^2[/tex]
Therefore, the y(t) is [tex]y(t) = 4 + 8t - \frac{1}{2} y^2[/tex].
Part III:
To find the time it takes for the amount of salt in the tank to be reduced to 20% of the initial amount, we can solve for t when y(t) = 0.2 * y0:
⇒[tex]y(t) = 4 + 8t -\frac{1}{2}y^2 = 0.2 * y0[/tex]
⇒[tex]4 + 8t - \frac{1}{2} y^2 = 0.2 * 20[/tex]
Solving for t, we get:
⇒t =[tex]\frac{1}{8}[/tex](4 - 0.2 * 20) = approximately 6.875 minutes.
Hence, it will take approximately 6.875 minutes for the amount of salt in the tank to be reduced to 20% of the initial amount.
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