Answer: 173 centimeters of ribbon left
Step-by-step explanation:
Antonio starts with 2 meters of ribbon, which is equal to 200 centimeters (since 1 meter = 100 centimeters). He cuts off 27 centimeters to decorate the gift, so he is left with:
200 - 27 = 173 centimeters
Therefore, Antonio has 173 centimeters of ribbon left.
Use the technique of linear regression to find the line of best fit for the given points. Round any intermediate calculations to no less than six decimal places, and round the coefficients to two decimal places.
(1,9)
, (2,9)
, (3,4)
, (4,2)
, (5,4)
, (6,2)
, (7,3)
The line of best fit for this set of points is y = -1.00x + 9.00.
What is coefficient?Coefficient is a value that is used to indicate the degree or amount of a certain property or characteristic. It is usually used in mathematics and physics, and is a numerical measure of how two variables are related. For example, the coefficient of friction is a number that indicates how much resistance there is between two surfaces when one is sliding over the other. In economics, the coefficient of elasticity is used to measure how much of a change in one variable affects a change in another.
Linear regression is a technique used to find the line of best fit for a given set of points. To use linear regression, we must first calculate the mean of all the x and y coordinates. The mean of the x coordinates is 4, and the mean of the y coordinates is 4.5.
We then calculate the sum of the squared differences between the x and y coordinates and their respective means. This sum is 36.5.
Next, we calculate the sum of the products of the differences between the x and y coordinates and their respective means. This sum is -25.
Using the formula for linear regression, we can calculate the slope of the line of best fit. The slope is -1.00.
To calculate the y-intercept of the line of best fit, we use the slope and the mean of the x and y coordinates. The y-intercept is 9.00.
The line of best fit for this set of points is y = -1.00x + 9.00.
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Consider the following function.
q(x)=3x−(2+5x)8
Step 2 of 2 : Find two points on the line to graph the function.
The two points on the line to graph the function is (0, -16) and (1, -53).
What are coordinates?A coordinate system in geometry is a method for determining the precise location of points or other geometrical objects on a manifold, such as Euclidean space, using one or more integers, or coordinates. The order of the coordinates is crucial, and they are sometimes recognised by their position in an ordered tuple and occasionally by a letter, as in "the x-coordinate". In elementary mathematics, the coordinates are assumed to be real numbers, although they might instead be complex numbers or components of a more abstract system, such a commutative ring.
The given function is:
q(x)=3x−(2+5x)8
To find two points on the line to graph the function we substitute different values of x.
For x = 0:
q(x)=3x−(2+5x)8
q(0) = 3(0) - (2 + 5(0))8
q(0) = -16
For x = 1:
q(x)=3x−(2+5x)8
q(1) = 3(1) - (2 + 5(1))8
q(1) = 3 - (7)8
q(1) = -53
Hence, the two points on the line to graph the function is (0, -16) and (1, -53).
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A right square pyramid has a height of 8.5 m and a base perimeter of 25 m. Calculate the surface area of the pyramid to the nearest square metre. (Don't forget to calculate the slant height)
SHOW WORK!!
The surface area of the right square pyramid is 106. 25 m²
What is a square pyramid?A square pyramid is seen as a pyramid with a square base.
If the apex is vertically above the center of the square, it is called a right square pyramid, and has four vertical symmetries.
Some properties of a right square pyramid are;
It has 5 FacesFour side faces which are trianglesThe base is a square8 edgesThe formula for the surface area of a square pyramid is half of the product of its base perimeter and its slant height:
SA = P × l / 2
where P is the base perimeter and l is the slant height.
Substitute the values;
Surface area = 25(8.5)/2 = 212.5/2 = 106. 25 m²
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substracts 5 - 3 1/3
Answer:
5/3
1 2/3
Step-by-step explanation:
5 - 3 1/3 can be simplified as follows:
5 - 3 1/3 = 5 - (3 + 1/3) [grouping the whole number and fraction together]
= 5 - (10/3) [converting the mixed number to an improper fraction]
= 15/3 - 10/3 [finding a common denominator for subtraction]
= 5/3
Therefore, 5 - 3 1/3 is equal to 5/3.
Jaylon works for a concrete company
His customer would be him to pour a new driveway. The concrete must be 3/4 fr deep.
Look at the drawing and calculate the number of cubic feet of concrete he will need to bring to the job.
Answer:
volume = length × width × depth
Assuming that the driveway has a rectangular shape, you can measure the length and width of the area to be covered and multiply by the depth of 3/4 ft (which is 0.75 ft) to get the volume in cubic feet.
For example, if the driveway is 20 ft long and 10 ft wide, the volume of concrete needed would be:
volume = length × width × depth
= 20 ft × 10 ft × 0.75 ft
= 150 cubic feet
Spelling List A-4
Short-o Sound - /o/
Spelling Words
1. dog
2.
top
3.
not
4.
box
5. jog
The answer after shorting are dog,top,not,box,jog.
What is sorting?
Sorting is the process of arranging or categorizing a set of items in a particular order or sequence according to specific criteria. It is a fundamental operation in computer science and is used extensively in various applications such as data analysis, information retrieval, and database management.
In general, sorting involves comparing elements of a collection, such as an array or a list, and rearranging them according to a predetermined order. The order can be based on a variety of factors, such as numerical value, alphabetical order, or date/time stamp. The two main categories of sorting algorithms are comparison-based sorting and non-comparison based sorting.
Some common sorting algorithms include bubble sort, insertion sort, selection sort, quicksort, merge sort, and heapsort. Each algorithm has its own advantages and disadvantages, and the choice of algorithm depends on the specific application and the characteristics of the data being sorted.
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Your complete question is:- Short "O" sound spelling in the following :-
Dog, Top, Not, Box, Jog
The Fort Salonga News charges $29.50 for a classified ad that is four or fewer lines long. Each line
above four lines costs an additional $5.25. Express the cost of an ad algebraically as a piecewise
function.
The cost of an ad algebraically as a piecewise function is 29.50x + 5.25
What is an algebraic function?An algebraic function is a polynomial equation that can be constructed using only algebraic operations such as addition, subtraction, multiplication, division and exponentiation
The given statement reads:
The Fort Salonga News charges $29.50
Each line above four lines costs an additional $5.25.
Let the expression be in x
y= 29.50x + 25.5
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Mathematical problem please help
Answer:
8
Step-by-step explanation:
Range is difference between max and min values:
20 - 12 = 8
Carla is deep-sea diving with two friends. Frank is exploring a coral reef 6.7 meters in
front of Carla, and Lola is floating on the surface directly above Carla. If Lola and Frank are 9.7 meters apart, how far apart are Carla and Lola? If necessary, round to the nearest tenth.
The distance between carla and Lola is approximately 7.01 meters
To find the distance between themLooking at the description given, we can conclude that the figure will resemble a right angle triangle.
And to solve this we use the Pythagoras theorem.
This states that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides “.
The sides of this triangle have been named Perpendicular, Base and Hypotenuse.
Here, the hypotenuse is the longest side, as it is opposite to the angle 90°
We can represent this as
H² = O² + A²
Where?
H = Hypotenuse (The longest side) 9.7mA = Adjacent side (The base of the triangle) 6.7mO = Opposite side (The perpendicular to the adjacent side)We are given the hypotenuse and the Adjacent side; we are to find the Opposite side.
Using the expression above, we will have.
9.7² = 6.7² + Opp²
94.09 = 44.89 + Opp²
Opp² = 94.09 - 44.89
Opp² = 49.2
Opp = √49.2
Opp = 7.01 meters approximately
Hence, the distance is 7.01 meters
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The multiplication of an integer by its consecutive odd integer is 11 more than the square of the number. Find the number
In the statement , the number is 5.5.
What is integer?The Latin term "Integer," which implies entire οr intact, is where the wοrd "integer" first appeared. Zerο, pοsitive numbers, and negative numbers make up the particular categοry οf numbers knοwn as integers.
Here let us take the given integer as x.
Then , cοnsecutive οdd integer is x+2.
Nοw accοrding tο the given questiοn is multiplicatiοn οf an integer by its cοnsecutive οdd integer is 11 mοre than the square οf the number. then, cοnverting intο expressiοn,
=> x [tex]\times[/tex] (x+2) = [tex]x^2+11[/tex]
=> [tex]x^2+2x=x^2+11[/tex]
=> 2x = 11
=> x = 11/2 = 5.5
Hence the number is 5.5.
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In a science class, 13.5% of the students have earned an A. If 3 students are chosen at random, what is the probability of choosing at least one student who has an A in the science class? Give your answer rounded to the nearest thousandth.
Answer:To introduce her class to binomial distributions, Mrs. Desai gives a
-item, multiple-choice quiz. The catch is, that students must simply guess an answer (A through E) for each question. Mrs. Desai uses her computer's random number generator to produce the answer key so that each possible answer has an equal chance to be chosen. Patti is one of the students in this class.
Let
the number of Patti's correct guesses.
To get a passing score on the quiz, a student must guess correctly at least
times. Would you be surprised if Patti earned a passing score? Compute an appropriate probability to support your answer.
Step-by-step explanation:
ASAPPPPP NEED HELP RN what is the value of this expression (-17) - (26) divided by 2
Thus, the solution of the expression [(-17) - (26) divided by 2] is found as the -21.5.
Explain about the subtraction of numbers?A quantity is subtracted from another quantity in subtraction, which is a mathematical process. Sutraction is expressed in word puzzles by phrases like "x smaller than 3" or "9 subtracted from x."
Add the terms first prior to utilizing the negative sign again for sum when subtracting negative numbers.Just one coefficients are added and subsequently the sign is retained when an algebraic term is subtracted from another.The given expression:
= [(-17) - (26)] / 2
Open the inner most brackets.
= [-17 - 26]/2
The numbers with the same sign (either positive or negative numbers gets added retaining the same sign).
= -43 / 2
Divide the number with 2.
= -21.5
Thus, the solution of the expression [(-17) - (26) divided by 2] is found as the -21.5.
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60 POINTS PLSS
Why is the quotient of three divided by one-fifth different from the quotient of one-fifth divided by three? Tell a story that could describe each situation.
HELP ASAP
After answering the presented question, we can conclude that They are equation still pleased with the quantity of pizza they received, although it is less than in the previous scenario.
What is Quotient?A quotient refers to the result obtained when one quantity is divided by another. For example, if you have 10 apples and you want to divide them equally among 2 people, each person would get 5 apples. The quotient in this case is 5, which represents the number of apples that each person would receive.
The quotient can be expressed in different forms, depending on the situation. If you are dividing two integers, the quotient can be a whole number (also known as an integer quotient) or a fraction (also known as a rational quotient). For example, if you divide 10 by 3, the quotient is 3 with a remainder of 1, which can be expressed as [tex]3 + 1/3[/tex] or as the fraction 10/3.
In algebra, the quotient can also refer to the result obtained when dividing two algebraic expressions. For example, if you divide [tex]x^2 + 3x + 2[/tex] by [tex]x + 1,[/tex] the quotient is [tex]x + 2[/tex], which represents the quotient of the division.
The concept of quotient is also used in other areas of mathematics, such as in modular arithmetic and in group theory, where it refers to the result of dividing one element of a group or a ring by another.
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What is the measure of angle B if the two triangles are similar? HELP PLS
1.) 35
2.) 45
3.) 113
4.) 22
The answer is 2) 45
According to the question
If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional.
Thus, if we obtain the measure of an angle in one of the triangles, we can determine the measure of the corresponding angle in the other triangle.
So here triangles ABC and DEF are similar so they will be congruent.
Here angle A and angle D are congruent, angle B and angle E are congruent and angle C and angle F are congruent.
So, angle B≅angle E= 45°
What is meant by congruent?
Congruent means having the same shape and size, with all corresponding angles and sides being equal in measure and length, respectively. It is denoted by the symbol ≅.
What is meant by similar?
Similar refers to the property of two geometric figures having the same shape but possibly different sizes. Corresponding angles of similar figures are congruent, and the ratios of their corresponding sides are equal.
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The two triangle is similar measure of angle B is 45.
What is meant by congruent?Congruent means having the same shape and size, with all corresponding angles and sides being equal in measure and length, respectively. It is denoted by the symbol ≅.
According to the question:If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional.
Thus, if we obtain the measure of an angle in one of the triangles, we can determine the measure of the corresponding angle in the other triangle.
So here triangles ABC and DEF are similar so they will be congruent.
Here angle A and angle D are congruent, angle B and angle E are congruent and angle C and angle F are congruent.
So, angle B≅angle E= 45°
Therefore,the two triangle is similar measure of angle B is 45.
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Use the quadratic formula f(x)=3x2−18x+23
to determine the coordinates of the vertex of the best-fitting parabola. Write the exact answer. Do not round.
Answer:
108x^2 +23
Step-by-step explanation:
i think...
3x×2×18x+23
108x×x+23
final answer: 108^2 +23
What is the 8th term for 2, 10, 50, 250, 1250,
If anyone sees this can they help me out?
The table shows the heights and the maximum speeds for various roller coasters throughout the world
Which value is closest to the linear correlation coefficient for the data?
A) 0.483
B) 0.205
C) 0.420
D) 0.695
The value that is closest to the linear correlation coefficient for the data include the following: D) 0.695.
How to find an equation of the line of best fit for the data?In order to determine a linear equation for the line of best fit (trend line) that models the data points contained in the table, we would have to use a graphing calculator (scatter plot).
In this scenario, the heights would be plotted on the x-axis of the scatter plot while the maximum speeds would be plotted on the y-axis of the scatter plot.
From the scatter plot (see attachment) which models the relationship between the heights and the maximum speeds, a linear equation for the line of best fit and correlation coefficient are as follows:
Equation: y = 0.205x + 33.42
Correlation coefficient, r = 0.695.
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are conditional probabilities always dependent?
They are dependent events because the outcome of B completely depends on what happened on A. Another notation to represent the probability of a series of dependent events is P(A|B), it is read as The probability of A given B.
Two events A and B are said to be independent if the fact that one event has occurred does not affect the probability that the other event will occur. If whether or not one event occurs does affect the probability that the other event will occur, then the two events are said to be dependent.
Define $f(x)=\frac{1+x}{1-x}$ and $g(x)=\frac{-2}{x+1}$. Find the value of\[g(f(g(f(\dotsb g(f(12)) \dotsb ))))\]where the function $f$ is applied 8 times, and the function $g$ is applied 8 times, alternating between the two.
The value of [tex]$g(f(g(f(g(f(g(f(12)))))))) = \frac{5926}{3335}$[/tex] when the function f is applied 8 times and g is applied 8 times alternatively.
How is function composition employed in this issue and what is it?The process of applying one function to the result of another is known as function composition. In other words, if we have two functions, f and g, we may combine them to create a new function that accepts an input, processes it though g, and then outputs f.
For the given composite function we start with the innermost function.
Find the value of f(12):
[tex]f(12) = \frac{1+12}{1-12} = -\frac{13}{11}$\\\$g(f(12)) = \frac{-2}{-\frac{13}{11}+1} = \frac{22}{13}$\\$f(g(f(12))) = \frac{1+\frac{22}{13}}{1-\frac{22}{13}} = -\frac{35}{9}$\\$g(f(g(f(12)))) = \frac{-2}{-\frac{35}{9}+1} = \frac{18}{13}$[/tex]
[tex]f(g(f(g(f(12))))) = -\frac{157}{139}$\\$g(f(g(f(g(f(12)))))) = \frac{278}{157}$\\$f(g(f(g(f(g(f(12))))))) = -\frac{3335}{2963}$\\$g(f(g(f(g(f(g(f(12)))))))) = \frac{5926}{3335}$[/tex]
Hence, the value of [tex]$g(f(g(f(g(f(g(f(12)))))))) = \frac{5926}{3335}$[/tex] when the function f is applied 8 times and g is applied 8 times alternatively.
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Find the vertex of the quadratic function f(x)=x^2-6x+1
(3, 8).
Step-by-step explanation:1. Write the expression in general form.General form is: y = ax² + bx + c.
[tex]y=x^2-6x+1[/tex]
2. Use the following formula to find the "x" coordinate of the vertex.Formula: [tex]x=-\frac{b}{2a}[/tex]
3. Identify "b" and "a" from the general form.So "b" should be the coefficient of the "x" variable, that value is: -6.
Now, the "a" value is implicit, since the x² doesn't have any numbers in front of it. Therefore, the value of "a" is 1.
4. Use the values to calculate the "x" coordinate of the vertex.[tex]x=-(\frac{(-6)}{2(1)})\\ \\x=-(\frac{-6}{2})\\ \\x=-(-3)\\ \\x=3[/tex]
5. Substitute "x" by the calculated value (3) in the original function formula to find the "y" coordinate of the vertex.[tex]y=x^2-6x+1\\ \\y=(3)^2-6(3)+1\\ \\y=9-18+1\\ \\y=-8[/tex]
6. Write the ordered pair of the vertex.(3, 8).
7. Graph.Take a look at the attached image to see the graph and vertex of this function.
On a number line ,how far apart are the numbers -5.5 and 7.5
Answer:
-5.5 and 7.5 are 13 units apart
Step-by-step explanation:
The distance between two numbers on a number line is found by subtracting the smaller number from the larger number.
7.5 - ( -5.5) = 7.5 + 5.5 = 13
A circle has the equation (* - 13)? + (y - 1)? = 4.
A: What is the center of the circle?
B: What is the radius of the circle?
Select two answers: one for question A, and one for question B.
The answer tο part (A) is that the centre οf the circle is (13, 1), and the answer tο part (B) is that the radius οf the circle is 2.
What is a circle?A circle is a twο-dimensiοnal shape that is defined as the set οf all pοints in a plane that are equidistant frοm a fixed pοint called the center. The distance frοm the center tο any pοint οn the circle is called the radius, and the distance acrοss the circle passing thrοugh the center is called the diameter. The equatiοn οf a circle with center (h, k) and radius r is:
(x - h)² + (y - k)² = r²
Cοmparing this with the given equatiοn:
(x- 13)² + (y - 1)² = 4
We can see that the center οf the circle is (13, 1), and the radius οf the circle is the square rοοt οf 4, which is 2.
Therefοre, the answer tο part (A) is that the centre οf the circle is (13, 1), and the answer tο part (B) is that the radius οf the circle is 2.
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Complete Question
A circle has the equation (x−13)2+(y−1)2=4.
A: What is the center of the circle?
B: What is the radius of the circle?
Select two answers: one for question A, and one for question B.
A: (13,1)
B: 4
A: (−13,−1)
A: (13,1)
A: (13,−1)
B: 2
B: 2–√
Help me plsss I’m in a lot of trouble rn
The missing value in the proportion 4:x = 20:10 is 2.
What is proportion?
In general, the term "proportion" refers to a part, share, or amount that is compared to a whole. According to the definition of proportion, two ratios are in proportion when they are equal.
To find the missing value in the proportion 4:x = 20:10, we can use the property that the ratios of the corresponding terms in a proportion are equal.
That is, 4:x = 20:10 is equivalent to the equation -
4 / x = 20 / 10
Simplifying this equation, we get -
4 / x = 2
Multiplying both sides of the equation by x, we get -
4 = 2x
Dividing both sides of the equation by 2, we get:
x = 2
Therefore, the value for x is obtained as 2.
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Find the missing value 4:x = 20:10.
How can you check to see if your two expressions from part A are equivalent?
Answer: wheres the question
Step-by-step explanation:
Help 100 points plus brainiest
Answer:
B
Step-by-step explanation:
For every 4 units moved to the side, it also moves up by 5 units.
Use the frequency table to answer the question Male: Do Believe in Ghosts 25 Don't believe in ghosts 12 Female: Do believe in ghosts 61 Don't believe in ghosts 26. Given that someone does not believe in ghosts, what is the probability they are male? Enter your answer as a fraction.
The probability that someone who doesn't believe in ghosts is male is 6/19.
Define the term probability?The measure of the likelihood or chance of an event occurring is known as Probability. It is expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty, and is often used in statistics, mathematics, and science.
The frequency table shows that 12 males don't believe in ghosts, and 26 females don't believe in ghosts. So, the total number of people who don't believe in ghosts is 12+26=38.
Out of these 38 people who don't believe in ghosts, 12 are male. Therefore, the probability that someone who doesn't believe in ghosts is male is:
P(Male | Don't believe in ghosts) = P(Male and Don't believe in ghosts) / P(Don't believe in ghosts)
= 12/38
= 6/19
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Suppose a survey indicated that the average American spends the following number of hours per week for each activity: 54 hours for sleep, 28 hours for work, 18 hours for family care, 20 hours for personal care, and 48 hours for leisure. In constructing a circle graph, how many degrees (to the nearest tenth) of the circle should be used for the “work” segment?
Work accounts for approximately 60° of the pie chart, or 15.48% of the total 360°.
How to solveTo calculate a pie chart, first, categorize the data and calculate the total.
Each slice of the pie chart represents a numerical value and has a size proportional to the value.
The sum of all the data is equal to 360°, with the total value always being 100%.
To find the percentage for each slice of the pie chart, divide the categories by the total and convert them into percentages.
Finally, calculate the degrees using the formula (Given Data/Total value of Data) × 360°.
For example, let's say we have data on sleep, work, family care, personal care, and leisure, with a total of 168 hours.
To find the percentage for each category, we can use the following calculations:
Sleep: 54/168 x 100 = 32.14%Work: 28/168 x 100 = 15.48%Family care: 18/168 x 100 = 10.71%Personal care: 20/168 x 100 = 11.90%Leisure: 48/168 x 100 = 28.57%Therefore, work accounts for approximately 60° of the pie chart, or 15.48% of the total 360°.
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in a certain community, 30% of the families own a dog, 20% of the families that own a dog also own a cat
Therefore, the probability that a randomly selected family owns a cat is 0.257. Therefore, the conditional probability that a randomly selected family doesn't own a dog given that it owns a cat is 0.778.
What is probability?Probability is a branch of mathematics that deals with the study of random events or situations. It involves the quantification of the likelihood or chance of an event occurring, expressed as a number between 0 and 1, with 0 representing impossibility and 1 representing certainty. The theory of probability provides a framework for making predictions, analyzing data, and making decisions in situations involving uncertainty or randomness.
Here,
Let's use the following events:
D: family owns a dog
C: family owns a cat
From the problem statement, we know:
P(D) = 0.3
P(C|D) = 0.2
P(C) = 0.34
1. To find the probability that a randomly selected family owns a cat, we can use the law of total probability:
P(C) = P(C|D) * P(D) + P(C|not D) * P(not D)
We don't know P(C|not D), but we can find it using the fact that P(not D) = 1 - P(D):
P(C|not D) = (P(C) - P(C|D) * P(D)) / P(not D)
P(C|not D) = (0.34 - 0.2 * 0.3) / (1 - 0.3)
P(C|not D) = 0.2857
Substituting the values:
P(C) = 0.2 * 0.3 + 0.2857 * 0.7
P(C) = 0.257
2. To find the conditional probability that a randomly selected family doesn't own a dog given that it owns a cat, we can use Bayes' theorem:
P(not D|C) = P(C|not D) * P(not D) / P(C)
Substituting the values:
P(not D|C) = 0.2857 * 0.7 / 0.257
P(not D|C) = 0.778
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Complete question:
In a certain community, 30% of the families own a dog, and 20% of the families that own a dog also own a cat. It is also known that 34% of all the families own a cat. What is the probability that a randomly selected family owns a cat? What is the conditional probability that a randomly selected family doesn't own a dog given that it owns a cat?
Jumping spiders can commonly be found in homes and barns throughout the United States. A jumping spider's jump can be modeled by the
equation h = 33.3t 16t2, where t represents the time in seconds and h is the height in feet.
A. When is the spider's height at 0 feet other than 0 seconds? Express your answer as a decimal. Do not round.
B. What is the spider's height after 1 second?
seconds
feet
BF
Answer:
A. To find when the spider's height is at 0 feet, we need to solve the equation h = 33.3t - 16t^2 for t when h = 0. This gives us:
0 = 33.3t - 16t^2
0 = t(33.3 - 16t)
t = 0 or t = 2.08125 (rounded to 5 decimal places)
So the spider's height is at 0 feet at 0 seconds and 2.08125 seconds.
B. To find the spider's height after 1 second, we substitute t = 1 into the equation:
h = 33.3t - 16t^2
h = 33.3(1) - 16(1)^2
h = 17.3 feet
So the spider's height after 1 second is 17.3 feet.
Solve the differential equation with
the given initial condition.
Dy/dx=3x/√y
(0,1)
The solution to the differential equation dy/dx = 3x/√(y) with initial condition (0,1) is y = ((9/4) × x² + 1)^(2/3).
Describe differential equation?Differential equations are used to model a wide variety of physical, biological, and economic systems, including the motion of objects under the influence of forces, the spread of disease in a population, and the behavior of financial markets.
There are many different types of differential equations, including ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve a single independent variable and its derivatives, while PDEs involve multiple independent variables and their derivatives.
To solve the differential equation dy/dx = 3x/√(y), we can separate the variables by multiplying both sides by √(y) and dividing both sides by 3x:
dy/dx = 3x/√(y)
√(y) dy/dx = 3x
√(y) dy = 3x dx
Integrating both sides:
∫√(y) dy = ∫3x dx
(2/3)*y^(3/2) = (3/2)*x² + C
where C is the constant of integration.
To find the value of C, we can use the initial condition (0,1), which means y=1 when x=0:
(2/3)*y^(3/2) = (3/2)*x² + C
(2/3)*1^(3/2) = (3/2)*0² + C
C = (2/3)
Substituting C = 2/3 back into the equation, we get:
(2/3)×y^(3/2) = (3/2) × x² + 2/3
y^(3/2) = (9/4) × x² + 1
y = ((9/4) × x² + 1)^(2/3)
Therefore, the solution to the differential equation dy/dx = 3x/√(y) with initial condition (0,1) is y = ((9/4) × x² + 1)^(2/3).
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