Step-by-step explanation:
Here we know that the formula to find Farhenheit is F= (9/5)×C +32
Then the degree by converting 67 would be 152.6 degree.
16 Paul buys a number of large sacks of fertiliser costing $x each. He spends $27. (a) Write down, in terms of x, an expression for the number of large sacks which Paul buys. Answer(a) 27 (b) Rula buys a number of small sacks of fertiliser. Each small sack costs $2 less than a large sack. Rula spends $25. Write down, in terms of x, an expression for the number of small sacks which Rula buys. Answer (b) 25 X-2 (c) Rula buys 4 more sacks than Paul. Write down an equation in x and show that it simplifies to 2x²-3x-27= 0.
The equation in x is 2[tex]x^{2}[/tex]-3x-$27=0, which can be solved using the quadratic formula to find the value(s) of x that satisfy the equation. The expressions derived in parts (a) and (b) are used to set up an equation involving x.
How will you write an equation in the form of x?To solve this problem, we need to use the information given in parts (a) and (b) to set up an equation involving x.
(a) Let the number of large sacks Paul buys be y. The cost of each large sack is given as x, so the total amount Paul spends is:
y * x = $27
Solving for y, we get:
y = 27/x
(b) Let the number of small sacks Rula buys be z. We are told that each small sack costs $2 less than a large sack, so the cost of each small sack is:
x - $2
The total amount Rula spends is $25, so we can set up an equation:
z * (x - $2) = $25
Simplifying this equation, we get:
z = 25/(x - $2)
(c) We are told that Rula buys 4 more sacks than Paul. Using the expressions we derived in parts (a) and (b), we can set up an equation:
z = y + 4
Substituting the expressions we derived in parts (a) and (b), we get:
25/(x - $2) = 27/x + 4
Multiplying both sides by x(x - $2), we get:
25x = 27(x - $2) + 4x(x - $2)
Simplifying this equation, we get:
25x = 27x - $54 + 4x² - 8x
Rearranging terms and simplifying, we get:
2x² - 3x - $27 = 0
So the equation in x is 2x²-3x-$27=0, which we can solve using the quadratic formula to find the value(s) of x that satisfy the equation.
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Come with the height of 8 and radius of 8
The volume of the cylinder with a height of 8 units and a radius of 8 units is approximately 1606.88 cubic units.
How to find volume of cylinder?The formula for the volume of a cylinder is:
V = πr²h , Where:
V is the volume of the cylinder
π is a mathematical constant approximately equal to 3.14159
r is the radius of the base of the cylinder
h is the height of the cylinder
To find the volume of a cylinder, you need to know the radius and height of the cylinder. Once you have those values, you can substitute them into the formula and solve for V.
Assuming this quesrtion referring to a cylinder, the dimensions you provided correspond to a cylinder with a height of 8 units and a radius of 8 units.
To find the volume of this cylinder, you can use the formula:
V = πr²h
where V is the volume, r is the radius, and h is the height. Plugging in the given values, we get:
V = π(8²)(8)
V = 1606.88 cubic units (rounded to two decimal places)
Therefore, the volume of the cylinder with a height of 8 units and a radius of 8 units is approximately 1606.88 cubic units.
complete question - Two cones with same base radius 8 cm and height 15 cm are joined together along their bases. Find the surface area of the shape so formed (answer to the nearest whole number).
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Complete qeustion:
What is volume of a cylinder that comes with the height of 8 and radius of 8.
Divide. State the quotient in simplest form.
4x5/ 12x²/
x2_3x-4 3x+3, x ≠ − 1, 4
The quotient in simplest form is [tex]x^3 * (x+1)/(x-4).[/tex]
What is fractiοn?A fractiοn represents a part οf a whοle οr, mοre generally, any number οf equal parts. When spοken in everyday English, a fractiοn describes hοw many parts οf a certain size there are, fοr example, οne-half, eight-fifths, three-quarters.
Numeratοrs and denοminatοrs are alsο used in fractiοns that are nοt cοmmοn, including cοmpοund fractiοns, cοmplex fractiοns, and mixed numerals.
In pοsitive cοmmοn fractiοns, the numeratοr and denοminatοr are natural numbers. The numeratοr represents a number οf equal parts, and the denοminatοr indicates hοw many οf thοse parts make up a unit οr a whοle.
Tο divide fractiοns, we invert the secοnd fractiοn and multiply it by the first. Sο:
4x⁵/12x² ÷ (x²-3x-4)/(3x+3)= (4x⁵/12x²) * (3x+3)/(x²-3x-4)
Inverted secοnd fractiοn and multiplied
= (4/12) * (x⁵/x²) * (3(x+1)/(x-4)(x+1))
Factοred the denοminatοr οf the secοnd fractiοn
= (1/3) * [tex]x^{(5-2)[/tex] * (3(x+1)/(x-4))
Simplified and cancelled οut cοmmοn factοrs[tex]= (1/3) * x^3 * (3(x+1)/(x-4))= x^3 * (x+1)/(x-4)[/tex]
Therefοre, the quοtient in simplest fοrm is [tex]x^3 * (x+1)/(x-4).[/tex]
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Two cars leave the same location traveling in opposite directions. One car leaves at 3:00 p.m. traveling at an average rate of 55 miles per hour. The other car leaves at 4:00 p.m. traveling at an average rate of 75 miles per hour. Let x represent the number of hours after the first car leaves.
How many hours after the first car leaves will the two cars be 380 miles apart?
ANSWER:55x+75(x−1)=380
The two cars will be 380 miles apart after 3.5 hours after the first car leaves based on distance formula.
The formula for distance, which is rate times time, must be used to address this issue. Let x be the number of hours following the departure of the first car. The distances between the two cars are increasing since they are moving in different directions.
The first thing we must do is determine how far the first car, which left at 3:00 p.m., drove. When the rate is 55 miles per hour and the time is x hours, we may use the calculation distance = rate x time. Hence, the first car's mileage was 55 miles.
The second step is to determine how far the second car, which left at 4:00 p.m., travelled. When the speed is 75 miles per hour and the time is x - 1 hours, we can apply the same calculation (since the second car left one hour later than the first). Thus, the second car's mileage is 75(x - 1) miles.
The sum of the distances covered by each car is the overall distance between the two vehicles. So that we can create an equation:
distance of first car + distance of second car = total distance
55x + 75(x - 1) = 380
Simplify:
55x + 75x - 75 = 380
Combining:
130x = 455
Divide sides by 130 to obtain:
x = 3.5
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Name one pair of congruent angles.
A.
B.
C.
A study called the Nelson study researched the number of text messages teenagers made from 2010 to 2012. In 2010, the study reported that teens sent an average of 2,380 text messages per month. In 2012, the study reported that teens sent an average of 3,213 texts per month. Complete this sentence: The relative change in the number of text messages from 2010 to 2012 was
The relative change in the number of text messages from 2010 to 2012 was a 35.04% increase.
What is Numbers?A number is a fundamental building block of mathematics. Numbers are used for indexing, counting, measuring, and a variety of other tasks. According to their characteristics, there are various sorts of numbers, including natural numbers, whole numbers, rational and irrational numbers, integers, real numbers, complex numbers, even and odd numbers, etc.
The relative change in the number of text messages from 2010 to 2012 was:
(3,213 - 2,380) / 2,380 x 100% = 35.04%
Therefore, the relative change in the number of text messages from 2010 to 2012 was a 35.04% increase.
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When treated with an antibiotic, 95% of all the dolphins are cured of an ear infection. If 6 dolphins are treated, find the probability that exactly 3 are cured.
The probability that exactly 3 out of 6 dolphins are cured is approximately 0.3342.
What is the probability?This is a binomial probability problem with the following parameters:
n = 6 (number of trials)
p = 0.95 (probability of success, which is being cured)
q = 1 - p = 0.05 (probability of failure, which is not being cured)
The probability of exactly 3 dolphins being cured is given by the binomial probability formula:
P(X = 3) = (6 choose 3) * 0.95^3 * 0.05^3
where "6 choose 3" represents the number of ways to choose 3 dolphins out of 6.
Using a calculator, we can evaluate this expression as:
P(X = 3) = (6 choose 3) * 0.95^3 * 0.05^3
P(X = 3) ≈ 0.3342
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Eric had a box of 38 candy bars that he planned to sell for 29 cents each. Unfortunately, his sister ate 7 of his candy bars before he started selling them. In order to bring in the same amount of money as he would have if his sister had not eaten any of the candy bars, what price (in cents) should he charge for each of the remaining candy bars?
Sorry for the late response but....
Answer:
36 cents
Step-by-step explanation:
First, we would want to calculate how much the box would sell for before the sister ate some.
We would get 38*29(in cents) and we can further get 1102 cents before the sister ate the candy bars.
The sister ate 7 candy bars leaving us with 31 candy bars. In order to deduct the price for each candy bar, it would be a situation like the following:
$6 for 3 boxes of 500 cookies (i know that's cheap)
6/3...
$2 per box.
Using our situation we get 1102 cents for 31 candy bars.
1102 cents/31...
35.5483871
That does seem unrealistic, so we round to the nearest whole number and we get...
36 cents
Can you help me answer this question
Answer:
B.a low mean and a high MAD
Kennedy drove 9494 miles in 2\tfrac{2}{3}2
3
2
hours. On average, how fast did she drive, in miles per hour? Enter your answer as a whole number, proper fraction, or mixed number in simplest form.
Kennedy's average speed was 3554 miles per hour by using the formula of average speed.
What is the average speed?Average speed is a measure of how fast something travels over a period of time, calculated by dividing the total distance traveled by the total time taken. It is usually expressed in units of distance per unit of time, such as miles per hour (mph) or kilometers per hour (km/h).
What does distance mean?Distance is a measure of the physical space between two points, objects, or locations. It is usually expressed in units such as meters, kilometers, miles, feet, or yards.
According to the given informationTo find Kennedy's average speed, we can use the formula:
average speed = total distance / total time
In this case, Kennedy drove 9494 miles in 2 2/3 hours. To convert the mixed number to an improper fraction, we can multiply the whole number by the denominator and add the numerator, then put the result over the denominator:
2 2/3 = (2 × 3 + 2) / 3 = 8/3
So Kennedy's total time was:
total time = 8/3 hours
Using the formula, we get:
average speed = 9494 miles / (8/3 hours)
To divide by a fraction, we can multiply by its reciprocal:
average speed = 9494 miles × (3/8 hours)
Simplifying, we get:
average speed = 3554.25 miles per hour
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Question 1 (1 point) Which scale factors result in a expansion? -3 0.25 09/7 0-3/10 4
Answer:
Any scale factor greater than 1 result in a expansion or increase in size. Any scale factor less than 1 result in a shrinkage or decrease in size.
What is a scale factor?
a scale factor is a ratio between the scale of a given original object(pre-image) and a new object, (after-image or image after transformation) which is the same shape but of a different size (bigger or smaller).
Example & how to use scale factor:
let's say we have a triangle with a side length of 2, 2, and 3.
then we take that triangle and dilate it with a scale factor of 2.
the new side lengths become 4, 4, and 6 because they are being multiplied by 2 because the scale factor is 2.
Explanation and answer:
1. A scale factor of 4 is an expansion because it's greater than 1
2. A scale factor of 9/7 or about 1.30 is an expansion because it's greater than one.
3. the rest are shrinkage scale factors because they're less than one.
Conclusion:
i hope this helps :)
geometry can be hard sometimes
Roselyn buys 6 notebooks and 4 dry erase markers. The price of each item is listed below
notebook: 1.95
dry erase markers:200
The total costs of the 6 notebooks and 4 dry-erase markers that Roselyn buys are $19.70.
How are the total costs determined?The total costs can be determined using mathematical operations.
The first mathematical operation used is the multiplication operation, involving the multiplicand and the multiplier, to determine the total cost of each item.
The second mathematical operation is the addition operation to determine the grand total.
Unit Prices:Notebook: 1.95
Dry erase markers: 2.00
Total Costs:Notebook: $11.70 ($1.95 x 6)
Dry erase markers: $8.00 ($2.00 x 4)
Total costs = $19.70
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Question Completion:The price of each item is listed below
notebook: 1.95
dry erase markers: 2.00
What are the total costs?
Find the surface area of this rectangular prism. Be sure to include the correct unit in your answer.
Answer:
Step-by-step explanation:
What is a rectangular prism?
A right rectangular prism is a box-shaped object, that is, a 3-dimensional solid which has six rectangular faces. Rectangular prisms can also be oblique - leaning to one side - but the side faces are parallelograms, not rectangles. A right rectangular prism is also called a cuboid, box, or rectangular hexahedron. Moreover, "rectangular prism" and "right rectangular prism" are often used interchangeably.
The most common math problems related to this solid are of the type right rectangular prism calc find V or find A, where the letters stand for the Volume and Area, respectively. Let's see the necessary rectangular prism formula and learn how to solve those problems quickly and easily.
How do I find the volume of a rectangular prism?
The rectangular prism volume formula is:
volume = h × w × l,
where h is prism height, w is its width, and l is its length. To calculate the volume of a cardboard box:
Find the box length. For example, it can be equal to 18 in.
Determine its width. Let's say you measured 12 in.
Find out the rectangular prism height. Assume it's 15 in.
Calculate the cuboid volume. Using the rectangular prism volume formula above, we get volume = (18 × 12 × 15) in = 3240 in³.
How do I find the area of a rectangular prism?
The surface area of the cuboid consists of 6 faces - three pairs of parallel rectangles. To find the rectangular prism surface area, add the areas of all faces:
surface_area = 2 × (h × w) + 2 × (h × l) + 2 × (l × w) = 2 × (h × w + h × l + l × w),
where h is prism height, w is its width, and l is its length.
Let's see an example of how to solve the right rectangular prism calc - find A problem. We'll come back to our example with the box and calculate its surface area:
Calculate the rectangular prism surface area. First rectangle area is 15in × 12in = 180in², second 15in × 18in = 270in² and third one 18in × 12in = 216in². Add all three rectangles' areas - it's equal to 666 in² (what a number!) - and finally multiply by 2. The surface area of our cardboard box is 1332in².
Or save yourself some time and use our rectangular prism calculator.
Finally, let's attack the right rectangular prism calc find d (that is, the diagonal) type of problem.
How do I calculate the diagonal of a rectangular prism?
To determine the diagonal of a rectangular prism, apply the formula:
diagonal = √(l² + h² + w²)
where h is prism height, w is its width, and l is its length.
Do you have the feeling that you saw the formula before? Yes, that's possible because this equation resembles the famous one from the Pythagorean theorem.
A young executive is going to purchase a vacation property for investment purposes. She needs to borrow $128,000.00 for 25 years at a 5.3% annual interest rate, with interest compounded monthly, and will make monthly payments of $770.82. (Round all answers to 2 decimal places.)
The requried initial loan amount is approximately $128,000.00.
We can use the formula for the present value of an annuity to find the initial loan amount:
[tex]PV = PMT * ((1 - (1 + r/n)^{(-nt))} / (r/n))[/tex]
where PV is the present value (initial loan amount), PMT is the monthly payment, r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the total number of years.
Substituting the given values, we get:
[tex]PV = 770.82 * ((1 - (1 + 0.053/12)^{(-12*25))} / (0.053/12))[/tex]
≈ $128,000.00
So the initial loan amount is approximately $128,000.00.
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suppose a charity received a donation of 14.2 million. if this represents 42% of the charity's donated funds, what is the total amount of its donated funds? round answer to nearest million
A 14.2 million donation was made to a charity. The overall amount of the charity's contributed funds is 34 million if which amounts to 42% of those monies.
Let x represent the total amount of donations received by the charity.
According to the problem, 14.2 million represents 42% of the charity's donated funds:
14.2 million = 0.42x
We must isolate x on one side of the equation in order to find it. We can do this by dividing both sides of the equation by 0.42:
x = 14.2 million / 0.42
x = 33.80952381 million
Rounding this answer to the nearest million gives:
x ≈ 34 million
Therefore, the total amount of the charity's donated funds is approximately 34 million.
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Please help I’m stuck
Answer: any negative value
Step-by-step explanation:
g(x)= ax², inc on x<0 & dec on x>0
Firstly we need to get the derivative of g(x)
g'(x)=2ax
for inc interval g'(x) should be +ve
x<0 (i.e. x= -ve), then a must be -ve
for dec interval g'(x) should be +ve
x>0 (i.e. x= +ve), then a must be -ve
so in both cases a must be a negative constant
Another one lol
What is the area, in square meters, of the shaded part of the rectangle below?
97 points
Answer: 210 square meters
Step-by-step explanation:
20 × 16 = 320
1/2 (20 × 11) = 110
320-110=210
Desmond: 4.5 Key features of square root functions:
Does a, h, and k act the same in the square root
function as they did in the quadratic function and/or
cubic functions from the last unit? Explain.
Answer:
No, a, h, and k behave differently in the square root function than they did in the quadratic function and/or cubic functions from the previous course. There is no variable (a) in front of the square root word in the square root function, and the shifts of the function are represented by h and k, respectively. Unlike in quadratic or cubic functions, where h stands for the vertex's horizontal shift and k for its vertical shift. The square root function's scope and range also vary from those of cubic and quadratic functions.
A tin has a height of 20 cm and a volume of 4850 cm³. Find the base area of the cylinder.
Answer:
242.50 [tex]cm^{2}[/tex]
Step-by-step explanation:
V = [tex]\pi r^{2}[/tex]h
4850 = [tex]\pi r^{2}[/tex] (20) Divide both sides by 20
242.5 = [tex]\pi r^{2}[/tex] The area of the base is [tex]\pi r^{2}[/tex]
Helping in the name of Jesus.
Mathew is mowing lawns to make money for his scouting trip. he charges 20 dollars to mow a lawn. his goal is to make it to at least 380$. whichinequality below could be used to find m, the number of lawns Matthew needs to mow to make his goal ?
The inequality used is 20m ≥ 380 and Matthew needs to mow at least 19 lawns to make his goal of $380.
What is an inequality?A mathematical comparison of two quantities using the symbols (less than), > (greater than), (less than or equal to), or is known as an inequality (greater than or equal to). Relationships between values that might not be equal, such the magnitude of two numbers or the range of values for a variable, are described as inequalities. Instead of a single value, the solutions to inequalities are sometimes stated as intervals of integers that meet the inequality. Every choice of x larger than 2 will meet the inequality, for instance, in the case of the inequality 2x + 3 > 7.
Let us suppose number of lawns mown = m.
The total earnings can be calculated as follows:
Total earnings = price per lawn x number of lawns
20m ≥ 380
m ≥ 19
Hence, Matthew needs to mow at least 19 lawns to make his goal of $380.
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Write the volume of the prism in a complete factor form
As a result, the rectangular prism's volume in fully factored form is:
V = (x - 1) * (l * w * h)
A 3D form with six rectangular faces is referred to as a rectangular solid. A rectangular solid's volume can be calculated using the formula V = lwh, where l denotes the solid's length, w its width, and h its height.
In this instance, we are informed that the rectangular solid's height is 17 units. Also, we are informed that the solid's base is a rectangle with an area of (x-1) square units. We can assume that the solid has length l and width w because its base is a rectangle.
What are some real-life example of rectangular solid?Solids with six rectangular faces are known as rectangular solids. They go by the name rectangular prisms as well. Following are some examples of rectangular solids in daily life:
Laptops, school notebooks, fish tanks, cargo containers, rooms, storage sheds, refrigerators, cereal boxes, tissue boxes, cars, and more.
As a result, we have:
l * w = x - 1
The rectangular solid's volume is:
V = lwh
With l * w = x - 1 and h = 17 as substitutes, we obtain:
V = (x - 1)lw
This phrase can be factored as follows:
V = (x - 1)l * w * h
Since l * w = x - 1 and h = 17, we have:
V = (x - 1) * l * w * h
V = (x - 1) * (l * w * h)
As a result, the rectangular prism's volume in fully factored form is:
V = (x - 1) * (l * w * h)
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For the function f(x)=√x-2, the average rate of change to the nearest hundredth over the interval-2 ≤ x ≤ 4 is Choose...
day
Step-by-step explanation:
To find the average rate of change of the function f(x) over the interval [-2,4], we need to calculate the change in the function value divided by the change in the input variable over that interval:
Average rate of change = (f(4) - f(-2))/(4 - (-2))
First, we calculate f(4):
f(4) = √(4 - 2) = √2
Next, we calculate f(-2):
f(-2) = √(-2 - 2) = √(-4)
Since the square root of a negative number is not a real number, the function f(x) is not defined for x < 2. Therefore, the interval [-2,4] is not entirely within the domain of the function.
However, we can still find the average rate of change over the part of the interval that is within the domain of the function, which is [2,4]. Therefore, we need to modify the formula accordingly:
Average rate of change = (f(4) - f(2))/(4 - 2)
f(2) = √(2 - 2) = 0
Plugging in the values we get:
Average rate of change = (√2 - 0)/(4 - 2) ≈ 0.71
Therefore, the average rate of change of f(x) over the interval [-2,4] (within the domain of the function) to the nearest hundredth is 0.71.
Six dollars is what percent of eight dollars
Answer:
75%
Step-by-step explanation:
Formula:
P = (x/y) * 100
where x is 6 dollars and y is 8 dollars.
P = (6/8) * 100
P = 0.75 * 100
P = 75%
What is the median?
2,6,7,7,8,8,9,15
Its a 2 mark question so i need working out including the answer.
Answer:
7.5
Step-by-step explanation:
Since the number of numbers is an even number, you're going to have to add 7 and 8 since they are both in the middle. After you added 7 and 8 your answer would be 15, you would have to divide 15 by 2 and your answer will be 7.5. Sorry if it's difficult to understand.
Answer:
7.5
(7+8)/2 = 7.5
Step-by-step explanation:
Hope this helps! =D
Mark me brainliest and Have a good day! =D
On average, for every 100 seventh grade students at a school, 48 students can play an instrument. If the school has 77 seventh grade students, about how many of those students would you expect to play an instrument?
Thus, number of students in seventh grade who can play instrument out of 77 are 37.
Define about the proportion:In general, the term "proportion" refers to a part, share, or amount that is compared to a whole. Depending on the definition of proportion, two ratios are in proportion when they are equal. Two ratios as well as fractions are equal when an equation or a declaration to that effect is utilised.
Whereas if ratio between the initial and second is identical to the ratio between of second and third, then each three quantities are referred to as having a continuing proportion.
Given data for the seventh grade students:
Out of 100 ---> 48 can play instruments.
Out of 77 ---> x (say) can play instruments.
So,
100/77 = 48/x
x = 48*77 / 100
x = 36.96
x = 37 (students)
Thus, number of students in seventh grade who can play instrument out of 77 are 37.
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A math class contains 10 females (three of whom speak French and the rest speak only English),and 12 males (two of whom speak French and the rest speak only English).
The probability that a randomly chosen student who speaks French is female is 33/110 and a female student speaks French is 15/41.
What is probability?
Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.
a) Let's use Bayes' theorem to calculate the probability that a randomly chosen female student speaks French. Let F represent the event that a student speaks French, and let F' represent the event that a student does not speak French. Similarly, let Fm and Ff represent the events that the student is male and female, respectively. Then we have:
P(F|Ff) = P(Ff|F)P(F) / P(Ff)
where P(Ff|F) = 3/10 is the probability that a female student speaks French, P(F) = 5/22 is the overall probability that a student speaks French, and P(Ff) = (3/10)(5/22) + (2/12)(17/22) = 41/220 is the probability that a randomly chosen student is female. Thus, we have:
P(F|Ff) = (3/10)(5/22) / (41/220) = 15/41
So the probability that a randomly chosen female student speaks French is 15/41.
b) Now we want to find the probability that a randomly chosen student who speaks French is female. Let's again use Bayes' theorem, with F and F' representing the events of speaking French and not speaking French, and M and F representing the events of being male and female, respectively. Then we have:
P(Ff|F) = P(F|Ff)P(Ff) / P(F)
where P(F|Ff) = 3/10 is the probability that a female student speaks French, P(Ff) = 41/220 is the probability that a randomly chosen student is female, and P(F) = (3/10)(5/22) + (2/12)(17/22) = 5/22 is the overall probability that a student speaks French. Thus, we have:
P(Ff|F) = (3/10)(41/220) / (5/22) = 33/110
So the probability that a randomly chosen student who speaks French is female is 33/110.
Hence, the probability that a randomly chosen student who speaks French is female is 33/110 and a female student speaks French is 15/41.
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what does x equal in this equation: x-3 + 60 =180
Answer:
Step-by-step explanation:
first to solve this you should make the numbers on one side after the equal sign with reversing the sign of each moved number like this :
X = 180 - 60 + 3 so X = 123- note that the number 3 was in a negative sign ( -3 ) before it's moved to the other side , and also the number 60 was in a positive sign ( + 60 ) before it's moved to the other side.
Find an equation of the parabola y = ax2 + bx + c that passes through (0, 1) and is tangent to the line y = 2x − 2 at (1, 0).
The equation of the parabola is y = 5x² - 6x + 1 .
What is parabola?
A parabola is a U-shaped plane curve. Any location on a parabola is at an equal distance from both the focus, a fixed point, and the directrix, a set straight line.
First, we figure out the value of c or the y intercept, we use the second point (0, 1) and substitute to the equation of the parabola. When x = 0, y = 1. So, c should be equal to 1. The parabola is y = ax^2 + bx + 1
Now, we can substitute the point (1,0) into the equation,
0 = a(1)² + b(1) + 1
0 = a + b + 1
a + b = -1
The slope at x = 1 is equal to 4 which is equal to the first derivative of the equation.
We take the derivative of the equation ,
y = ax² + bx + 1
y' = 2ax + b
x = 1, y' = 2
4 = 2a(1) + b
4 = 2a + b
So, we have two equations and two unknowns,
2a + b = 4
a + b = -1
Solving simultaneously,
a = 5
b = -6
Therefore, the equation of the parabola is y = 5x² - 6x + 1 .
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Write the absolute value inequality in the form |x−b| < c that has the solution set 5 < x < 7
We can write the absolute value inequality in the form |x−b| < c: |x−6| < 1
What is inequalities?
In mathematics, an inequality is a mathematical statement that indicates that two expressions are not equal. It is a statement that compares two values, usually using one of the following symbols: "<" (less than), ">" (greater than), "≤" (less than or equal to), or "≥" (greater than or equal to).
The solution set 5 < x < 7 can be rewritten as:
x > 5 (since x must be greater than 5)
x < 7 (since x must be less than 7)
To put this in the form |x−b| < c, we first find the midpoint between 5 and 7:
b = (5+7)/2 = 6
Then, we find the distance between b and either endpoint:
c = |7-6| = 1 (or |5-6| = 1)
Finally, we can write the absolute value inequality in the form |x−b| < c:
|x−6| < 1.
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Solve the systems by substitution.
-6x - 2y = 20
3y=x
Answer: The solution to the system of equations by substitution is (x, y) = (-3, -1).